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Strichartz Estimates for the (, )-Generalized Laguerre Operators

In this paper, we prove Strichartz estimates for the (, )-generalized Laguerre operators ⁻¹(−||²⁻ᵃ Δₖ + ||ᵃ) which were introduced by Ben Saïd-Kobayashi-Ørsted, and for the operators ||²⁻ᵃ Δₖ. Here k denotes a non-negative multiplicity function for the Dunkl Laplacian Δₖ, and denotes a positive rea...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2025
Main Authors: Taira, Kouichi, Tamori, Hiroyoshi
Format: Article
Language:English
Published: Інститут математики НАН України 2025
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/212877
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Cite this:Strichartz Estimates for the (, )-Generalized Laguerre Operators. Kouichi Taira and Hiroyoshi Tamori. SIGMA 21 (2025), 014, 37 pages

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author Taira, Kouichi
Tamori, Hiroyoshi
author_facet Taira, Kouichi
Tamori, Hiroyoshi
citation_txt Strichartz Estimates for the (, )-Generalized Laguerre Operators. Kouichi Taira and Hiroyoshi Tamori. SIGMA 21 (2025), 014, 37 pages
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container_title Symmetry, Integrability and Geometry: Methods and Applications
description In this paper, we prove Strichartz estimates for the (, )-generalized Laguerre operators ⁻¹(−||²⁻ᵃ Δₖ + ||ᵃ) which were introduced by Ben Saïd-Kobayashi-Ørsted, and for the operators ||²⁻ᵃ Δₖ. Here k denotes a non-negative multiplicity function for the Dunkl Laplacian Δₖ, and denotes a positive real number satisfying certain conditions. The cases = 1, 2 were studied previously. We consider more general cases here. The proof depends on symbol-type estimates of special functions and a discrete analog of the stationary phase theorem inspired by the work of Ionescu-Jerison.
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 014, 37 pages Strichartz Estimates for the (k, a)-Generalized Laguerre Operators Kouichi TAIRA a and Hiroyoshi TAMORI b a) Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, Japan E-mail: taira.kouichi.800@m.kyushu-u.ac.jp URL: https://sites.google.com/view/the-home-page-of-kouichi-taira/home b) Department of Mathematical Sciences, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama, 337-8570, Japan E-mail: tamori@shibaura-it.ac.jp Received June 24, 2024, in final form February 12, 2025; Published online March 02, 2025 https://doi.org/10.3842/SIGMA.2025.014 Abstract. In this paper, we prove Strichartz estimates for the (k, a)-generalized Laguerre operators a−1 ( −|x|2−a∆k + |x|a ) which were introduced by Ben Säıd–Kobayashi–Ørsted, and for the operators |x|2−a∆k. Here k denotes a non-negative multiplicity function for the Dunkl Laplacian ∆k and a denotes a positive real number satisfying certain conditions. The cases a = 1, 2 were studied previously. We consider more general cases here. The proof depends on symbol-type estimates of special functions and a discrete analog of the stationary phase theorem inspired by the work of Ionescu–Jerison. Key words: Strichartz estimates; oscillatory integrals; representation theory; Schrödinger equations 2020 Mathematics Subject Classification: 35Q41; 22E45 1 Introduction For the usual Laplacian ∆ on Rn (n ≥ 1), the following inequalities hold:∥∥eit∆u0∥∥Lp(R;Lq(Rn)) ≤ C∥u0∥L2(Rn), where (p, q) ∈ [2,∞]2 satisfies 2/p + n/q = n/2 with (p, q, n) ̸= (2,∞, 2). These estimates are called Strichartz estimates and have been widely studied in the past thirty years. Strichartz [29] proved them for p = q by using the Fourier restriction estimates and a duality argument. The most difficult part, that is the end-point case (p, q) = (2, 2n/(n − 2)) with n ≥ 3, was proved by Keel and Tao [18]. These are used for well-posedness of linear and non-linear time- dependent Schrödinger equations [14, 34]. See also the book [31]. For the Harmonic oscilla- tor Hos = −∆+|x|2 2 , a similar estimates hold:∥∥e−itHosu0 ∥∥ Lp([−T,T ];Lq(Rn)) ≤ CT ∥u0∥L2(Rn), where the region [−T, T ] cannot be replaced by R essentially due to the existence of L2- eigenfunctions. Given a root system R in Rn (we assume reducedness and do not assume crystallographic condition for the definition of root system, see [2, Definition 2.1]), a [0,∞)-valued function on R which is invariant under the finite reflection group C associated with R is called a non- negative multiplicity function. For a non-negative multiplicity function k and a > 0, we define mailto:taira.kouichi.800@m.kyushu-u.ac.jp https://sites.google.com/view/the-home-page-of-kouichi-taira/home mailto:tamori@shibaura-it.ac.jp https://doi.org/10.3842/SIGMA.2025.014 2 K. Taira and H. Tamori the (k, a)-generalized Laguerre operator by Hk,a := −|x|2−a∆k + |x|a a on Rn. Here ∆k denotes the Dunkl Laplacian (see [2, formula (2.9)]). If k ≡ 0, the Dunkl Laplacian coincides with the usual Laplacian: ∆0 = ∆ = ∑n j=1 ∂ 2 xj . The (k, a)-generalized Laguerre operator Hk,a is the generator of the (k, a)-generalized La- guerre semigroup which is a holomorphic semigroup introduced by Ben Säıd, Kobayashi and Ørsted [2]. For the case k ≡ 0 and a = 2 (resp. a = 1), the semigroup is the Hermite semi- group [11, 16] (resp. the Laguerre semigroup [19, 20, 21]). These two semigroups are associated with some realization (called the Schrödinger model) of minimal representations of the meta- plectic group Mp(n,R) and a double cover of the indefinite orthogonal group O(n + 1, 2). As unitary representations of S̃L(2,R)× C, they deformed these two representations with parame- ters k and a, and obtained a family of unitary representations. The (k, a)-generalized Laguerre semigroups are associated with them. In [2], the Fourier transforms associated with the (k, a)-generalized Laguerre semigroups are introduced and these various properties are studied. Recently, there have been several studies related to these operators such as real Paley–Wiener theorem [23], Lp-Lq-boundedness of Fourier multipliers [22], Hardy inequality [32] and wavelet transform [3]. The aim of this paper is to prove Strichartz estimates of Schrödinger equations associated with the (k, a)-generalized Laguerre operators. This problem is proposed in [2] and solved in [1] and [24] for a = 1 and a = 2 (see also [25], where they deal with orthonormal Strichartz estimates). The (0, 2)-generalized Laguerre operator H0,2 is just the Harmonic oscillator Hos and so their results are a generalization of the classical result for Hos. Here, we deal with more general cases. For a = 1, 2, the integral kernel of the Schrödinger propagator has a nice expression (due to [2, formula (4.58)]), which immediately implies the dispersive estimate [2, Proposition 4.26]. Hence the Strichartz estimates for a = 1, 2 are a direct consequence of this estimate and the result in [18] by Keel and Tao, see also [1, 25]. One of the difficulties to extend it to general a is a lack of such a nice expression of the Schrödinger propagator. Actually, this is just expressed in terms of an infinite sum of a product of special functions (see (1.5) and (1.6)). Therefore, we need to control this sum uniformly with respect to some parameters. To overcome this difficulty, we use a strategy inspired by the proof of Carleman estimates due to Ionescu and Jerison [17]. To be precise, we reduce estimates of the sum to those of integrals and use the theory of oscillatory integrals such as the stationary phase theorem with several parameters. One difference from [17] is that we avoid using the dyadic decomposition which is used there many times. Instead, we employ an appropriate scaling and simplify some arguments. In a sequel work [30], we will give another approach based on a deformation of integrals developed in the proof for the Strichartz estimates on flat cones [12]. In the last few decades, numerous works have focused on the dispersive estimates or the Strichartz estimates for Schrödinger operators with critical electromagnetic potentials such as Aharonov–Bohm magnetic fields (see [8, 9, 10, 13]) and Laplace–Beltrami operators on conic manifolds ([12, 35]). Due to the spherical symmetry of their operator, the integral kernel of their propagator has the form∑ ν : eigenvalues on the sphere Kν(t, r1, r2)Hν(θ1, θ2), where Kν is the propagator in the radial direction and Hν is the projection in the spherical direction. To achieve optimal estimates for this integral kernel, one has to use the oscillatory behavior of Kν and Hν , in other words, some cancelation of the sum much like our case. In Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 3 their paper, it is accomplished through the use of the complex contour deformation and certain functional equations of the special functions. Therefore, their method appears to be inapplicable when the radial propagator or the spherical projection is not expressed by special functions. Typical scenarios involving these situations arise in the studies of the Schrödinger equation with the degenerate trapping [6] or the wave equation on the Schwartzshild spacetime [7]. In their works, the dispersive or Strichartz estimates for initial values with a fixed angular momentum are considered, that is, they studied single modes only and did not consider the sum possibly because of the difficulty to treat the oscillation of the sum. In this paper, we deduce the asymptotic expansions of the radial direction and the spherical projection first and then sum up them by exploiting their oscillatory behavior. Specifically, we use the properties of the special functions in the first step only. Hence the authors believe that the method employed here specifically in the second step remains applicable even when Kν and Hν are not expressed in terms of special functions as is the case in [6, 7] (although we might need more precise analysis of the radial propagators). The second contribution of this paper is to give a naive application of the stationary phase theorem (see Proposition 2.4). This is used to prove improved dispersive estimates for Hk,a (near the diagonal) under the restriction 0 < a < 2. Our integral kernel has multiple parameters and it seems important to consider when a similar statement the usual stationary phase theorem holds uniformly with respect to additional parameters and when it is improved. Our Proposition 2.4 addresses intermediate cases between two scenarios where the decay order of an oscillatory integral is improved, see Section 2.3. Finally, we also obtain symbol-type estimates of higher-order derivatives for J-Bessel func- tions, which was done in [17] up to second derivatives and was anticipated to be true for higher- order derivatives there (see [17, Remark after Theorem 9.1]). It seems that the method used in [17] via complex counter deformation cannot be applied to the estimates for higher-order derivatives. Here we use an alternative method based on the stationary phase type theorem and partially solve them at the cost of loss of estimates for some parameters. See Proposition 3.1 for the precise statement and Appendix A.4 for its proof. We also mention a recent work [26], where the precise asymptotic behavior of the Bessel function is given although the authors do not know whether our symbol-type estimates follow from the results in [26]. 1.1 Main theorem Let us state our main theorem. For a nonnegative multiplicity function k and a > 0, we write ϑk,a(x) := |x|a−2 ∏ α∈R |⟨α, x⟩|k(α) (1.1) and assume that the homogeneous degree of the measure ϑk,a(x)dx on Rn is positive σk,a := n+ ∑ α∈R k(α) + a− 2 a > 0. (1.2) Then it is shown in [2, Corollary 3.22] that the operator Hk,a with domain Wk,a(Rn) defined in [2, equation (3.29)] is essentially self-adjoint on the Hilbert space L2(Rn;ϑk,a(x)dx). We denote the unique self-adjoint extension of Hk,a by the same symbol Hk,a. We write Lq = Lq(Rn;ϑk,a(x)dx) and L p(I, Lq) = Lp(I;Lq(Rn;ϑk,a(x)dx)) for I ⊂ R. Recall that an exponent pair (p, q) ∈ [2,∞]2 is called σk,a-admissible if 1 p + σk,a q = σk,a 2 and (p, q, σk,a) ̸= (2,∞, 1). When n = 1, a nonnegative multiplicity function k is a constant function. In this case, we regard k as a nonnegative real number k(α) (α ∈ R), and we see ∑ α∈R k(α) = 2k. 4 K. Taira and H. Tamori Theorem 1.1. We assume one of the following, which implies (1.2): � n = 1 and a ≥ 2− 4k, � n ≥ 2 and (0 < a ≤ 1 or a = 2), � n ≥ 2, 1 < a < 2 and k ≡ 0. Let (p, q), (p1, q1), (p2, q2) ∈ [2,∞]2 be σk,a-admissible exponents. Then, for T > 0, there ex- ists C > 0 such that∥∥e−itHk,au ∥∥ Lp([−T,T ];Lq) ≤ C∥u∥L2 , (1.3)∥∥∥∥∫ t 0 e−i(t−s)Hk,af(s)ds ∥∥∥∥ Lp1 ([−T,T ];Lq1 ) ≤ C∥f∥ Lp∗2 ([−T,T ];Lq∗2 ) , (1.4) where r∗ denotes the Hölder conjugate of r: r∗ = r/(r− 1) and in addition (pj , qj) ̸= ( 2, 2σk,a σk,a−1 ) when all of the conditions n ≥ 2, 1 < a < 2 and k ≡ 0 hold. Remark 1.2. (1) The a = 1 case was treated in [1], where an additional assumption σk,a ≥ 1 is necessary in order to use an upper estimate of I ( 2, σk,a−1 2 ;w; t ) [2, Proposition 4.26] (see (1.5) for the definition of I ) although it is not explicitly written there. When n = 1, the assumption σk,a ≥ 1 implies our assumption a ≥ 2 − 4k. Moreover, the end-point case( 2, 2σk,a σk,a−1 ) is excluded in [1]. The end-point case follows from the result in [18]. (2) For n ≥ 2 with a ≤ 1, we also obtain a dispersive estimate, see the proof of Theo- rem 1.1 in Section 6.2. On the other hand, the dispersive estimate might break for n ≥ 2 with 1 < a < 2 (see Theorem 1.4 (i) and Remark 1.5). Nevertheless, the Strichartz esti- mates still hold if k ≡ 0 since its proof just relies on the dispersive estimate around the diagonal of the integral kernel due to the nature of the TT ∗ argument. The authors believe that the Strichartz estimates do not hold for a > 2 although they do not know its proof. (3) We exclude an inhomogeneous end-point estimate for 1 < a < 2 with n ≥ 2 and k ≡ 0 since a global dispersive estimate is absent. A technique used in [15, 35] might be available, however, our estimates are not sufficient to apply their method. (4) In the above estimates, we cannot replace the time interval [−T, T ] by R. In fact, we take u ̸= 0 be an L2-eigenfunction of Hk,a and we denote the corresponding eigenvalue by λ ∈ R (note that its spectrum is discrete, see [2, Corollary 3.22]). Then (1.3) im- plies u ∈ Lq since ∣∣e−itHk,au(x) ∣∣ = ∣∣e−itλu(x) ∣∣ = |u(x)|. On the other hand,∥∥e−itHk,au ∥∥ Lp(R;Lq) = ∥u∥Lp(R;Lq) = ∞ although ∥u∥L2 <∞. Moreover, we can deduce global in time Strichartz estimates for −|x|a∆k. Theorem 1.3. Under the same assumptions and notation as Theorem 1.1, there exists C > 0 such that ∥∥eit|x|a∆ku ∥∥ Lp(R;Lq) ≤ C∥u∥L2 , ∥∥∥∥∫ t 0 ei(t−s)|x|a∆kf(s)ds ∥∥∥∥ Lp1 (R;Lq1 ) ≤ C∥f∥ Lp∗2 (R;Lq∗2 ) . Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 5 1.2 Key theorem We write Jλ(z) for the J-Bessel function, Ĩλ(w) for the normalized I-Bessel function, and Cν m(t) for the Gegenbauer polynomial of degree m. These functions are defined by Ĩλ(w) := e− π 2 iλ (w 2 )−λ Jλ(iw), Cν m(t) := (−2)m m! Γ(m+ ν)Γ(m+ 2ν) Γ(ν)Γ(2m+ 2ν) (1− t2)−ν+ 1 2 dm dtm ( 1− t2 )m+ν− 1 2 . Now we define I (b, ν;w; t) := Γ(bν + 1) ∞∑ m=0 (w 2 )bm Ĩb(m+ν)(w)(m+ ν)ν−1Cν m(t) (1.5) = Γ(bν + 1) ∞∑ m=0 ( iw 2 )−bν e− π 2 bmiJb(m+ν)(iw)(m+ ν)ν−1Cν m(t) for b > 0, ν ≥ −1 2 , w ∈ C \ (−∞, 0) and t ∈ [−1, 1], where we interpret (m + ν)ν−1Cν m(t) as limν↘0(m+ ν)ν−1Cν m(t) when ν = 0. Moreover, we take a branch of wbm such that wbm ∈ (0,∞) for w ∈ (0,∞) and wbm|w=0 = 0. Then the sum in (1.5) absolutely converges, and I is a continuous function (see [2, Lemma 4.17 (1)] or Section 6.1). In [2, Theorem C, equations (4.50) and (4.52)], it is shown that the integral kernel e−itHk,a(x, x′) of e−itHk,a for 0 < |t| < π is given by ck,a ei |x|a+|x′|a a cot(t) (i sin(t))σk,a ∫ Rn I ( 2 a , a(σk,a − 1) 2 ;−i 2|x| a 2 |x′| a 2 a sin(t) ; ξ · x̂′ ) dµkx̂(ξ), (1.6) where x̂ = x/|x|, ck,a is the constant defined in [2, equation (1.6)], · denotes the standard inner product on Rn and dµkx̂ is the probability measure introduced in [2, equation (2.5)]. Moreover, for k ≡ 0, we have a more explicit expression e−itH0,a(x, x′) = c0,a ei |x|a+|x′|a a cot(t) (i sin(t)) n+a−2 a I ( 2 a , n− 2 2 ;−i 2|x| a 2 |x′| a 2 a sin(t) ; x̂ · x̂′ ) . (1.7) Theorem 1.1 for n ≥ 2 is a consequence of uniform bounds for I (b, ν;w; t): Theorem 1.4. Let ν ≥ 0. (i) Suppose 0 < b < 2 and ε > 0. Then there exists Cb,ν,ε > 0 such that |I (b, ν;−iy; cosφ)| ≤ Cb,ν,ε(1 + |y|)(1−b)ν for y ∈ R, φ ∈ [0, π − ε]. When b = 1, we can take ε = 0. (ii) Suppose b > 0. Then there exists Cb,ν > 0 such that |I (b, ν;−iy; cosφ)| ≤ Cb,ν(1 + |y|)(2−b)ν for y ∈ R, φ ∈ [0, π]. In particular, when 1 < b < 2, the sum I (b, ν;−iy; cosφ) is uniformly bounded with respect to y ∈ R and φ ∈ [0, π − ε]. Moreover, when b = 1, b ≥ 2 or ν = 0, the sum I (b, ν;−iy; cosφ) is uniformly bounded with respect to y ∈ R and φ ∈ [0, π]. Remark 1.5. These estimates are sharp for b = 1, 2 with respect to the growth in y. In fact, [2, equations (4.45), (4,46)] show |I (1, ν;−iy; cosφ)| = 1 and I (2, ν;−iy;−1) = Γ ( ν + 1 2 ) Ĩν− 1 2 (0) = 1, which do not decay in y. The authors believe that they are sharp also for general b > 0. We will pursue it in our sequel work [30]. 6 K. Taira and H. Tamori 1.3 Idea of the proof Here we give an idea of the proof of Theorem 1.4. We remark that the difficulty lies in the uniformity with respect to the parameters y and φ. For the case |y| ≲ 1, the results are an immediate consequence of the estimates given in [2] (for rigorous treatment, see Section 6.1). Hence we consider the case |y| ≳ 1. Let us assume y ≳ 1 for simplicity. First, we try to estimate the sum I using optimal estimates for the Bessel functions and the Gegenbauer polynomials. We write I (b, ν;−iy; cosφ) =Lb,νy −bν ∞∑ m=0 (m+ ν)e− π 2 bmiJb(m+ν)(y)ν −1Cν m(cosφ). Since the estimates for finite m are easy to prove, we only consider the sum over m ≫ 1. By the bounds for the Bessel functions |Jµ(y)| ≤ Cµ− 1 3 in (3.2) and the Gegenbauer polynomi- als ∣∣ν−1Cν m(cosφ) ∣∣ ≤ Cm2ν−1 in (3.3), we have∣∣∣∣∣y−bν ∞∑ m≫1 (m+ ν)e− π 2 bmiJb(m+ν)(y)ν −1Cν m(cosφ) ∣∣∣∣∣ ≤ Cy−bν ∑ m≫1 m ·m− 1 3 ·m2ν−1 = Cy−bν ∑ m≫1 m2ν− 1 3 . Since the sum ∑ m≫1m 2ν− 1 3 is not convergent for ν ≥ −1/3 at all and since the upper bounds for special functions are optimal, such a direct method cannot be applied. Similarly, the estimate based on |Ĩλ(−iy)| ≤ Γ(λ+ 1)−1 is far from a uniform estimate. The drawback of the above strategy is to use the triangle inequality | ∑ ·| ≤ ∑ | · |. To obtain a better estimate of the sum I (b, ν;−iy; cosφ), we need to make use of some cancellation of the sum instead of using the triangle inequality. To do this, we write it as a sum of a WKB form (m+ ν)e− π 2 bmiJb(m+ν)(y)ν −1Cν m(cosφ) =: ζ(m, y, φ)eiS(m,y,φ), where a phase function S is a real-valued function and an amplitude ζ does not oscillate as m→ ∞ (uniformly in other parameters y, φ). Roughly speaking, it follows from the classical formula (2.1) that we can replace the discrete sum by a sum of integrals of the form∫ m≫1 ζ(m, y, φ)eiS(m,y,φ)+2πimqdm, q ∈ Z. (1.8) The problem on convergence (or uniform boundedness in some parameters) of such an integral has a long history and is related to the stationary phase theorem [28]. We can anticipate that a uniform estimate for the sum I can be proved by this theorem. On the other hand, we have to be careful to justify it because of the following reasons: � To use cancelation of the oscillation via the stationary or non-stationary phase theorems, we need symbol-type (derivative) estimates of ζ. To do this, we have to study those of the special functions Jµ and Cν m. � The integral (1.8) has a lot of parameters and we have to estimate it in a uniform way. � The J-Bessel function Jb(m+ν)(y) has different asymptotic behaviors as m, y → ∞ on the regions m ≪ y, m ≈ y, m ≫ y (see Proposition 3.1) and therefore the WKB form ζ(m, y, φ)eiS(m,y,φ) has different properties on each region. � The decay rate of Cν m(cosφ) as m → ∞ becomes worse as φ → 0, π (see Proposition 3.3) and hence estimates for φ = 0, π are also worse in general. Nevertheless, at φ = 0 with 0 < b < 2, we can improve the estimate since the first derivative of S(m, y, 0) vanishes only on regions where m is small enough (see Theorem 1.4 (i) and Section 5.2). Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 7 1.4 Related problems In this subsection,1 we mention a few related problems, including those concerning Strichartz estimates and fractional operators. � Prove the optimality of our Strichartz estimates (see Theorem 1.1). � Prove the Strichartz estimates for fractional operators such as Hα k,a or (−∆k,a) α (α > 0). When α = 1/2, this corresponds to the Strichartz estimates for the wave equation. � Study properties of the integral kernel E(x, y) of the spectral projection forHk,a and−∆k,a. In [15], the Strichartz estimates on asymptotically conic manifolds are proved by using the expression of E(x, y) and the stationary phase theorem. 1.5 Notation Let N = {0, 1, 2, . . .}, N∗ = {1, 2, 3, . . .}, R+ = (0,∞), R− = (−∞, 0). For a parameter i ∈ I and Ai, Bi, we write Ai ≲ Bi for i ∈ I if there is a constant C > 0 independent of i ∈ I such that Ai ≤ CBi. We denote Ai ∼ Bi if both Ai ≲ Bi and Bi ≲ Ai hold for i ∈ I. 2 Preliminary 2.1 A sum of monotonic functions The next lemma is elementary and follows from the piecewise quadrature. Lemma 2.1. Let f : R → [0,∞) be a continuous monotonically decreasing function. Then∑ m∈bN,m1≤m≤m2 f(m) ≤ ∫ m2 m1−b f(x)dx for b > 0 and m1,m2 ∈ R with m1 < m2, where bN = {bn | n ∈ N}. 2.2 Discrete oscillatory integrals In this paper, the following classical formula plays an important role:∑ m∈Z eiS(m)ζ(m) = ∫ R eiS(m)ζ(m)dm− 1 2πi ∑ q∈Z\{0} 1 q ∫ R ei(S(m)+2πqm)ζ̃(m)dm, (2.1) where ζ̃(m) = ζ ′(m)+iS′(m)ζ(m) (for example, see [17, Proof of Lemma 5.3]). In this paper, we use this formula for a compactly supported smooth function ζ and a smooth function S, so the convergence of the sum does not matter. From this formula, we can deduce a discrete analogue of the non-stationary phase theorem. Proposition 2.2. Let Cα > 0 for any nonnegative integer α and M ≥ 1, 0 < r ≤ 1, 0 < ρ ≤ 1, k ∈ R. Suppose that supp ζ ⊂ [0,M ], |∂αmζ(m)| ≤ CαM k−ρα. Suppose that a smooth real-valued function S ∈ C∞(R;R) satisfies dist(∂mS(m), 2πZ) ≥ r, ∣∣∂α+1 m S(m) ∣∣ ≤ CαM −ρα|∂mS(m)|. 1The authors would like to thank the anonymous referees for suggesting several unsolved problems and inspiring the writing of this subsection. 8 K. Taira and H. Tamori Then, for each N > 0,∣∣∣∣∣ ∞∑ m=1 eiS(m)ζ(m) ∣∣∣∣∣ ≤ CNM k+1(1 + rMρ)−N , where CN > 0 depends only on 0 < ρ ≤ 1, N > 0 and a finite number of Cα. Remark 2.3. In [17], the fact that CN is independent of r is important since they avoid to use the stationary phase theorem there. In this paper, we do not use this fact, that is, we use the case r = 1 only in this paper. Proof of Proposition 2.2. The proof is a slight modification of [17, Lemma 5.3] and we discuss briefly here. By the assumption and the intermediate value theorem, there exists a unique k ∈ Z such that ∂mS(m) ∈ (2πk, 2π(k + 1)). This implies r ≤ ∂mS(m)− 2πk ≤ 2π − r. By replacing S(m) by S(m)− 2πk, we may assume k = 0 and r ≤ ∂mS(m) ≤ 2π − r. From the formula (2.1), the problem reduces to the estimates for each integrals appearing in (2.1). By the change of variable from m to Mρm, we have∫ R eiS(m)ζ(m)dm =Mρ ∫ R eiSM (m)ζM (m)dm, (2.2) where we set SM (m)= S(Mρm) and ζM (m)= ζ(Mρm). Then the assumptions imply |∂αmζM (m)| ≤ CαM k, ∣∣∂α+1 m SM (m) ∣∣ ≤ Cα|∂mSM (m)|, ∂mSM (m) ≥ rMρ and supp ζM ⊂ [ 0, 4M1−ρ ] . We in- tegrate by partsN times in the right-hand side of (2.2) (use the identity (iS′ M (m))−1∂meiSM (m) = eiSM (m) and Lemma A.1) and obtain∣∣∣∣∫ R eiS(m)ζ(m)dm ∣∣∣∣ ≤Mρ · CNM k(1 + rMρ)−N ·M1−ρ = CNM k+1(1 + rMρ)−N , where the term M1−ρ comes from the volume of the support supp ζM . The second terms of the right-hand side of (2.1) are similarly dealt with as in the proof of [17, Lemma 5.3]. ■ 2.3 A variant of the stationary phase theorem Here we give a variant of the stationary phase theorem which is used for an improvement of estimates of I described in Section 5.2. We consider the following integral with parameters λ, φ: I(λ, φ) := ∫ R eiλS(µ,φ)γ(µ, λ, φ)dµ and its decay rate with respect to λ≫ 1 If we freeze the parameter φ, then the stationary phase theorem just implies that if µ 7→ S(µ, φ) is a Morse function (that is, all critical points are non-degenerate in the sense that ∂2µS ̸= 0 there), the optimal decay rate of I is λ− 1 2 . To obtain an improved decay, here we consider the following two scenarios: � If we assume that γ vanishes with order 2ν at all the critical points of S in addition, then the optimal decay rate is improved to be λ− 1 2 −ν . � If the symbol γ itself decays like (1 + λ)−ν , then the decay order becomes λ− 1 2 −ν . The following proposition interpolates the above two situations in a uniform way with respect to the parameter φ. A typical example of such a phase function is S(µ, φ) = (µ − φ)2. This simple case could simplify the reading of the proof below. Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 9 Proposition 2.4. Let ν ≥ 0 and c, C > 0. Suppose that S ∈ C∞([0, 1]2;R) and γ(·, λ, ·) ∈ C∞(R× [0, 1]) satisfy supp γ(·, λ, φ) ⊂ [0, 1], |∂αµγ(µ, λ, φ)| ≤ Cαµ 2ν−α(1 + λφµ)−ν ,∣∣∂2µS(µ, φ)∣∣ ≥ c, |∂µ∂φS(µ, φ)| ≥ C uniformly in µ ∈ [0, 1], φ ∈ [0, 1] and λ ≫ 1. Suppose that S(·, φ) has a unique critical point µ0(φ) which is smooth with respect to φ and that µ0(0) = 0. Then |I(λ, φ)| ≲ λ−ν− 1 2 uniformly in λ≫ 1 and φ ∈ [0, 1]. Remark 2.5. (1) Without the assumption µ0(0) = 0, we obtain I(λ, φ) ≲ λ− 1 2 only. The additional de- cay λ−ν comes from the factor (1 + λφµ)−ν in the assumption of γ and the fact that µ0(0) = 0 as is mentioned before. (2) We observe that the critical point µ0(φ) of S is close to 0 if φ ∼ 0. There we take advantage of the assumption that the symbol γ vanishes at zero with order 2ν like |γ(µ, λ, φ)| ≲ µ2ν . On the other hand, if φ is away from zero, then the critical point µ0(φ) is also away from zero. In this case, γ itself decays like λ−ν near the critical point (due to |φ| ≳ 1 and |µ0(φ)| ≳ 1), which makes us to prove the improved decay. The difficulty here is to deal with the case where φ is small enough but depending on λ. (3) The critical point of S(·, φ) is unique due to the condition ∣∣∂2µS(µ, φ)∣∣ ≥ c in this case. Proof of Proposition 2.4. We may assume supp γ(·, λ, φ) ⊂ [ λ− 1 2 , 1 ] . In fact, we set γ′(µ, λ, φ) = χ ( λ 1 2µ ) γ(µ, λ, φ), where χ ∈ C∞([0,∞),R) satisfies χ(µ) = 1 on µ ≤ 1 and χ(µ) = 0 for µ ≥ 2. Using the bound |γ(µ, λ, φ)| ≲ µ2ν , we have∣∣∣∣∫ R eiλS(µ,φ)γ′(µ, λ, φ)dµ ∣∣∣∣ ≲ λ−ν ∣∣∣∣∫ R χ ( λ 1 2µ ) dµ ∣∣∣∣ ≲ λ−ν− 1 2 . Hence, we can replace γ by γ − γ′, where we note that γ − γ′ satisfies supp(γ − γ′)(·, λ, φ) ⊂ [ λ− 1 2 , 1 ] , |∂αµ (γ − γ′)(µ, λ, φ)| ≤ Cαµ 2ν−α(1 + λφµ)−ν . In the following, we assume supp γ(·, λ, φ) ⊂ [ λ− 1 2 , 1 ] . (i) First, we consider the case φ ∈ [0, 1] satisfying µ0(φ) ≤ 2−1λ− 1 2 . Since the signature of ∂2µS(µ, φ) does not change for µ ∈ [ λ− 1 2 , 1 ] and φ ∈ [0, 1], we have |∂µS(µ, φ)| = ∣∣∣∣∣ ∫ µ µ0(φ) ∂2µS(µ ′, φ)dµ′ ∣∣∣∣∣ = ∫ µ µ0(φ) ∣∣∂2µS(µ′, φ)∣∣dµ′ ≥ c(µ− µ0(φ)) ≥ 2−1cµ for µ ∈ [ λ− 1 2 , 1 ] . By integrating by parts and using supp γ(·, λ, φ) ⊂ [ λ− 1 2 , 1 ] , we have |I(λ, φ)| = ∣∣∣∣(−iλ)−N ∫ R eiλS(µ,φ)LNγ(µ, λ, φ)dµ ∣∣∣∣ ≤ λ−N ∫ 1 λ− 1 2 |LNγ(µ, λ, φ)|dµ, where L = ∂µ ◦ (∂µS)(µ, φ) −1 and we take N > ν + 1. We observe from the assumption and Lemma A.1 that ∣∣LNγ(µ, λ, φ) ∣∣ ≲ µ2ν−2N , which leads to |I(λ, φ)| ≲ λ−N ∫ 1 λ− 1 2 µ2ν−2Ndµ ≲ λ−ν− 1 2 . 10 K. Taira and H. Tamori (ii) Next, we consider the case φ ∈ [0, 1] satisfying µ0(φ) ∈ [ 2−1λ− 1 2 , 1 ] . Differentiating (∂µS)(µ0(φ), φ) = 0 with respect to φ, we have |µ′0(φ)| = ∣∣−(∂µ∂φS)(µ0(φ), φ)/∂ 2 µS(µ0(φ), φ) ∣∣ ∼ 1, and hence µ0(φ) ∼ φ for φ ∈ [0, 1] due to µ0(0) = 0. Now the assumption µ0(φ) ≥ 2−1λ− 1 2 yields φ ≳ λ− 1 2 . By scaling, we have I(λ, φ) = ∫ R eiλS(µ,φ)γ(µ, λ, φ)dµ = µ0(φ)λ −ν ∫ R eiλµ0(φ)2Sφ(µ)γφ(µ, λ)dµ, where we set Sφ(µ) = µ0(φ) −2S(µ0(φ)µ, φ), γφ(µ, λ) = λνγ(µ0(φ)µ, λ, φ). We note that µ0(φ) is a unique critical point of S(µ, φ). Let ψ1, ψ2, ψ3 ∈ C∞(R; [0, 1]) such that ψ1 + ψ2 + ψ3 = 1, suppψ1 ⊂ ( −∞, 34 ] , suppψ2 ⊂ [ 1 2 , 3 2 ] and suppψ3 ⊂ [ 5 4 ,∞ ) . We write I(λ, φ) = µ0(φ)λ −ν 3∑ j=1 ∫ R eiλµ0(φ)2Sφ(µ)γφ(µ, λ)ψj(µ)dµ = µ0(φ)λ −ν 3∑ j=1 Ij(λ, φ). First, we deal with I2(λ, φ). Now we see |∂µS(µ, φ)| ≥ c|µ−µ0(φ)| for µ ∈ [0, 1] and φ ∈ [0, 1]. Then we have |∂µSφ(µ)| = µ0(φ) −1 · |(∂µS)(µ0(φ)µ, φ)| ≥ c|µ− 1| for µ ∈ supp γφ(·, λ) and φ ∈ (0, 1]. Moreover, for α ∈ N and α′ ∈ N \ {0}, we have |∂αµγφ(µ, λ)| = λνµ0(φ) α|(∂αµγ)(µ0(φ)µ, λ, φ)| ≲ λνµ0(φ) α(µ0(φ)µ) 2ν−α ( 1 + λµ0(φ) 2µ )−ν ≲ µν−α, supp(γφ(·, λ)) ⊂ { µ0(φ) −1λ− 1 2 ≤ µ ≤ µ0(φ) −1 } , ∣∣∂1+α′ µ Sφ(µ) ∣∣ ≲ |µ0(φ)|α ′−1 Thus the stationary phase theorem (see Lemma A.4 with k = 2, j = 0 and x0 = 1) implies µ0(φ)λ −ν |I2(λ, φ)| ≲ µ0(φ)λ −ν · ( λµ0(φ) 2 )− 1 2 = λ− 1 2 −ν , where we use λµ0(φ) 2 ≳ 1. Next, we consider the estimate of I1(λ, φ). We note that |∂µSφ(µ)| ≳ 1 for µ ∈ suppψ1. Then the non-stationary phase theorem (see Lemma A.3 with µ = ν) implies µ0(φ)λ −ν |I1(λ, φ)| ≲ µ0(φ)λ −ν · ( λµ0(φ) 2 )−ν−1 ≲ φλ−ν · ( λφ2 )− 1 2 = λ− 1 2 −ν , where we use λµ0(φ) 2 ≳ 1. Finally, we consider the term I3(λ, φ). We note that |∂µSφ(µ)| ≳ (|µ| + 1) for µ ∈ suppψ3. Then, by integrating by parts N(> 1) times, we have µ0(φ)λ −ν |I3(λ, φ)| ≲ µ0(φ)λ −ν · ( λµ0(φ) 2 )−N ≲ µ0(φ)λ −ν · ( λµ0(φ) 2 )− 1 2 = λ− 1 2 −ν , where we use λµ0(φ) 2 ≳ 1. This completes the proof. ■ Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 11 3 Asymptotic expansion of special functions and decomposition of the sum 3.1 Asymptotic expansion of Bessel functions Here we discuss asymptotic behavior of Bessel functions. For our purpose, we need symbol-type estimates, which is the estimates for higher order derivatives of the amplitudes. Such estimates are studied in [17, Theorem 9.1] up to the second derivatives. However, their method cannot be applied to estimates for higher-derivatives. Their theorem is not sufficient for our purpose since we need estimates whose order depends on ν here. Now we write down the statement here and give its proof in Appendix A.4. We define h1(z) = √ 1− z2 − z cos−1 z, (3.1) where cos−1 z ∈ [0, π/2] for 0 ≤ z ≤ 1. Proposition 3.1. (i) There are smooth functions a±,y : [ 0, y − 1 2y 1 3 ] → C such that Jµ(y) = y− 1 4 (y − µ)− 1 4 ( a+,y(µ)e iyh1( µ y ) + a−,y(µ)e −iyh1( µ y )) and for each α ∈ N, there exists Cα > 0 such that |∂αµa±,y(µ)| ≤ Cα(y − µ)−α, for y ≥ 8, µ ∈ [ 0, y − 1 2y 1 3 ] . (ii) Let N > 0 and α ∈ N. Then there exist Cα > 0 and CαN > 0 such that |∂αµJµ(y)| ≤  Cαy − 1+α 3 for y ≥ 8, µ ∈ [ y − 2y 1 3 , y + 2y 1 3 ] , CαNµ − 1 4 (µ− y)− 1 4 −α ( y−1(µ− y)3 )−N for y ≥ 8, µ ∈ [ y + 1 2y 1 3 ,∞ ) . Remark 3.2. (1) The estimate for µ ∈ [ y + 1 2y 1 3 ,∞ ) is weaker than the one in [17, Theorem 9.1] at least for α = 0, 1, 2. In fact, [17, Theorem 9.1] shows that Jµ(y) and its derivatives decay exponentially with respect to y−1(µ− y)3 (due to the term e−yh2(µ/y) in [17]) although we just obtain the polynomial decay here. (2) Another drawback of this proposition compared to [17, Theorem 9.1] is not to give symbol- type estimates of the Y -Bessel functions. In particular, we have a uniform bound |Jµ(y)| ≤ Cy− 1 3 y ≥ 8, µ ≥ 0. (3.2) 3.2 Asymptotic expansion of Gegenbauer polynomials The following proposition is a refinement of [17, Lemma 10.2] and its proof is given in Ap- pendix A.6. Proposition 3.3. Let ν ≥ 0. There are functions g+(m,φ), g−(m,φ) which are smooth respect to m ∈ [1,∞) and a function r(m,φ) defined for m ∈ N∗ such that ν−1Cν m(cosφ) = ∑ ± gν,±(m,φ)e ±imφ + r(m,φ) 12 K. Taira and H. Tamori for m ∈ N∗ (we interpret the left-hand side for ν = 0 as limν↘0 ν −1Cν m(cosφ)), and |∂αmgν,±(m,φ)| ≤ Cα,νm 2ν−1−α(1 +m sinφ)−ν , m ≥ 1, |r(m,φ)| ≤ CN,νm −N , m ∈ N∗ for each N > 0, α ∈ N, and φ ∈ [0, π] with constants Cα,ν , CN,ν > 0. Remark 3.4. (1) In particular, for each ε > 0, we have |∂αmgν,±(m,φ)| ≤ Cα,ν,εm ν−1−α uniformly in φ ∈ [ε, π − ε]. (2) We do not impose that gν,± are continuous with respect to φ ∈ [0, π]. In this paper, we just need uniform estimates of gν,± in φ ∈ [0, π]. (3) In [17, Lemma 10.2], a similar estimate for ν−1Cν m(cosφ) divided by Cν m(1) is given. Here we also deal with the estimate for Cν m(1). Moreover, the ranges of the parameters are extended compared to there. In particular, we have a uniform bound: For ν ≥ 0, there exists C > 0 such that∣∣ν−1Cν m(cosφ) ∣∣ ≤ Cm2ν−1 m ∈ N∗, φ ∈ [0, π], (3.3) where the left-hand side is interpreted as ∣∣ limν↘0 ν −1Cν m(cosφ) ∣∣ when ν = 0. 3.3 Decomposition of the sum In this subsection, we divide the sum I (b, ν;−iy; cosφ) into three parts according to the asymp- totic behavior of the Bessel function. We fix b > 0 and set Lb,ν := Γ(bν + 1)2bν . (3.4) From (1.5), we obtain the formula I (b, ν;−iy; cosφ) =Lb,νy −bν ∞∑ m=0 e− π 2 bmiJb(m+ν)(y)(m+ ν)ν−1Cν m(cosφ). (3.5) In the following, we consider the case y ≫ 1 for simplicity. The case y ≪ −1 is similarly dealt with (see proof of Theorem 1.4 in Section 6.1). We define µ(m) := b(m+ ν). (3.6) Corresponding to the asymptotic behavior of the Bessel functions, we define Ω1 = { m ∈ [1,∞) | 1 ≤ µ(m) ≤ y − 1 2 y 1 3 } , Ω2 = { m ∈ [1,∞) | y − 2y 1 3 ≤ µ(m) ≤ y + 2y 1 3 } , Ω3 = { m ∈ [1,∞) | µ(m) ≥ y + 1 2 y 1 3 } . Lemma 3.5. For y ≫ 1, there exist χ0 ∈ C∞ c (R; [0, 1]) and χj,y ∈ C∞(R; [0, 1]) (1 ≤ j ≤ 3) such that χ0(m) = 1 for µ(m) ≤ 2 or m ≤ 1 and χ0(m) + 3∑ j=1 χj,y(m) = 1 for m ≥ 0, suppχj,y ⊂ Ωj , |∂αmχ2,y(m)| ≤ Cαy −α 3 , |∂αmχ1,y(m)| ≤ Cαmax(m−α, (y − µ(m))−α), |∂αmχ3,y(m)| ≤ Cα(µ(m)− y)−α. (3.7) Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 13 Proof. Let χ0 ∈ C∞ c (R; [0, 1]) and χj ∈ C∞(R; [0, 1]) for j = 1, 2, 3 such that χ0(m) = 1 for µ(m) ≤ 2 or m ≤ 1, ∑3 j=1 χj = 1 on R, suppχ1 ⊂ ( −∞,−1 2 ] , suppχ2 ⊂ [−2, 2] and suppχ3 ⊂ [ 1 2 ,∞ ) . We define χj,y(m) = (1− χ0(m))χj ( y− 1 3 (µ(m)− y) ) for j = 1, 2, 3. They satisfies the desired properties. ■ For σ = (σ1, σ2) ∈ {±} × {±} and j = 2, 3, we set S1,σ(m, y, φ) = σ1yh1 ( µ(m) y ) + ( σ2φ− π 2 b ) m, Sσ2(m, y, φ) = ( σ2φ− π 2 b ) m, where h1 is defined in (3.1). We remark that S1,σ(m, y, φ) is defined for 0 ≤ µ(m) ≤ y. We take y ≫ 1 such that Ω2 ∪ Ω3 ⊂ {µ(m) ≥ 8} in order to apply Proposition 3.1. We define ζ1,σ(m, y, φ) = Lb,νy −bν− 1 4χ1,y(m)(m+ ν)(y − µ(m))− 1 4 gν,σ2(m,φ)aσ1,y(µ(m)), ζj,σ2(m, y, φ) = Lb,νy −bνχj,y(m)(m+ ν)Jµ(m)(y)gν,σ2(m,φ), j = 2, 3, where a±,y and gν,± are defined in Propositions 3.1 and 3.3 respectively and Lb,ν is a constant defined in (3.4). Now it follows from Propositions 3.1, 3.3, the formula (3.5) and Lemma 3.5 that I (b, ν;−iy; cosφ) = ∑ σ∈{±}×{±} I1,σ(y, φ) + 3∑ j=2 ∑ σ2∈{±} Ij,σ2(y, φ) +R(y, φ), (3.8) where we set I1,σ(y, φ) := ∞∑ m=1 ζ1,σ(m, y, φ)e iS1,σ(m,y,φ), Ij,σ2(y, φ) := ∞∑ m=1 ζj,σ2(m, y, φ)e iSσ2 (m,y,φ), R(y, φ) := Lb,νy −bν ∞∑ m=0 χ0(m)e− π 2 bmiJµ(m)(y)(m+ ν)ν−1Cν m(cosφ) + Lb,νy −bν ∞∑ m=0 (1− χ0(m))e− π 2 bmiJµ(m)(y)(m+ ν)r(m,φ) for j = 2, 3. The remainder term R(y, φ) is easy to handle. Proposition 3.6. There exists C > 0 such that |R(y, φ)| ≤ Cy−bν− 1 3 for y ≫ 1 and φ ∈ [0, π]. Proof. Since χ0 is compactly supported, then the first term in the definition of R(y, φ) is bounded by a constant times y−bν · y− 1 3 = y−bν− 1 3 due to (3.2) and (3.3). Moreover, the second term has a similar bound since r(m, y) is rapidly decreasing by Proposition 3.3. ■ In the following, we focus on studying I1,σ and Ij,σ2 . 14 K. Taira and H. Tamori 3.4 Properties of the phase functions Here we prove some estimates of the phase functions defined in the last subsection. We recall h1(z) = √ 1− z2 − z cos−1 z and h′1(z) = − cos−1 z. For σ = (σ1, σ2) ∈ {±} × {±} and j = 2, 3, we have ∂mS1,σ(m, y, φ) = −σ1b cos−1 ( µ(m) y ) + σ2φ− π 2 b, ∂mSσ2(m, y, φ) = σ2φ− π 2 b, where µ(m) is defined in (3.6). Lemma 3.7. Let 0 < δ < 1 and σ ∈ {±} × {±}. (i) For α ∈ N∗, there exists Cα > 0 such that ∣∣∂α+1 m S1,σ(m, y, φ) ∣∣ ≤ {Cαy −α for 1 ≤ µ(m) ≤ δy, Cαy − 1 2 (y − µ(m))−α+ 1 2 for δy ≤ µ(m) ≤ y − 1 2y 1 3 and for m ∈ Ω1, y ≫ 1 and φ ∈ [0, π]. (ii) For α ∈ N∗ \ {1}, we have ∂αmSσ2(m, y, φ) = 0. Proof. Since ∂z cos −1 z = −(1− z2)− 1 2 , we have ∂2mS1,σ(m, y, φ) = σ1b 2 ( y2 − µ(m)2 )− 1 2 . We have ∣∣∂α+1 m S1,σ(m, y, φ) ∣∣ = ∣∣∂α−1 m ∂2mS1,σ(m, y, φ) ∣∣ ≲ yα−1 ( y2 − µ(m)2 )−α+ 1 2 ≤ y− 1 2 (y − µ(m))−α+ 1 2 . If 1 ≤ µ(m) ≤ δy, then we have (y− µ(m))−1 ≲ y−1. Combining these estimates, we obtain the part (i). The part (ii) is easy to prove. ■ 3.5 Estimates of the amplitudes In this subsection, we deduce estimates for ζ1,σ and ζj,σ2 which are defined in Section 3.3. Recall from (3.6) that µ(m) = b(m+ ν). Lemma 3.8. Let σ = (σ1, σ2) ∈ {±} × {±}, N > 0, 0 < ε < π/2 and α ∈ N. Then (i) supp ζ1,σ(·, y, φ) ⊂ Ω1 and supp ζj,σ2(·, y, φ) ⊂ Ωj for j = 2, 3. (ii) Let 0 < δ < 1. We have |∂αmζ1,σ(m, y, φ)| ≲  y−bν− 1 2 (1 +m)2ν−α(1 +m sinφ)−ν when µ(m) ≤ δy, y(2−b)ν− 1 4 (y − µ(m))− 1 4 −α(1 +m sinφ)−ν when δy ≤ µ(m) ≤ y − 1 2y 1 3 for m ≥ 1, y ≫ 1 and φ ∈ [0, π]. In particular, for fixed ε > 0, we have |∂αmζ1,σ(m, y, φ)| ≲ { y−bν− 1 2 (1 +m)ν−α when µ(m) ≤ δy, y(1−b)ν− 1 4 (y − µ(m))− 1 4 −α when δy ≤ µ(m) ≤ y − 1 2y 1 3 for m ≥ 1, y ≫ 1 and φ ∈ [ε, π − ε]. Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 15 (iii) We have |ζ2,σ2(m, y, φ)| ≲ { y(1−b)ν− 1 3 for φ ∈ [ε, π − ε], y(2−b)ν− 1 3 for φ ∈ [0, π] and for m ≥ 1, y ≫ 1. Moreover, |∂αmζ2,σ2(m, y, φ)| ≲ y(2−b)ν− 1+α 3 for m ≥ 1, y ≫ 1 and φ ∈ [0, π]. (iv) We have |ζ3,σ2(m, y, φ)| ≲  y(2−b)ν+ 1 12 (µ(m)− y)− 5 4 for y + 1 2y 1 3 ≤ µ(m) ≤ 4y, φ ∈ [0, π], y(1−b)ν+ 1 12 (µ(m)− y)− 5 4 for y + 1 2y 1 3 ≤ µ(m) ≤ 4y, φ ∈ [ε, π − ε], (1 +m)−N for µ(m) ≥ 2y, φ ∈ [0, π] and for m ≥ 1, y ≫ 1. Moreover, |∂αmζ3,σ2(m, y, φ)| ≲ y(2−b)ν− 1 4 −α 3 for m ≥ 1 y ≫ 1 and φ ∈ [0, π]. Proof. (i) This follows from the definition of ζ1,σ, ζj,σ2 and the support properties of χj,y (3.7). (ii) It turns out from Lemmas 3.1, 3.3 and 3.5 that |∂αmζ1,σ(m, y, φ)| ≲ y−bν− 1 4 (1 +m)2ν(y − µ(m))− 1 4 (1 +m sinφ)−ν ×max((y − µ(m))−α,m−α). Now the claim in (ii) follows from the support property of ζ1,σ. (iii) We note that Ω2 = { m ∈ R | y − 2y 1 3 ≤ µ(m) ≤ y + 2y 1 3 } . This part directly follows from Lemmas 3.1, 3.3 and 3.5. (iv) We recall µ(m) = b(m+ν) and that ζ3,σ2(m, y, φ) is supported in Ω3 = { m ∈ R | µ(m) ≥ y + 1 2y 1 3 } . By Lemmas 3.1, 3.3 and 3.5, for each N > 0 and α ∈ N, we have |∂αmζ3,σ2(m, y, φ)| ≲ y−bν(1 +m)2ν− 1 4 (1 +m sinφ)−ν(µ(m)− y)− 1 4 −α ( y−1(µ(m)− y)3 )−N for m ≥ 1, y ≫ 1 and φ ∈ [0, π]. We note that y−1(µ(m) − y)3 ≥ 1/8 and µ(m)− y ≥ 1 2y 1 3 for m ∈ Ω3. All the estimates we desire are proved by these inequalities. ■ 4 Estimates of I2,σ2 and I3,σ2 In this section, we prove bounds for I2,σ2 and I3,σ2 appearing in (3.8). They are easier to handle than I1,σ. In this section, we assume b > 0, ν ≥ 0 and let σ2 ∈ {±}. 4.1 Intermediate region Proposition 4.1. We have |I2,σ2(y, φ)| ≲ { y(1−b)ν if b /∈ 2Z, y(2−b)ν if b ∈ 2Z for y ≫ 1 and φ ∈ [0, π]. 16 K. Taira and H. Tamori Proof. By Lemma 3.8 (i) and (iii), we have |ζ2,σ2(m, y, φ)| ≲ y(2−b)ν− 1 3 and |I2,σ2(y, φ)| ≲ y(2−b)ν− 1 3 ∑ m∈Ω2∩N 1 ≲ y(2−b)ν , where we recall Ω2 = { m ∈ R | y − 2y 1 3 ≤ µ(m) ≤ y + 2y 1 3 } and use the number of element of Ω2 ∩ N is bounded by y 1 3 times a constant. We next consider the case b /∈ 2Z. A similar argument implies |I2,σ2(y, φ)| ≲ y(1−b)ν uniformly in φ ∈ [ε, π−ε] (with a constant 0 < ε < π/2), where we use the first estimate in Lemma 3.8 (iii). Thus, it remains to prove the bound of I2,σ2(y, φ) for φ ∈ [0, ε] ∪ [π − ε, π] with ε > 0 small enough when b /∈ 2Z. In this case, we have −π 2 b, σ2π − π 2 b /∈ 2πZ. Then we find ε > 0 small enough such that there is cε > 0 satisfying dist ( σ2φ − π 2 b, 2πZ ) ≥ cε for φ ∈ [0, ε) ∪ (π − ε, π]. This implies dist(∂mSσ2(m, y, φ), 2πZ) ≥ cε for φ ∈ [0, ε) ∪ (π − ε, π]. Moreover, it follows from Lemma 3.8 (iii) that |∂αmζ2,σ2(m, y, φ)| ≲ y(2−b)ν− 1 3 −α 3 . Clearly, we have ∂α+1 m Sσ2(m, y, φ) = 0 for α ≥ 1. Applying Proposition 2.2 with k = (2 − b)ν − 1 3 , r = cε, M ∼ y and ρ = 1 3 , for each N > 0, we obtain |I2,σ2(y, φ)| ≲ y−N for φ ∈ [0, ε)∪(π−ε, π]. This completes the proof. ■ 4.2 Decaying region Proposition 4.2. We have |I3,σ2(y, φ)| ≲ { y(1−b)ν if b /∈ 2Z, y(2−b)ν if b ∈ 2Z for y ≫ 1 and φ ∈ [0, π]. Proof. We recall µ(m) = b(m + ν). Taking χ ∈ C∞(R; [0, 1]) such that χ(µ) = 1 for µ ≤ 2 and χ(µ) = 0 for µ ≥ 4 and setting χ = 1− χ, we write I3,σ2(y, φ) = ( ∞∑ m=1 (χ(µ(m)/y) + χ(µ(m)/y))ζ3,σ2(m, y, φ)e iSσ(m,y,φ) ) =: I3,1,σ2(y, φ) + I3,2,σ2(y, φ). First, we deal with the second term I3,2,σ2(y, φ). Let N > 0. Then Lemma 3.8 (iv) implies |ζ3,σ2(m, y, φ)| ≲ (1 +m)−N−1 for m ∈ suppχ(µ(m)/y). Thus we obtain |I3,2,σ2(y, φ)| ≲ ∞∑ µ(m)≥2y,m≥1 (1 +m)−N−1 ≲ y−N . Next, we deal with the first term I3,1,σ2(y, φ). Lemma 3.8 (iv) implies |ζ3,σ2(m, y, φ)| ≲ y(2−b)ν+ 1 12 (µ(m)− y)− 5 4 for m ∈ suppχ(µ(m)/y). (4.1) Hence, |I3,1,σ2(y, φ)| ≲ y(2−b)ν+ 1 12 ∑ µ(m)∈[y+ 1 2 y 1 3 ,4y] (µ(m)− y)− 5 4 ≲ y(2−b)ν+ 1 12 ∫ 4y y+ 1 2 y 1 3−b (µ− y)− 5 4dµ ≲ y(2−b)ν , Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 17 where we use Lemma 2.1 in the second inequality. Consequently, we obtain |I3,1,σ2(y, φ)| ≲ y(2−b)ν , which completes the proof for b ∈ 2Z. When b /∈ 2Z, a similar argument implies |I3,1,σ2(y, φ)| ≲ y(1−b)ν uniformly in φ ∈ [ε, π − ε] (with a constant 0 < ε < π/2), where we use the second estimate in Lemma 3.8 (iv) instead of (4.1). Thus, it remains to prove the bound of I3,1,σ2(y, φ) when b /∈ 2Z for φ ∈ [0, ε] ∪ [π − ε, π] with ε > 0 small enough. Finally, we suppose b /∈ 2Z and prove the bound for I3,1,σ2(y, φ) for φ ∈ [0, ε] ∪ [π − ε, π] with ε > 0 small enough. In this case, we have −π 2 b, σ2π− π 2 b /∈ 2πZ. Then we find ε > 0 small enough such that there is cε > 0 satisfying dist(σ2φ − π 2 b, 2πZ) ≥ cε for φ ∈ [0, ε) ∪ (π − ε, π]. This implies dist(∂mS3,σ2(m, y, φ), 2πZ) ≥ cε for φ ∈ [0, ε)∪ (π− ε, π]. Moreover, it follows from Lemma 3.8 (iv) that |∂αmζ3,1,σ2(m, y, φ)| ≲ y(2−b)ν− 1 4 −α 3 . Clearly, we have ∂α+1 m S3,σ2(m, y, φ) = 0 for α ≥ 1. Now applying Proposition 2.2 with k = (2 − b)ν − 1 4 , r = cε, M ∼ y and ρ = 1 3 , for each N > 0, we obtain |I3,1,σ2(y, φ)| ≲ y−N for φ ∈ [0, ε) ∪ (π − ε, π]. This completes the proof. ■ 5 Estimates of I1,σ In this section, we prove estimates for I1,σ, which are more delicate than those of I2,σ2 , I3,σ2 . We assume b > 0, ν ≥ 0 and let σ ∈ {±} × {±}. 5.1 General bounds for b > 0 Proposition 5.1. Let ε > 0. Then |I1,σ(y, φ)| ≲ { y(1−b)ν for φ ∈ [ε, π − ε], y(2−b)ν for φ ∈ [0, π] and for y ≫ 1. Proof. We deal with the case φ ∈ [0, π] only since the proof is almost same if we use the first estimate of Lemma 3.8 (ii) instead of the second one. The identity (2.1) implies I1,σ(y, φ) = ∫ R ζ1,σ(m, y, φ)e iS1,σ(m,y,φ)dm − 1 2πi ∑ q∈Z\{0} 1 q ∫ R ζ̃1,σ(m, y, φ)e iS1,σ(m,y,φ)+2πiqmdm, (5.1) where we set ζ̃1,σ(m, y, φ) = ∂mζ1,σ(m, y, φ) + i(∂mS1,σ)(m, y, φ)ζ1,σ(m, y, φ). By Lemmas 3.7 and 3.8 (ii), for each ζ ∈ { ζ1,σ, ζ̃1,σ } we have |∂αmζ(m, y, φ)| ≲ { y−bν− 1 2 (1 +m)2ν−α when µ(m) ≤ 3 4y, y(2−b)ν− 1 4 (y − µ(m))− 1 4 −α when 3 4y ≤ µ(m) ≤ y − 1 2y 1 3 (5.2) for m ≥ 1, y ≫ 1 and φ ∈ [0, π]. Moreover, supp ζ ⊂ Ω1 holds. Now we consider an integral Iq := ∫ R ζ(m, y, φ)eiS1,σ(m,y,φ)+2πiqmdm for q ∈ Z and ζ ∈ {ζ1,σ, ζ̃1,σ}. Taking c0 > 0 such that |∂mS1,σ(m, y, φ)| ≤ c0 for m ≥ 1, y ≫ 1 and φ ∈ [0, π]. Now we show that |Iq| ≲ { y(2−b)ν for |q| ≤ c0/π, y(2−b)ν(1 + |q|)−1 for |q| > c0/π, 18 K. Taira and H. Tamori and for y ≫ 1 and φ ∈ [0, π]. These estimates immediately imply the bound |I1,σ(y, φ)| ≲ y(2−b)ν due to the identity (5.1). First, we deal with the case |q| > c0/π. Since |∂m(S1,σ(m, y, φ) + 2πqm)| ≥ 2π|q| − c0 ≳ (1 + |q|), for ν > 0, the integration by parts yields |Iq| = ∣∣∣∣∫ R ∂m ( 1 ∂m(S1,σ(m, y, φ) + 2πqm) ζ(m, y, φ) ) eiS1,σ(m,y,φ)+2πiqmdm ∣∣∣∣ ≤ ∫ R ( |∂2mS1,σ(m, y, φ)||ζ(m, y, φ)| |∂m(S1,σ(m, y, φ) + 2πqm)|2 + |∂mζ(m, y, φ)| |∂m(S1,σ(m, y, φ) + 2πqm)| ) dm ≲ (1 + |q|)−1y−bν− 1 2 ∫ 1≤µ(m)≤ 3 4 y,m≥1 (1 +m)2ν−1dm + (1 + |q|)−1y(2−b)ν− 1 4 ∫ 3 4 y≤µ(m)≤y− 1 2 y 1 3 (y − µ(m))− 5 4dm ≲ (1 + |q|)−1y(2−b)ν , where we use (5.2), Lemma 3.7 (i) and the support property supp ζ ⊂ Ω1. The case ν = 0 can be proved similarly if we use the integration by parts twice. Next, we consider the case |q| ≤ c0/π. By the change of variable µ(m)(= bm+ bν) = yµ, Iq =b −1e−i(σ2 φ b −π 2 + 2πq b )bνy(2−b)ν+ 1 2 ∫ R eiySq(µ,φ)γy,φ(µ)dµ, where we set Sq(µ, φ) = σ1h1(µ) + ( σ2 φ b − π 2 + 2πq b ) µ, γy,φ(µ) = y(b−2)ν+ 1 2 ζ (yµ b − ν, y, φ ) . Thus it remains to show∣∣∣∣∫ R eiySq(µ,φ)γy,φ(µ)dµ ∣∣∣∣ ≲ y− 1 2 for y ≫ 1, φ ∈ [0, π]. (5.3) By (5.2), |∂αµγy,φ(µ)| ≲ µ2ν−α(1− µ)− 1 4 −α, supp γy,φ ⊂ { µ ∈ R | 0 ≤ µ ≤ 1− 1 2 y− 2 3 } for y ≫ 1 and φ ∈ [0, π]. We write∫ R eiySq(µ,φ)γy,φ(µ)dµ = ∫ R eiySq(µ,φ)γy,φ(µ)ψ1(µ)dµ+ ∫ R eiySq(µ,φ)γy,φ(µ)ψ2(µ)dµ = I1 + I2, where ψ1, ψ2 ∈ C∞ c (R; [0, 1]) satisfy ψ1(µ)+ψ2(µ) = 1 for µ ∈ [0, 1], ψ1(µ) = 1 for 0 ≤ µ ≤ 1−2δ and ψ2(µ) = 1 for 1− δ ≤ µ ≤ 1 where δ > 0 is determined later. Since∣∣∂2µSq(µ, φ)∣∣ = ∣∣σ1∂2µh1(µ)∣∣ = ∣∣(1− µ2 )− 1 2 ∣∣ ≥ 1 for µ ∈ suppψ1 ∩ [0, 1], the stationary phase theorem (see Lemma A.5 (i) with λ = y) implies |I1| ≲ y− 1 2 . On the other hand, using the change of variable µ′ = √ 1− µ (with µ = 1 − µ′2), we have dµ = −2µ′dµ′ and I2 = ∫ R eiyS̃q(µ′,φ)γ2(µ ′)dµ′, where we set S̃q(µ ′, φ) = Sq ( 1−µ′2, φ ) and γ2(µ ′) = 2µ′γy,φ ( 1− µ′2 ) ψ2 ( 1− µ′2 ) . We note that |∂αµ′γ2(µ ′)| ≲ µ′ 1 2 −α, supp γ2 ⊂ { 1√ 2 y− 1 3 ≤ µ′ ≤ √ 2δ } Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 19 for y ≫ 1 and φ ∈ [0, π] and that S̃q is smooth with respect to µ′ close to 0 and φ ∈ [0, π]. It follows from the identity ∂3µ′S̃q(µ ′, φ) = σ1∂ 3 µ′ ( h1 ( 1− µ′2 )) = 12σ1µ ′h′′1 ( 1− µ′2 ) − 8σ1µ ′3h (3) 1 ( 1− µ′2 ) = 4 √ 2σ1 +O(µ′) as µ′ → 0 that ∣∣∂3µ′S̃q(µ ′, φ) ∣∣ ≳ 1 for µ′ ∈ supp γ2 and φ ∈ [0, π] if δ > 0 is small enough (here we note that h ( 1− µ′2 ) is smooth at µ′ = 0 although h(µ) is not smooth at µ = 1 ). Thus, the stationary phase theorem (see Lemma A.5 (ii) with λ = y) implies |I2| ≲ y− 1 2 . This proves (5.3) and completes the proof of Proposition 5.1. ■ 5.2 Improvement for 0 < b < 2 In this subsection, we improve Proposition 5.1 near φ = 0 when 0 < b < 2. Proposition 5.2. We assume 0 < b < 2. Then there exist φ0 > 0 such that |I1,σ(y, φ)| ≲ y(1−b)ν for y ≫ 1, φ ∈ [0, φ0]. Remark 5.3. Combining with Proposition 5.1, we obtain the uniform estimates for φ ∈ [0, π−ε] with arbitrary ε > 0. In order to prove it, we need a more information about the phase function S1,σ. In the part (i) of the next lemma, we use the assumption 0 < b < 2 crucially. This is used to prove that the second term of the right-hand side in (2.1) is harmless. Lemma 5.4. (i) There exists ε0 > 0 such that |∂mS1,σ(m, y, φ)| ≤ 2π − ε0 for all m ∈ Ω1 and y ≥ 1 and φ ∈ [ 0, (1− b 2)π ] . (ii) There exists φ1, c0 > 0 such that |∂mS1,σ(m, y, φ)| ≥ c0 for m ∈ Ω1, y ≥ 1 with µ(m) ≥ 1 2y and φ ∈ [0, φ1]. Remark 5.5. If b ≥ 2, then ∂mS1,σ(m, y, φ) can take a value in 2πZ\{0}. This prevents better estimates of I1,σ. Proof of Lemma 5.4. We recall ∂mS1,σ(m, y, φ) = −bσ1 cos−1 (µ(m) y ) + σ2φ− π 2 b. (i) This follows from a direct calculation: ∣∣−bσ1 cos−1(z)+σ2φ− π 2 b ∣∣ ≤ πb+φ ≤ (1+ b 2)π for φ ∈ [ 0, ( 1− b 2 ) π ] and 0 ≤ z ≤ 1. Then we set ε0 = ( 1− b 2 ) π, which is positive since 0 < b < 2. (ii) This also follows from a direct calculation∣∣∣−bσ1 cos−1(z) + σ2φ− π 2 b ∣∣∣ ≥ (π 2 − cos−1 z ) b− |φ| ≥ π 6 b− |φ| for 1 2 ≤ z ≤ 1. Taking φ1 = c0 = πb 12 , we obtain the bound in (ii). ■ Proof of Proposition 5.2. Define φ0 := min (( 1− b 2 ) π, φ1 ) > 0, where φ1 is as in Lemma 5.4. Taking χ ∈ C∞(R; [0, 1]) such that χ(µ) = 1 for µ ≤ 1/2 and χ(µ) = 0 for µ ≥ 3/4 and setting χ = 1− χ, we write I1,σ(y, φ) = ∞∑ m=1 (χ(µ(m)/y) + χ(µ(m)/y))ζ1,σ(m, y, φ)e iS1,σ(m,y,φ) =: I1,1,σ(y, φ) + I1,2,σ(y, φ). 20 K. Taira and H. Tamori By the support property of χ and Lemma 3.8 (ii), we have |∂αm(χ(µ(m)/y)ζ1,σ(m, y, φ))| ≲ y(2−b)ν− 1 3 −α 3 . Moreover, by virtue of Lemma 3.7 (i) and Lemma 5.4 (i), (ii), the phase function S1,σ satisfies the assumption of Proposition 2.2 with r = min(c0, ε0),M ∼ y and ρ = 1 3 . Applying Proposition 2.2 with k = (2 − b)ν − 1 3 , for each N > 0, we obtain |I1,2,σ(y, φ)| ≲ y−N . Thus, it remains to estimate I1,1,σ(y, φ). We write I1,1,σ as in (5.1) and consider the integral Iq = ∫ R ζ(m, y, φ)eiS1,σ(m,y,φ)+2πiqmdm, for ζ ∈ {ζ ′1,σ, ζ ′′1,σ}, where we set ζ ′1,σ(m, y, φ) =χ(µ(m)/y)ζ1,σ(m, y, φ), ζ ′′1,σ(m, y, φ) =∂mζ ′ 1,σ(m, y, φ) + i(∂mS1,σ)(m, y, φ)ζ ′ 1,σ(m, y, φ). As in the proof of Proposition 5.1, it suffices to prove the existence of φ0 > 0 such that |Iq| ≲ { y(1−b)ν for q = 0, y(1−b)ν(1 + |q|)−1 for |q| ≥ 1, (5.4) and for m ≥ 1, y ≫ 1 and φ ∈ [0, φ0]. By Lemmas 3.7 (i), 3.8 (ii) and the support property of χ, for ζ ∈ {ζ ′1,σ, ζ ′′1,σ}, we have |∂αmζ(m, y, φ)| ≲ y−bν− 1 2m2ν−α(1 +m sinφ)−ν (5.5) and supp ζ(·, y, φ) ⊂ { m ≥ 1 | µ(m) ∈ [ µ(1), 34y ]} . First, we consider the case |q| ≥ 1. In this case, we have |∂m(S1,σ(m, y, φ) + 2πqm)| ≳ (1 + |q|) by Lemma 5.4 (i). By using integration by parts many times, for each N > 0, we have |Iq| ≲ y−N (1 + |q|)−N due to Lemma 3.7 (i) and the estimate (5.5). This proves the second estimates of (5.4). Next, we consider the case q = 0. By the change of variable µ(m)(= bm+ bν) = yµ, I0 = b−1e−i(σ2 φ b −π 2 )bνy(2−b)ν+ 1 2 ∫ R eiyS(µ,φ)γy,φ(µ)dµ, where we set S(µ, φ) = σ1h1(µ) + ( σ2 φ b − π 2 ) µ, γy,φ(µ) = y(b−2)ν+ 1 2 ζ (yµ b − ν, y, φ ) . By (5.5), the support property of χ and φ ≲ sinφ, we have |∂αµγy,φ(µ)| ≲ µ2ν−α(1 + yφµ)−ν , supp γy,φ ⊂ [ µ(1)y−1, 3 4 ] . Moreover, the phase function S(µ, φ) satisfies the assumption of Proposition 2.4 with the critical point µ(φ) = cos ( σ1 ( −π 2 + σ2 φ b )) . Now Proposition 2.4 with λ = y implies |I0| ≲ y(2−b)ν+ 1 2 · y−ν− 1 2 = y(1−b)ν . This completes the proof. ■ Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 21 6 Proof of the main theorem 6.1 Proof of Theorem 1.4 The claim (i) for b = 1 directly follows from [2, equation (4.45)]. Hence we may assume b ̸= 1. Let R0 ≫ 1. First, we prove the theorem for |y| ≤ R0. To do this, we follow the argument in [2, Lemma 4.17]. Using the bound ∣∣Ĩλ(w)∣∣ ≤ e|Rew|Γ(λ+ 1)−1, Cν 0 (t) = 1 and ∣∣ν−1Cν m(t) ∣∣ ≲ (1 +m)2ν−1 for m ≥ 1 (see [2, equation (4.16)], [2, Fact 4.8] and (3.3)), we have |I (b, ν;−iy; cosφ)| ≲ ∞∑ m=0 (1 +m) · |y|bm · 1 Γ(bm+ bν + 1) · (1 +m)2ν−1 = ∞∑ m=0 (1 +m)2ν Γ(bm+ bν + 1) · |y|bm, which is convergent and bounded by a constant independent of y ∈ R with |y| ≤ R0. Now the estimate for y ≥ R0 follows from the decomposition (3.8) of I and Propositions 3.6, 4.1, 4.2, 5.1, 5.2. The case y ≤ −R0 is similarly dealt with due to the formula I (b, ν; iy; cosφ) =Lb,νe ibνπy−bν ∞∑ m=0 e π 2 bmiJb(m+ν)(y)(m+ ν)ν−1Cν m(cosφ), which in turn follows from the identity Jµ ( eiπy ) = eiµπJµ(y) and (3.5). 6.2 Proof of Theorem 1.1 Now we use the following general result due to Keel–Tao. Theorem 6.1 ([18]). Let X be a measure space and {U(t)}t∈R be a bounded family of continuous linear operators on L2(X) such that there are C > 0 and σ > 0 such that ∥U(t)U(s)∗∥L1(X)→L∞(X) ≤ C|t− s|−σ for all t, s ∈ R with t ̸= s. Let (p, q) ∈ [2,∞]2 such that 1 p + σ q = σ 2 and (p, q, σ) ̸= (2,∞, 1). (6.1) Then there exists C > 0 such that ∥U(t)u∥Lp(R;Lq(X)) ≤ C∥u∥L2(X),∥∥∥∥∫ t −∞ U(t)U(s)∗f(s)ds ∥∥∥∥ Lp1 (R;Lq1 (X)) ≤ C∥f∥ Lp∗2 (R;Lq∗2 (X)) , where (p1, q1), (p2, q2) satisfy (6.1) and r∗ denotes the Hölder conjugate of r: r∗ = r/(r − 1). First we note that the operator norm of an operator from L1 to L∞ is equal to the L∞-norm of its integral kernel. We may assume T ≤ π 4 due to the argument in Appendix B. First, we consider the cases � n = 1 and a ≥ 2− 4k, � n ≥ 2 and (0 < a ≤ 1 or a = 2). 22 K. Taira and H. Tamori We claim∣∣e−itHk,a(x, x′) ∣∣ ≲ |t|−σk,a for |t| ≤ π 2 . (6.2) Let us prove the claim for the case n = 1 and a ≥ 2− 4k. In this case, we have σk,a ≥ 1 2 and [2, Proposition 4.29] implies e−itHk,a(x, x′) = Γ(σk,a) ei |x|a+|x′|a a cot(t) (i sin(t))σk,a × ( Ĩσk,a−1 ( 2|xx′| 2 a ai sin(t) ) + xx′ (ai sin(t)) 2 a Ĩσk,a−1+ 2 a ( 2|xx′| 2 a ai sin(t) )) . (6.3) From the asymptotic expansion of the I-Bessel function [33, Section 7.23 (2), (3)], we see Iν(±iy) = e±iy (±2πiy)1/2 ( 1 +O ( y−1 )) (6.4) as y → ∞. Hence (6.4) and Ĩν(z) = ( z2) −νIν(z) imply that there exists C > 0 such that∣∣Ĩσk,a−1(iy) ∣∣+ ∣∣y 2 a Ĩσk,a−1+ 2 a (iy) ∣∣ ≤ C(1 + |y|)−σk,a+ 1 2 ≤ C for any y ∈ R. Therefore, our claim (6.2) follows from (6.3). Let us consider the case n ≥ 2 and (0 < a ≤ 1 or a = 2). Since dµkx̂ is a probability measure, we see∥∥∥∥∫ Rn f ( ξ · x̂′ ) dµkx̂(ξ) ∥∥∥∥ L∞(Rn×Rn) ≤ ∥f∥L∞([−1,1]). Using this estimate and (1.6), we obtain ∣∣e−itHk,a(x, x′) ∣∣ ≤ck,a 1 | sin(t)|σk,a sup η∈[−1,1] ∣∣∣∣I(2 a , a 2 (σk,a − 1);−i 2|x| a 2 |x′| a 2 a sin(t) ; η )∣∣∣∣. Therefore, the claim (6.2) follows from Theorem 1.4 (ii). By the claim and Theorem 6.1 with U±(t) := 1[0,T ](t)e ∓itHk,a , we obtain the homogeneous Strichartz estimates (1.3) and∥∥∥∥∫ t 0 e−i(t−s)Hk,af(s)ds ∥∥∥∥ Lp1 ([0,T ];Lq1 ) ≤ C∥f∥ Lp∗2 ([0,T ];Lq∗2 ) ,∥∥∥∥∫ t 0 ei(t−s)Hk,af(−s)ds ∥∥∥∥ Lp1 ([0,T ];Lq1 ) ≤ C∥f∥ Lp∗2 ([−T,0];Lq∗2 ) . Since ∥∥∥∥∫ t 0 ei(t−s)Hk,af(−s)ds ∥∥∥∥ Lp1 ([0,T ];Lq1 ) = ∥∥∥∥∫ t 0 e−i(t−s)Hk,af(s)ds ∥∥∥∥ Lp1 ([−T,0];Lq1 ) , we also obtain the inhomogeneous Strichartz estimates (1.4). Next, we consider the case when 1 < a < 2 and k ≡ 0. Let 0 < φ0 < 2π. Take a finite partition of unity {χj}Nj=1 ⊂ C∞(Sn−1; [0, 1] ) on the sphere such that ω, η ∈ suppχj ⇒ ω ·η ≥ cosφ0. Due to (1.7) and Theorem 1.4 (i), there exists C > 0 such that ∣∣χj(x̂)e −itHk,a(x, x′)χj ( x̂′ )∣∣ ≤ C|t|−σ0,a for |t| ≤ π 2 and x, x′ ∈ Rn \ {0}, where we recall x̂ = x/|x|. Hence,∥∥χj(x̂)e −i(t−s)Hk,a(x, x′)χj ( x̂′ )∥∥ L∞(Rn x×Rn x′ ) ≤ C|t− s|−σ0,a Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 23 for |t|, |s| ≤ π 4 with t ̸= s. Applying Theorem 6.1 with U(t) := 1[−T,T ](t)χj(x̂)e −itHk,a , we obtain∥∥χj(x̂)e −itHk,au ∥∥ Lp([−T,T ];Lq) ≤ C∥u∥L2 . Since the number N is finite and since ∑N j=1 χj = 1, we obtain the homogeneous Strichartz estimates (1.3). The inhomogeneous Strichartz estimates (1.4) when 1 < a < 2 and k ≡ 0 follow from the Christ–Kiselev lemma [5] and a complex interpolation since the end-point case (pj , qj) = (2, 2σ0,a σ0,a−1) is excluded (see the proof of [4, Theorem 6]). 6.3 Proof of Theorem 1.3 Let us define elements of sl(2,R) as follows e+ := ( 0 1 0 0 ), h := ( 1 0 0 −1 ) , e− := ( 0 0 1 0 ). As- sume σk,a > 0. Then it is proved in [2, Theorem 3.30] that there exists a unitary represen- tation Ωk,a of the universal cover S̃L(2,R) of SL(2,R) on the Hilbert space L2(Rn, ϑk,a(x)dx) (see (1.1) for the definition of ϑk,a) satisfying Ωk,a ( exp ( te+ )) u(x) = e it|x|a a u(x), Ωk,a(exp(th))u(x) = etσk,au ( e 2t a x ) , Ωk,a(exp(te −))u(x) = e it|x|2−a∆k a u(x), Ωk,a(exp(t(e − − e+)))u(x) = e−itHk,au(x) for any u ∈ L2(Rn, ϑk,a(x)dx). Let θ = arctan(t). We apply Ωk,a to the identity exp ( θ ( e− − e+ )) = exp ( −te+ ) exp ( log ( 1 + t2 ) 2 h ) exp(te−) in S̃L(2,R), and we obtain e−iθHk,au(x) = e −it|x|a a ( 1 + t2 )σk,a 2 ( e it|x|2−a∆k a u )(( 1 + t2 )1/a x ) (6.5) for any u ∈ L2(Rn, ϑk,a(x)dx). Proof of Theorem 1.3. From (6.5), the integral kernel e it|x|a−2∆k a (x, y) of e it|x|a−2∆k a equals e it|x|a (1+t2)a ( 1 + t2 )−σk,a 2 e−i arctan(t)Hk,a (( 1 + t2 )−1 a x, y ) . Therefore, the similar arguments as the proof of Theorem 1.1 and the equation( 1 + t2 ) 1 2 sin(arctan(t)) = t imply ∣∣e it|x|a−2∆k a (x, y) ∣∣ ≲ |t|−σk,a for t ∈ R, and Theorem 1.3 follows. ■ A Asymptotic behavior of special functions A.1 Leibniz’s rule We frequently use the following formula, which directly follows from Leibniz’s rule. Lemma A.1. Let r be a smooth function and N ∈ N\{0}. Then there exist smooth functions bjN such that ( ∂x ◦ r(x)−1 )N = 1 r(x)N N∑ j=0 bjN (x)∂jx with |bjN (x)| ≲ N−j∑ k=1 ∑ ℓ1+···+ℓk=N−j, ℓ1,...,ℓk≥1 k∏ i=1 ∣∣r(ℓi)(x)∣∣ |r(x)| . 24 K. Taira and H. Tamori Remark A.2. More precisely, we have ( ∂x ◦ r(x)−1 )N = 1 r(x)N ∂Nx + 1 r(x)N N−1∑ j=0 ( N−j∑ k=1 ∑ ℓ1+···+ℓk=N−j, ℓ1,...,ℓk≥1 CjN ℓ1ℓ2...ℓk k∏ i=1 r(ℓi)(x) r(x) ) ∂jx with constants CjN ℓ1ℓ2...ℓm > 0. A.2 Non-stationary phase theorem with a singular amplitude The following lemma is more or less well known and an easy consequence of integration by parts. We give a proof for completeness of the paper. Lemma A.3. Let µ > −1, γ ∈ C∞((0,∞)) which is supported in x ≤ 10 such that for α ∈ N, we have |∂αx γ(x)| ≲ |x|µ−α for x ∈ (0,∞). Let f ∈ C∞(R) satisfy Im f(x) ≥ 0, f ′(x) ̸= 0 for x ∈ supp γ and f(0) ∈ R. We define bµ(λ) := e−iλf(0) ∫ ∞ 0 γ(x)eiλf(x)dx for λ ≳ 1. Then for each α ∈ N |∂αλ bµ(λ)| ≲ λ−µ−1−α, λ ≳ 1. (A.1) In particular, ∣∣ ∫∞ 0 γ(x)eiλf(x)dx ∣∣ ≲ λ−µ−1 for λ ≳ 1. Proof. We only need to prove these estimates for sufficiently large λ. First, we prove (A.1) for α = 0. Let χ ∈ C∞ c (R) such that χ(x) = 1 for |x| ≤ 1/2 and χ(x) = 0 for |x| ≥ 1. Define χλ(x) = χ(λx) and χλ(x) = 1− χλ(x). Then we have∣∣∣∣∫ ∞ 0 γ(x)χλ(x)e iλf(x)dx ∣∣∣∣ ≲ ∫ 1/λ 0 xµdx ≲ λ−µ−1. (A.2) On the other hand, the integration by parts yields∣∣∣∣∫ ∞ 0 γ(x)χλ(x)e iλf(x)dx ∣∣∣∣ = λ−N ∣∣∣∣∫ ∞ 0 LN (γ(x)χλ(x))e iλf(x)dx ∣∣∣∣ , where we define L = ∂x ◦ (if ′(x))−1. By using Leibniz’s rule, Lemma A.1 and the assumption f ′ ̸= 0 on supp γ, we obtain |LN (γ(x)χλ(x))| ≲ |x|µ−N . Hence, for N > µ+ 1,∣∣∣∣∫ ∞ 0 γ(x)χλ(x)e iλf(x)dx ∣∣∣∣ ≲ λ−N ∫ 10 1 2λ xµ−Ndx ≲ λ−µ−1. (A.3) The inequalities (A.2) and (A.3) imply (A.1) for α = 0. To deal with the case α ≥ 1, we write ∂αλ bµ(λ) = iαe−iλf(0) ∫ ∞ 0 (f(x)− f(0))αγ(x)eiλf(x)dx. By Taylor’s theorem, we have ∂Nx (γ(x)(f(x)−f(0))α) = O ( |x|µ+α−N ) for x ∈ supp γ. Moreover,∣∣e−iλf(0) ∣∣ = 1 due to the assumption f(0) ∈ R. Hence the same argument as the case α = 0 gives (A.1) for α ≥ 1. ■ Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 25 A.3 Stationary phase theorem In this paper, we use the following versions of the stationary phase theorem (the van der Corput lemma). The proof is same as the proof of [27, Lemma 1.1.2] (see also in [28, p. 334]). The statement about uniformity in a parameter follows from its proof and Lemma A.1. We consider an integral I(λ) := ∫ R γ(x)eiλf(x)dx for λ ≳ 1. Lemma A.4. Let k ≥ 2 be an integer and j ≥ 0, x0 ∈ R, γ ∈ C∞(R \ {x0}) ∩ Cc(R) and f ∈ C∞(R;R) such that |∂αx γ(x)| ≲ |x−x0|j−α, |f(x)| ≲ |x−x0|k and |f ′(x)| ≳ |x−x0|k−1 for x ∈ supp γ \ {x0}. Then |I(λ)| ≲ λ− j+1 k for λ ≳ 1. As a result, we also obtain the following version. Lemma A.5. Let γ ∈ C∞(R \ {0}) ∩ Cc(R) and f ∈ C∞(R;R). (i) Let ν ≥ 0. Suppose that |∂βxγ(x)| ≲ |x|ν−β for x ∈ R \ {0} and f ′′(x) ̸= 0 for x ∈ supp γ. Then |I(λ)| ≲ λ− 1 2 for λ ≳ 1. (ii) Suppose that ∣∣∂βxγ(x)∣∣ ≲ |x| 1 2 −β for x ∈ R \ {0}, f ′(0) = 0 and f ′′′(0) ̸= 0. If supp γ is sufficiently close to 0, then |I(λ)| ≲ λ− 1 2 for λ ≳ 1. Remark A.6. (1) The main difference between Lemma A.4 and this lemma is that we do not assume f ′(0) = 0 in (i) for example. (2) The estimates in (i) and (ii) are optimal. If in addition we assume that supp γ is close to 0 for the case (i), then the optimal estimate would be |I(λ)| ≲ λ− 1 2 (1+ν). Proof of Lemma A.5. The proof of the part (i) is almost same as the case (ii). Thus we only deal with the case (ii). (ii) We may assume f(0) = 0 since ∣∣eiλf(0)∣∣ = 1. Moreover, we may assume that a := f ′′(0) is sufficiently small since our result directly follows from [28, p.334, Corollary] for a away from 0. For simplicity, we also assume a > 0 and f ′′′(0) ≥ 8. First, we consider the case where a3λ ≪ 1. We write f ′(x) = ax + f ′′′(0) 2 x2 + g(x) with g(x) = O ( |x|3 ) . Then we can take supp γ small enough such that |f ′(x)| ≳ |x|2 for x ∈ supp γ with |x| ≥ λ− 1 3 due to the assumptions a3λ ≪ 1 and f ′′′(0) ̸= 0. Take χ ∈ C∞ c ((−2, 2); [0, 1]) such that χ(x) = 1 for |x| ≤ 1 and set χλ(x) = χ ( λ 1 3xv) and χλ = 1− χλ. Then∣∣∣∣∫ R γ(x)χλ(x)e iλf(x)dx ∣∣∣∣ ≲ ∫ |x|≤2λ− 1 3 |x| 1 2dx ≲ λ− 1 3 (1+ 1 2 ) = λ− 1 2 . On the other hand, the integration by parts yields∣∣∣∣∫ R γ(x)χλ(x)e iλf(x)dx ∣∣∣∣ ≤ λ−1 ∫ R |∂x ( f ′(x)−1γ(x)χλ(x) ) |dx ≲ λ−1 ∫ |x|≥λ− 1 3 |x|− 5 2dx ≲ λ− 1 2 . Thus we obtain |I(λ)| ≲ λ− 1 2 . Next we consider the case where a3λ ≳ 1. Taking supp γ sufficiently close to 0, we may assume the following three conditions hold: Critical points of f (on supp γ) are x = 0 or x = x0(a) with 26 K. Taira and H. Tamori |x0(a)| ∈ [ a 4 , 3 4a ] ( due to the factorization f ′(x) = x ( a+ f ′′′(0) 2 x+ h(x) ) with h(x) = O ( x2 )) . The second condition is |f ′′(x0(a))| ≳ |a|. The third condition is |f ′(x)| ≳ |ax| for |x| ≥ a. By scaling, we have I(λ) = a 3 2 ∫ R γa(x)e iλa3fa(x)dx, fa(x) := a−3f(ax), γa(x) := a− 1 2γ(ax). (A.4) We observe fa(x) = 1 2x 2 + f ′′′(0) 6 x3 +O ( a|x|4 ) and ∣∣∂βxγa(x)∣∣ ≲ |x| 1 2 −β. Moreover, critical points of fa are x = 0 or x = a−1x0(a) with ∣∣a−1x0(a) ∣∣ ∈ [14 , 34], |f ′′a (x)| ≳ 1 and |f ′a(x)| ≳ |x| for |x| ≥ 1. Then we divide the integral (A.4) into three parts I1(λ), I2(λ), I3(λ), where the integrand of Ij(λ) is supported close to 0 for j = 1, a−1x0(a) −1 for j = 2 and |f ′′a (x)| ≳ 1 on the supports of the integrands of I1(λ) and I2(λ) for sufficiently small a. Then Lemma A.4 with the large param- eter λa3 gives |I1(λ)| ≲ a 3 2 ( λa3 )− 1 2 (1+ 1 2 ) ≲ λ− 1 2 and |I2(λ)| ≲ a 3 2 ( λa3 )− 1 2 = λ− 1 2 (for the latter case, we use f ′(x(a)) = 0 and use this lemma for the phase function f(x)− f(x(a))). Moreover, integrating by parts many times, we have |I3(λ)| ≲ a 3 2 ( λa3 )−N for all N > 0. Taking N = 1 2 , we obtain |I3(λ)| ≲ λ− 1 2 and hence |I(λ)| ≲ λ− 1 2 . ■ A.4 Asymptotics of the Bessel function In this appendix, we prove Proposition 3.1. First, we seek a nice integral representation approx- imating the Bessel function Jµ(y). Lemma A.7. There exists χ1 ∈ C∞ c (( −3π 4 , 3π 4 ) ; [0, 1] ) such that χ1(w) = 1 for |w| ≤ 2π 3 and χ1(w) = χ1(−w) such that Jµ(y) = 1 2π ∫ R ei(y sinw−µw)χ1(w)dw +R0(µ, y) for µ ≥ 0, y ≥ 1, (A.5) where R0(µ, y) satisfies |∂αµR0(µ, y)| ≤ CNα(1 + y + µ)−N for each N ∈ N. Proof. It is known (see [33, Section 6.21 (3)], see also [17, equation (9.7)]) that the Hankel function can be written as Hµ(y) = 1 πi (∫ 0 −∞ + ∫ πi 0 + ∫ ∞+πi πi ) ey sinhw−µwdw (A.6) and ReHµ(y) = Jµ(y). Since the curve [−∞, 0] ∪ [0, πi] ∪ [πi,∞ + πi] is not smooth, we will deform it to remove the corner at πi by using the Cauchy integral formula, where we note that the corner at 0 is not involved with the asymptotics of Jν(y). More precisely, we take a curve γ : R → C such that γ is smooth and |γ′(r)| = 1 except at 0, and γ(r) =  r for r ≤ 0, ir for 0 ≤ r ≤ 3 4π, r − 2 3π − L+ iπ for r ≥ 3 4π + L,{ Re γ(r) ≥ 0, Im γ(r) ∈ [ 2π 3 , π ] , for 2 3 π ≤ r ≤ 3 4 π + 2L with a constant L > 0. Since Re(y sinhw−µw) = y sinh(Rew) cos(Imw)−µRew and y, µ ≥ 0, we have Re(y sinhw − µw)|w=γ(r) ≤ 0 for r ≥ 0. (A.7) Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 27 Now we take χ1 ∈ C∞ c (( −3π 4 , 3π 4 ) ; [0, 1] ) , χ2 ∈ C∞ c ([ 2π 3 , 3π 4 +2L ] ; [0, 1] ) and χ3 ∈ C∞(R; [0, 1]) such that χ1 + χ2 + χ3 = 1 on [0,∞) and χ1(r) = χ1(−r), χ1(r) = 1 for |r| ≤ 2π 3 , χ3(r) = { 0 for r ≤ 3 4π + L, 1 for r ≥ 3 4π + 2L. By the Cauchy’s integral formula, we rewrite (A.6) as Hµ(y) = 1 πi ∫ 0 −∞ ey sinhw−µwdw + 1 π ∫ π 0 ei(y sinw−µw)χ1(w)dw + R̃0(µ, y), R̃0(µ, y) = 1 πi ∫ R ey sinh γ(r)−µγ(r)χ2(r)γ̃ ′(r)dr + e−iπµ πi ∫ R e−y sinhw−µwχ3 ( w + 2 3 π + L ) dw. Since the first term is purely imaginary, we obtain Jµ(y) = ReHµ(y) = 1 2π ∫ R ei(y sinw−µw)χ1(w)dw +Re R̃0(µ, y), where we use χ1(r) = χ1(−r), suppχ1 ⊂ ( −3 4π, 3 4π ) and y sin(−w)− µ(−w) = −(y sinw− µw). Now we set R0(µ, y) := Re R̃0(µ, y). It suffices to prove |∂αµ R̃0(µ, y)| ≤ CNα(1 + y + µ)−N for each N ∈ N, µ ≥ 0 and y ≥ 1. We note that |∂r(y sinh γ(r) − µγ(r))| ≥ |y cosh(Re γ(r)) cos(Im γ(r)) − µ| ≳ y + µ for r ∈ suppχ2, where we use |γ′(r)| = 1 and Re γ ≥ 0, Im γ ∈ [23π, π] on suppχ2. Moreover, |∂w(−y sinhw − µw)| = y coshw + µ ≳ y + µ. Thus, the integration by parts with (A.7) yields ∣∣∂αµ R̃0(µ, y) ∣∣ ≤ CNα(1 + y + µ)−N for each N ∈ N, µ ≥ 0 and y ≥ 1. ■ Remark A.8. This lemma might be proved also by integration by parts and by Schläfli’s formula Jµ(y) = 1 π ∫ π 0 cos(y(sinw)− µw)dw − sin(πµ) π ∫ ∞ 0 e−y(sinhw)−µwdw, which follows from (A.6). However, to do this, we need to calculate the boundary terms at w = π in the first term and the one at w = 0 in the second term explicitly. In the above proof, we avoid it and use the deformation of the integral instead. Now we study the first term (we call it F1(y, µ)) of the right-hand side in (A.5). To do this, we use a variant of the stationary phase theorem. Roughly speaking, the phase function y sinw − µw of F1(y, µ) has two non-degenerate critical points when µ≪ y and no critical points when µ≫ y. When µ ≈ y, the critical points can become degenerate although the third derivative of the phase function does not vanish there. The difficulty here is to deal with them in a uniform way. We will use the following lemma with a = y − µ, b = y, f(w) = sinw − w. Lemma A.9. Let χ ∈ C∞ c ((−3, 3)). Let f ∈ C∞([−3, 3];R) such that f (j)(0) = 0 for j = 0, 1, 2, ∓f ′′(w) ≤ 0 and c1|w| ≤ |f ′′(w)| ≤ c2|w| for 0 ≤ ±w ≤ 3 with a constant 0 < c1 < c2. For a ∈ R, b ≥ 1 and j ∈ N, set Ij(a, b) = ∫ R χ(w)wjeiaw+ibf(w)dw. (A.8) 28 K. Taira and H. Tamori (i) Suppose a, b ≥ 1, |a| 1 2 b− 1 2 ≤ 1, |a| 3 2 b− 1 2 ≥ 1/8. Then the map [−3, 3] ∋ w 7→ aw + bf(w) has just two critical points w±(a, b) with ±w±(a, b) > 0. Moreover, we can write I0(a, b) = ∑ ± I0,±(a, b)e iaw±(a,b)+ibf(w±(a,b)), |∂αa I0,±(a, b)| ≤ Cαa − 1 4 −αb− 1 4 uniformly in a, b ≥ 1, |a| 1 2 b− 1 2 ≤ 1, |a| 3 2 b− 1 2 ≥ 1/8. (ii) For each j ∈ N and N > 0, there exist Cj > 0 and CjN > 0 such that |Ij(a, b)| ≲ { Cjb − j+1 3 , a ∈ R, b ≥ 1, |a| 1 2 b− 1 2 ≤ 1, CjN ( |a| 3 2 b− 1 2 )−N |a|−j− 1 4 b− 1 4 , a ≤ −1, b ≥ 1, |a| 3 2 b− 1 2 ≥ 1/8. Proof. When a = 0, then the claim in (ii) directly follows from the stationary phase theorem (see Lemma A.4 with k = 3). Hence in the following, we assume a ̸= 0. Set s = |a| 1 2 b− 1 2 , λ = |a| 3 2 b− 1 2 , F (w, s) = (sgn a)w + bλ−1f(sw) = (sgn a)w + s−3f(sw). By the change of variable w 7→ sw, we have Ij(a, b) = sj+1 ∫ R χ(sw)wjeiλF (w,s)dw. (A.9) By the assumption on f , we have − c2 2 w 2 ≤ f ′(w) ≤ − c1 2 w 2 for |w| ≤ 3. We note that there is Cα > 0 such that |∂αwF (w, s)| ≤ Cα for α ≥ 2 uniformly as long as s ≤ 1. Moreover, |∂wF (w, s)| = ∣∣(sgn a) + s−2f ′(sw) ∣∣ ≥ c ( 1 + |w|2 ) for |w| ∈ [ 2c − 1 2 1 ,∞ ) (A.10) uniformly in a. Moreover, when a ≥ 0, the derivative ∂wF (w, s) = 1 + s−2f ′(sw) has just the two critical points w±(s) with ±w±(s) ∈ [√ 2c−1 2 , √ 2c−1 1 ] and c1 √ 2c−1 2 ≤ ∣∣∂2wf(w±(s), s) ∣∣ ≤ c2 √ 2c−1 1 . When a < 0, then F (·, s) has no critical points. Moreover, w±(a, b) := sw±(s) are critical points of the map w 7→ aw + bf(w). (i) Suppose a, b ≥ 1, 0 < s ≤ 1 and λ ≥ 1/8. Let χ+ ∈ C∞ c ((0,∞); [0, 1]) such that χ+(w) = 1 for w ∈ [√ 2c−1 2 , √ 2c−1 1 ] , suppχ+ ⊂ ( 0, 2 √ c−1 1 ] . Set χ−(w) := χ+(−w) and χ0 = 1− χ+ − χ−. We write (A.9) for j = 0 as I0(a, b) = ∑ ± eiλF (w±(s),s)I0,±(a, b) +R1(a, b), I0,±(a, b) := s ∫ R χ(sw)χ±(w)e iλ(F (w,s)−F (w±(s),s))dw, R1(a, b) := s ∫ R χ(sw)χ0(w)e iλF (w,s)dw. We prove that for α, β,N ∈ N,∣∣∂αλ (s∂s)βI0,±(a, b)∣∣ ≲ sλ− 1 2 −α, ∣∣∂αλ (s∂s)βR1(a, b) ∣∣ ≲ sλ−N . (A.11) Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 29 First, we observe |(s∂s)αw±(s)| ≲ 1 for 0 < s ≤ 1. (A.12) The estimate for α = 0 follows from the fact ±w±(s) ∈ [√ 2c−1 2 , √ 2c−1 1 ] as we have proved. To see (A.12) for α ≥ 1, we recall w±(s) are the critical points of F (·, s), that is, 1 + s−2f ′(sw±(s)) = 0 ⇔ f ′(sw±(s)) = s2. Thus we have f ′′(sw±(s))(w±(s) + s∂sw±(s)) = 2s. (A.13) Since |f ′′(sw±(s))| ∼ s|w±(s)| ∼ s, we have s|w±(s)+ s∂sw±(s)| ≲ 2s, which implies |s∂sw±(s)| ≲ 1. This proves (A.12) for α = 1. Differentiating (A.13) many times and using induction on α, we obtain (A.12) for general α ≥ 0. By Taylor’s theorem and the assumption on f , the function F (w, s) = w + s−3f(sw) is smooth with respect to w and s ∈ [0, 1], where the point is that F is smooth even near s = 0. It turns out from this fact, (A.12) and the Taylor’s theorem that F (w + w±(s), s)− F (w±(s), s) = g±(w, s)w 2, where g± is smooth with respect to w ∈ suppχ± and s ∈ (0, 1] and satisfies∣∣∂αw(s∂s)βg±(w, s)∣∣ ≲ 1 for w ∈ suppχ±, s ∈ (0, 1]. Moreover, we have g±(0, w) = ( ∂2wF ) (w±(s), s) = s−1f ′′(sw±(s)), which satisfies |g±(0, s)| ≳ 1 uniformly in 0 < s ≤ 1. Consequently, I0,±(a, b) can be written as I0,±(a, b) = s ∫ R ψ0,±(w, s)e iλg±(w,s)w2 dw, where ψ0,±(w, s) = χ(s(w + w±(s)))χ±(w + w±(s)). We also note that∣∣∂αw(s∂s)βψ0,±(w, s) ∣∣ ≲ 1 for w ∈ R, s ∈ (0, 1]. Now we prove the first estimate of (A.11). For simplicity, we consider the case (α, β) = (1, 0) or (α, β) = (0, 1) only. To do this, we write ∂λI0,±(a, b) = is ∫ R ψ0,±(w, s)g±(w, s)w 2eiλg±(w,s)w2 dw, s∂sI0,±(a, b) = s ∫ R ( ψ0,±(w, s) ( 1 + iλs∂sg±(w, s)w 2 ) + s∂sψ0,±(w, s) ) eiλg±(w,s)w2 dw. Now we apply the stationary phase theorem (see Lemma A.4 with k = 2, j = 0, 2, x0 = 0) and obtain the first estimate of (A.11) for (α, β) = (1, 0), (0, 1). The estimates for its higher deriva- tives are similarly proved. To prove the second estimate of (A.11), we use the fact that ∂wF (w, s) does not vanish for w ∈ suppχ0 ∩ suppχ(s·) with the uniform estimate (A.10). Using the inte- gration by parts many times with (A.10) with the estimates for its higher-order derivatives, we obtain the second estimate of (A.11). Next, we show |∂γaI0,±(a, b)| ≤ Cγa − 1 4 −γb− 1 4 , |∂γaR1(a, b)| ≤ CγN ( a 3 2 b− 1 2 )−N a− 1 4 −γb− 1 4 (A.14) 30 K. Taira and H. Tamori for γ ∈ N and N > 0. Set Iγ0,±(a, b) = ∂γaI0,±(a, b). We can deduce ∣∣∂αλ (s∂s)βIγ0,±(a, b)∣∣ ≤ Cαβs 1+γλ− 1 2 −α−γ by (A.11) and induction due to Iγ+1 0,± = ∂aI γ 0,±, ∂a = (∂aλ)∂λ + (∂as)∂s = s ( 3/2∂λ + 2−1λ−1s∂s ) and s ≤ 1, λ ≥ 1/8. In particular, we obtain |∂γaI0,±(a, b)| ≤ Cγs 1+γλ− 1 2 −γ = Cγa − 1 4 −γb− 1 4 . The estimate for R1(a, b) is similarly proved. Since w±(a, b) = sw±(s), we have λfa,b(w±(s)) = λw±(s) + bf(sw±(s)) = aw±(a, b) + bf(w±(a, b)) and hence I0(a, b) = ∑ ± I0,±(a, b)e i(aw±(a,b)+bf(w±(a,b))) +R1(a, b). Finally, we show that R1(a, b) can be absorbed into either I0,+ or I0,−. To do this, it suffices to show ∣∣∂αa (e−i(aw±(a,b)+bf(w±(a,b)))R1(a, b) )∣∣ ≤ CαNa − 1 4 −αb− 1 4 . We observe∣∣∂αa eiλfa,b(w±(s)) ∣∣ ≤ Cαs −α = Cαa −α 2 b α 2 . since ∂a(λfa,b(w±(s))) = s−1w±(s). Thus the second inequality of (A.14) with N ≫ 1 gives the desired estimate. (ii, 1) First, we consider the case a ∈ R, b ≥ 1 and s ≤ 1. Let χ1 ∈ C∞ c (R; [0, 1]) such that χ1(w) = 1 for |w| ≤ √ 2c−1 1 , χ1(w) = 0 for |w| ≥ 2 √ c−1 1 . Set χ2 = 1− χ1. We write (A.9) for j = 0 as Ij(a, b) =s j+1 ∫ R χ(sw)wj(χ1(w) + χ2(w))e iλfa,b(w)dw =: Ij,1(a, b) + Ij,2(a, b). The second term is easy to handle: The integration by parts with (A.10) yields |Ij,2(a, b)| ≤ CjNs j+1λ−N for N > 0. Taking N = j+1 3 , we have |Ij,2| ≲ sj+1λ− j+1 3 = b− j+1 3 . Thus, we focus on the estimate for I1,j . When λ ≤ 1 (which is equivalent to |a| ≤ b 1 3 ), then |Ij,1(a, b)| ≲ sj+1 = |a| j+1 2 b− j+1 2 ≤ b− j+1 3 . On the other hand, when λ ≥ 1, then the stationary phase theorem (see Lemma A.4 with k = 3) implies |Ij,1(a, b)| ≲ sj+1λ− j+1 3 = b− j+1 3 . Thus we obtain |Ij(a, b)| ≲ b− j+1 3 . (ii, 2) Suppose a ≤ −1, b ≥ 1, s ≤ 1 and λ ≥ 1/8. The case s ≥ 1 is dealt with later. Since λ ≥ 1/8, it suffices to prove the inequality for large integer N . By the assumption on f , we have f ′a,b(w) ≥ 1 + cw2 and ∣∣f (α)a,b (w)/f ′ a,b(w) ∣∣ ≤ Cα(1 + |w|)−1 for sw ∈ suppχ with a con- stant Cα independent of s, λ for α ≥ 2. In fact,∣∣f ′′a,b(w)∣∣ = bλ−1s ∣∣f ′′(sw)∣∣ ≤ bλ−1s2 · |sw| ≲ (1 + |w|),∣∣f (α)a,b (w) ∣∣ = bλ−1sα ∣∣f (α)(sw)∣∣ ≤ bλ−1s3 ∣∣f (α)(sw)∣∣ = O(1) for α ≥ 3. Now we set L = Dw ◦ (f ′s,λ(w)). Then the integration by parts yields Ij(a, b) = λ−Nsj+1 ∫ R LN1 ( χ(sw)wj ) eiλfa,b(w)dw, for N1 ∈ N. Taking N1 large enough, we have∣∣LN1 ( χ(sw)wj )∣∣ ≤ C ( 1 + w2 )−1 Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 31 with a constant C > 0 independent of s, λ and w ∈ suppχ. Here the independence with respect to s, λ follows from the estimates for fa,b(w). Thus we have |Ij(a, b)| = O ( λ−Nsj+1 ) . Taking N1 = N + j + 1 2 , we have λ−N1sj+1 = ( |a| 3 2 b− 1 2 )−N · |a|− 3 2 j− 3 4 b 1 2 j+ 1 4 · |a| j+1 2 b− j+1 2 = ( |a| 3 2 b− 1 2 )−N |a|−j− 1 4 b− 1 4 . (ii, 3) Suppose a ≤ −1, b ≥ 1, s ≥ 1 and λ ≥ 1/8. To deal with this case, we do not use (A.9) but the definition (A.8). Since a ≤ −1 and f ′(w) < 0, we have |∂w(aw + bf(w))| ≥ a. Thus the integration by parts yields |Ij(a, b)| ≤ CN ′ |a|−N ′ for all N ′ > 0. Now take N ′ > 0 large enough such that |a|−N ′ ≤ ( |a| 3 2 b− 1 2 )−N |a|−j− 1 4 b− 1 4 , which is possible since s ≥ 1 implies |a| ≥ b. This completes the proof. ■ Proof of Proposition 3.1. Set a = y−µ, b = y and f(w) = sinw−w. Then it turns out that aw + bf(w) = y sinw − µw and f satisfies the assumption of Lemma A.9. We observe ∂µ = (∂µa)∂a + (∂µb)∂b = −∂a, |a| 3 2 b− 1 2 = |µ− y| 3 2 y− 1 2 , s ≤ 1 ⇔ |µ| ≤ 2|y|, λ ≥ 1/8 ⇔ |y − µ| ≥ 1 2 y 1 3 , ∂αµ I0(a, b) = (−i)α ∫ R eiy sinw−iµwwαχ1(w)dw. Then the part (ii) follows from Lemmas A.7 and A.9 (ii) when µ ≤ 2y holds. The case µ ≥ 2y follows from the integration by parts in the above expression of ∂αµ I0(a, b), where we use |∂y(y sinw − µw)| ≥ µ− |y| ≥ 1 4(µ+ y). Since w±(a, b) are critical points of aw + bf(w), that is, w±(a, b) = ± cos−1( b−a b ), we have aw±(a, b) + bf(w±(a, b)) = ±yh1 ( µ y ) , where we recall h1(z) = √ 1− z2 − z cos−1 z. This proves the part (i) if we set a+,y(µ) = 1 2π I0,+(a, b) + e −yh1( µ y ) R0(µ, y), a−,y(µ) = 1 2π I0,−(a, b). ■ A.5 Asymptotics of the Beta function For m ∈ [1,∞) and ν > 0, we define Fν,1(m) := ν−1Γ ( ν + 1 2 ) √ πΓ(ν) · Γ(m+ 2ν) Γ(m+ 1)Γ(2ν) . (A.15) By [2, Fact 4.8, equation (4.30)], we have Cν m(1) = Γ(m+2ν) Γ(m+1)Γ(2ν) and hence Fν,1(m) = ν−1Γ ( ν + 1 2 ) Cν m(1) √ πΓ(ν) for m ∈ N. (A.16) We denote the Beta function by B(x, y). Then it is known that B(x, y) = Γ(x)Γ(y) Γ(x+y) . We may write Fν,1(m) = ν−1Γ ( ν + 1 2 ) √ πΓ(ν) · 1 mB(m, 2ν) . (A.17) 32 K. Taira and H. Tamori Proposition A.10. Let α ≥ 0 be an integer and ν > 0. Then there exists Cν,α > 0 such that |∂αmB(m, 2ν)| ≤ Cν,αm −2ν−α, |∂αmFν,1(m)| ≤ Cν,αm 2ν−1−α for all m ∈ [1,∞). Proof. First, we note that it suffices to prove these estimates for sufficiently large m. As is well known (essentially due to Stirling’s formula), we have B(m, 2ν) ∼ Γ(2ν)m−2ν as m → ∞. This implies that the estimates for Fν,1(m) follow from the estimates for B(m, 2ν) and (A.17). The case α = 0 follows from B(m, 2ν) ∼ Γ(2ν)m−2ν as m → ∞. Hence, in the following, we consider the estimates for α ≥ 1. Let ψ ∈ C∞(R; [0, 1]) satisfy ψ(t) = 1 for t ≥ 1/2 and ψ(t) = 0 for t ≤ 1 4 . Since B(m, 2ν) =∫ 1 0 t m−1(1− t)2ν−1dt, we have ∂αmB(m, 2ν) = ∫ 1 0 tm−1gα,1(t)dt+ ∫ 1 0 tm−1gα,2(t)dt, where we set gα,1(t) = (log t)α(1− t)2ν−1ψ(t) and gα,2(t) = (log t)α(1− t)2ν−1(1− ψ(t)). First, we deal with the second term. Since supp gα,2 ⊂ {t ≤ 1/2} and since |(log t)α| ≲ |t|−1 for 0 < t ≤ 1/2, we have∣∣∣∣∫ 1 0 tm−1gα,2(t)dt ∣∣∣∣ ≲ ∫ 1 2 0 tm−2dt ≲ 2−m ≲ m−2ν−α for sufficiently large m ≥ 1. Next, we consider the first term. We note∣∣∂βt gα,1(t)∣∣ ≤ Cα|1− t|2ν−1+α−β for t ∈ [0, 1]. Now we write tm−1 = e(m−1)(log t) = ei(m−1)f̃(t), f̃(t) = −i log t. Then f̃ ′(t) = −i/t ̸= 0 and Im f̃(t) ≥ 0 hold for t ∈ suppψ ∩ (−∞, 1]. Now it follows from Lemma A.3 with λ = m, µ = 2ν − 1 and the change of variable x = 1− t that∣∣∣∣∫ 1 0 tm−1gα,1(t)dt ∣∣∣∣ ≲ m−2ν−α holds for sufficiently large m ≥ 1. This completes the proof. ■ A.6 Asymptotics of the Gegenbauer polynomials In this appendix, we prove Proposition 3.3 using non-stationary phase theorem. The case ν = 0 is easy to deal with (see the proof of Proposition 3.3 below) and hence we focus on the case ν > 0. By [2, equation (4.30)] (see also [17, Section 10]) and (A.16), ν−1Cν m(cosφ) =Fν,1(m) ∫ 1 −1 (cosφ+ i(sinφ)u)m ( 1− u2 )ν−1 du (A.18) for ν > 0, m ∈ N∗ and φ ∈ [0, π], where Fν,1 is defined in (A.15). To get an asymptotics of Cν m(cosφ), we extend N∗ ∋ m 7→ Cν m(cosφ) to a function on m ∈ [1,∞) (up to a negligible term). However, since the integrant (cosφ+ i(sinφ)u)m is not a single-valued function, we have to do it a bit carefully. Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 33 First, we deal with the case φ ∈ ( π 4 , 3 4π ) . We consider two branches of logarithmic smooth functions log+ z : C \ {iy | y ≤ 0} → C and log− z : C \ {iy | y ≥ 0} → C such that log+(cosφ+ i sinφ) = iφ and log−(cosφ− i sinφ)) = −iφ for all φ ∈ [0, π]. We define smooth functions f± by f±(φ, u) = −i log±(cosφ+ i(sinφ)u)∓ φ for (φ, u) such that cosφ + i(sinφ)u belongs to the domain of log±. We note that ([0, π] × R±) ∪ (( v[0, π4 ] ∪ [ 3 4π, π ]) × [ −1 2 , 1 2 ]) is included in the domain of f±. Then eimf±(φ,u) = (cosφ+ i(sinφ)u)me∓imφ for all m ∈ N∗ and ±u > 0. Now let χ±, χ0 ∈ C∞(R; [0, 1]) such that χ+ + χ0 + χ− = 1 on R, suppχ0 ⊂ [ −1 2 , 1 2 ] , suppχ+ ⊂ [ 1 4 ,∞ ) and suppχ− ⊂ ( −∞,−1 4 ] . We define Hν,±(m,φ) := ∫ 1 −1 eimf±(φ,u)χ±(u) ( 1− u2 )ν−1 du, Eν(m,φ) := ∫ 1 −1 (cosφ+ i(sinφ)u)mχ0(u) ( 1− u2 )ν−1 du, where Hν,±(m,φ) is defined for all m ∈ [1,∞) although Eν(m,φ) is defined only for m ∈ N∗ whenever φ ∈ ( π 4 , 3 4π ) (note that cosφ + i(sinφ)u may be zero for u ∈ suppχ0). By these construction and (A.18), we have ν−1Cν m(cosφ) =Fν,1(m) ( eimφHν,+(m,φ) + e−imφHν,−(m,φ) + Eν(m,φ) ) (A.19) for m ∈ N∗ and φ ∈ ( π 4 , 3 4π ) . Next, we consider the case φ ∈ [ 0, π4 ] ∪ [ 3 4π, π ] . We define E′ ν(m,φ) := ∫ 1 −1 eimf+(φ,u)χ0(u) ( 1− u2 )ν−1 du, m ∈ [1,∞), φ ∈ [ 0, π 4 ] ∪ [ 3 4 π, π ] , where we note that cosφ + i(sinφ)u ∈ C \ {iy | y ≤ 0} whenever φ ∈ [ 0, π4 ] ∪ [ 3 4π, π ] and u ∈ suppχ0. Then ν−1Cν m(cosφ) =Fν,1(m) ( eimφHν,+(m,φ) + e−imφHν,−(m,φ) + eimφE′ ν(m,φ) ) (A.20) holds for m ∈ N∗ and φ ∈ [ 0, π4 ] ∪ [ 3 4π, π ] . Now we show that Hν,±, Eν and E′ ν satisfy symbol-type estimates. The basic idea of the proof for Hν,± and Eν is to use the non-stationary phase theorem with a singular amplitude (see Lemma A.3). To do this, we observe that our phase function f±(φ, u) satisfies |f ′±(φ, u)| ≳ | sinφ| on u ∈ suppχ± (see the proof below). Although ∂uf±(φ, u) = 0 when φ = 0, π, we consider a large parameter λ = m sinφ rather than m and can apply Lemma A.3. Lemma A.11. Let α ≥ 0 be an integer and ν,N > 0. (i) There exists CN,ν > 0 such that |Eν(m,φ)| ≤ CN,νm −N for m ∈ N∗ and φ ∈ ( π 4 , 3 4π ) . (ii) There exists Cα,N,ν > 0 such that |∂αmE′ ν(m,φ)| ≤ Cα,N,νm −α(1 + m sinφ)−N for m ∈ [1,∞) and φ ∈ [ 0, π4 ] ∪ [ 3 4π, π ] . (iii) There exists Cα,ν > 0 such that |∂αmHν,±(m,φ)| ≤ Cα,νm −α(1+m sinφ)−ν for m ∈ [1,∞) and φ ∈ [0, π]. 34 K. Taira and H. Tamori Proof. (i) Since | cosφ+ i(sinφ)u| ≤ √ 5 8 for u ∈ suppχ0 and φ ∈ ( π 4 , 3 4π ) , we have | cosφ+ i(sinφ)u|m ≤ ( 5 8 )m 2 ≲ m−N for m ∈ N∗. The estimate for Eν directly follows from this inequality. (ii) Setting ψ(u) := χ0(u) ( 1− u2 )ν−1 , we write E′ ν(m,φ) = ∫ R eimf+(φ,u)ψ(u)du. First, we consider the case α = 0 by using the integration by parts. Since log+(cosφ + i sinφ) = iφ, we have f+(φ, u) = i ∫ 1 u d dr log+(cosφ+ i(sinφ)r)dr = − ∫ 1 u sinφ(cosφ− i(sinφ)r) cos2 φ+ r2 sin2 φ dr (A.21) and hence Im f+(φ, u) = ∫ 1 u (sin2 φ)r cos2 φ+r2 sin2 φ dr. This implies Im f+(φ, u) ≥ 0 for u ∈ [−1, 1]. In fact, this is trivial for u ≥ 0. For u ≤ 0, we write∫ 1 u ( sin2 φ ) r cos2 φ+ r2 sin2 φ dr = ∫ 1 0 ( sin2 φ ) r cos2 φ+ r2 sin2 φ dr + ∫ 0 u ( sin2 φ ) r cos2 φ+ r2 sin2 φ dr = ∫ 1 −u ( sin2 φ ) r cos2 φ+ r2 sin2 φ dr, which is non-negative. Now we set L = ∂u ◦ (i∂uf+(φ, u))−1. By integrating by parts, we have |E′ ν(m,φ)| = m−N ∣∣∣∣∫ R eimf+(φ,u)LNψ(u)du ∣∣∣∣ ≤ m−N ∫ R ∣∣LNψ(u) ∣∣du by virtue of Im f+(φ, u) ≥ 0. Since ∂αu f+(φ, u) = (−1)α−1(α − 1)!(sinφ)α(cosφ + iu sinφ)−α, for each α ≥ 2, we obtain |∂uf+(φ, u)|−1 ≲ | sinφ|−1 and |∂αu f+(φ, u)||∂uf+(φ, u)|−1 ≲ 1 for u ∈ [−1, 1] and φ ∈ [ 0, π4 ] ∪ [ 3 4π, π ] . By Lemma A.1, we conclude ∣∣LNψ(u) ∣∣ ≲ | sinφ|−N and hence |E′ ν(m,φ)| ≲ m−N | sinφ|−N for φ ∈ [ 0, π4 ] ∪ [ 3 4π, π ] (we note that ψ is compactly supported). On the other hand, the inequality Im f+(φ, u) ≥ 0 also implies |E′ ν(m,φ)| ≲ 1. Combining these estimates, we have proved (ii) for α = 0. Next, we consider the case α ≥ 1. We note ∂αmE ′ ν(m,φ) = iα ∫ R eimf+(φ,u)f+(φ, u) αψ(u)du. As in the case of α = 0, we can deduce ∣∣LN (f+(φ, u) αψ(u)) ∣∣ ≲ |sinφ|α−N , which leads to the part (ii) for α ≥ 1 since f+(0, u) = f+(π, u) = 0. (iii) We deal with the case + only. There exists c > 0 such that |cosφ+ iu sinφ| ≥ c for u ∈ suppχ+ and φ ∈ [0, π]. Since Im f+(φ, u) = ∫ 1 u (sin2 φ)r cos2 φ+r2 sin2 φ dr, which is proved above, we have Im f+(φ, u) ≥ 0 for u ∈ suppχ+ and φ ∈ [0, π]. Moreover, we have f+(φ, 1) = 0. Now we prove (iii) for α = 0. Set m′ = m sinφ and g(φ, u) = (sinφ)−1f+(φ, u). Then we write Hν,+(m,φ) = ∫ 1 −1 e im′g(φ,u)χ+(u) ( 1− u2 )ν−1 du. Since ∂αu g(φ, u) = (−1)α−1(α− 1)!(sinφ)α−1(cosφ+ iu sinφ)−α for α ≥ 1 and since |cosφ+ iu sinφ| ∈ [ 1√ 2 , 1 ] for φ ∈ [ 0, π4 ] ∪ [ 3 4π, π ] , we have |∂ug(φ, u)| ≥ c, |∂αu g(φ, u)||∂ug(φ, u)|−1 ≤ C ′ α with a constant c > 0 and C ′ α > 0. By Lemma A.3 with λ = m sinφ and the change of variable x = 1− u, we obtain |Hν,+(m,φ)| ≲ (m′)−ν for m′ = m sinφ ≥ 1. On the other Strichartz Estimates for the (k, a)-Generalized Laguerre Operators 35 hand, it follows from Im f+(φ, u) ≥ 0 that |Hν,+(m,φ)| ≲ 1. Combining these estimates, we obtain |Hν,+(m,φ)| ≲ (1 +m′)−ν = (1 +m sinφ)−ν for m sinφ ≥ 1. For m sinφ ≤ 1, this estimate is easy to prove. Finally, we consider the case α ≥ 1. We observe that ∂αmHν,+(m,φ) = iα ∫ R eim ′g(φ,u)f+(φ, u) α ( 1− u2 )ν−1 χ+(u)du and ∣∣∂βuf+(φ, u)α∣∣ ≲ (sinφ)α(1− u)max(α−β,0) since f+(φ, u) = O((1 − u)) by (A.21). Thus, a similar argument as the case α = 0 shows that |∂αmHν,+(m,φ)| ≲ (sinφ)α(1 +m sinφ)−ν−α. Using an inequality sinφ(1 +m sinφ)−1 ≤ m−1, we obtain (iii) for α ≥ 1. ■ Proof of Proposition 3.3. First, we consider the case ν = 0. From [2, equation (4.28)], we have lim ν→0 ν−1Cν m(cosφ) = 2 cos(mφ) m = eimφ + e−imφ m . Thus, we can take gν,±(m,φ) = 1/m and r(m,φ) = 0. Next, we consider the case ν > 0. For φ ∈ ( π 4 , 3 4π ) , we set gν,±(m,φ) = Fν,1(m)Hν,±(m,φ) and r(m,φ) = Fν,1(m)Eν(m,φ). For φ ∈ [ 0, π4 ] ∪ [ 3 4π, π ] , we set gν,+(m,φ) = Fν,1(m)(Hν,+(m,φ) + E′ ν(m,φ)), gν,−(m,φ) = Fν,1(m)Hν,−(m,φ) and r(m,φ) = 0. Then Proposition 3.3 directly follows from the identities (A.19), (A.20), Proposition A.10 and Lemma A.11. ■ B Finite time Strichartz estimates Here, we show that the Strichartz estimates hold for finite time T assuming that they hold for each time t with |t| ≤ T0. The following argument is more or less well known. For simplicity, assuming T0 < T < 2T0, (1.3) and (1.4) hold for T0 and T−T0, we shall show that (1.3) and (1.4) hold for T . Set U(t) = e−itHk,a . Since U(t + s) = U(t)U(s), [−T, T ] = [−T0, T0] ∪ [−T,−T0] ∪ [T0, T ] and T < 2T0, ∥U(t)u0∥pLp([−T,T ];Lq) = ∥U(t)u0∥pLp([−T0,T0];Lq) + ∥U(t)U(−T0)u0∥pLp([−T+T0,0];Lq) + ∥U(t)U(T0)u0∥pLp([0,T−T0];Lq) ≲ ∥u0∥pL2 + ∥U(−T0)u0∥pL2 + ∥U(T0)u0∥pL2 = 3∥u0∥pL2 , where we use the fact that U(t) is unitary in the last line. This proves (1.3) for T . Set Γf(t) = ∫ t 0 U(t− s)f(s)ds. To see that (1.4) holds for T , it suffices to prove ∥Γf∥Lp1 ([−T,−T0]∪[T0,T ];Lq2 ) ≲ ∥f∥ Lp∗2 ([−T,T ];Lq∗2 ) . (B.1) We firstly observe∥∥∥∥∫ T0 0 U(−s)f(s)ds ∥∥∥∥ L2 ≤ C∥f∥ Lp∗2 ([0,T0];L q∗2 ) (B.2) due to the duality of ∥U(t)u0∥Lp([0,T0];Lq) ≲ ∥u0∥L2 which in turn follows from (1.3) for T0. Setting Γ1f(t) = ∫ T0 0 U(t− s)f(s)ds and Γ2f(t) = ∫ t T0 U(t − s)f(s)ds, we have Γ = Γ1 + Γ2. 36 K. Taira and H. Tamori Since Γ1f(t) = U(t) ∫ T0 0 U(−s)f(s)ds, the homogeneous estimate ∥U(t)u0∥Lp([−T,T ];Lq) ≲ ∥u0∥L2 implies ∥Γ1f∥Lp1 ([T0,T ];Lq2 ) ≲ ∥∥∥∥∫ T0 0 U(−s)f(s)ds ∥∥∥∥ L2 ≲︸︷︷︸ (B.2) ∥f∥ Lp∗2 ([0,T0];L q∗2 ) . On the other hand, setting g(t) = f(t + T0), we have Γ2f(t) = ∫ t−T0 0 U(t − T0 − s)g(s)ds. Using (1.4) for T − T0(< T0), we obtain ∥Γ2f∥Lp1 ([T0,T ];Lq1 ) = ∥∥∥∥∫ t 0 U(t− s)g(s)ds ∥∥∥∥ Lp1 ([0,T−T0];Lq1 ) ≲ ∥g∥ Lp∗2 ([−T+T0,T−T0];L q∗2 ) . Combining them, we conclude ∥Γf∥Lp1 ([T0,T ];Lq1 ) ≲ ∥f∥ Lp∗2 ([−T,T ];Lq∗2 ) . Similarly, we have ∥Γf∥Lp1 ([−T,−T0];Lq1 ) ≲ ∥f∥ Lp∗2 ([−T,T ];Lq∗2 ) . Thus we have proved (B.1). Acknowledgment KT was supported by JSPS KAKENHI Grant Number 23K13004, and HT was supported by JSPS KAKENHI Grant Numbers 20J00024 and 23K12947. HT is grateful to Toshiyuki Kobayashi for helpful comments. The authors would like to appreciate Hatem Mejjaoli for let- ting us know the papers [3, 24]. The authors would like to thank the anonymous referees for their valuable suggestions and improvements. References [1] Ben Säıd S., Strichartz estimates for Schrödinger–Laguerre operators, Semigroup Forum 90 (2015), 251–269. 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Ann. 376 (2020), 525–581, arXiv:1804.02390. https://doi.org/10.1016/j.jfa.2021.109350 https://doi.org/10.1016/j.jfa.2021.109350 https://arxiv.org/abs/2003.03086 https://doi.org/10.1016/S0294-1449(16)30399-7 https://doi.org/10.1016/S0294-1449(16)30399-7 https://doi.org/10.2140/apde.2016.9.151 https://arxiv.org/abs/1310.0909 https://doi.org/10.1090/pspum/048/974332 https://doi.org/10.1007/s00039-003-0439-2 https://doi.org/10.1007/s10440-005-0464-2 https://doi.org/10.1142/9789812770790_0006 https://arxiv.org/abs/math.RT/0607007 https://doi.org/10.1090/S0065-9266-2011-00592-7 https://arxiv.org/abs/0712.1769 https://doi.org/10.1093/imrn/rnab256 https://doi.org/10.1002/mma.6449 https://doi.org/10.1002/mma.6449 https://doi.org/10.1080/00036811.2012.692780 https://arxiv.org/abs/2208.12015 https://arxiv.org/abs/2203.06329 https://doi.org/10.2140/paa.2023.5.461 https://doi.org/10.1017/CBO9780511530029 https://doi.org/10.1017/CBO9780511530029 https://doi.org/10.1215/S0012-7094-77-04430-1 https://doi.org/10.1090/cbms/106 https://arxiv.org/abs/2008.00804 https://doi.org/10.1007/BF01212420 https://doi.org/10.1007/s00208-019-01892-7 https://arxiv.org/abs/1804.02390 1 Introduction 1.1 Main theorem 1.2 Key theorem 1.3 Idea of the proof 1.4 Related problems 1.5 Notation 2 Preliminary 2.1 A sum of monotonic functions 2.2 Discrete oscillatory integrals 2.3 A variant of the stationary phase theorem 3 Asymptotic expansion of special functions and decomposition of the sum 3.1 Asymptotic expansion of Bessel functions 3.2 Asymptotic expansion of Gegenbauer polynomials 3.3 Decomposition of the sum 3.4 Properties of the phase functions 3.5 Estimates of the amplitudes 4 Estimates of I_{2,s_2} and I_{3,s_2} 4.1 Intermediate region 4.2 Decaying region 5 Estimates of I_{1,s} 5.1 General bounds for b>0 5.2 Improvement for 0<b<2 6 Proof of the main theorem 6.1 Proof of Theorem 1.4 6.2 Proof of Theorem 1.1 6.3 Proof of Theorem 1.3 A Asymptotic behavior of special functions A.1 Leibniz's rule A.2 Non-stationary phase theorem with a singular amplitude A.3 Stationary phase theorem A.4 Asymptotics of the Bessel function A.5 Asymptotics of the Beta function A.6 Asymptotics of the Gegenbauer polynomials B Finite time Strichartz estimates References
id nasplib_isofts_kiev_ua-123456789-212877
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2026-03-21T11:38:30Z
publishDate 2025
publisher Інститут математики НАН України
record_format dspace
spelling Taira, Kouichi
Tamori, Hiroyoshi
2026-02-13T13:49:30Z
2025
Strichartz Estimates for the (, )-Generalized Laguerre Operators. Kouichi Taira and Hiroyoshi Tamori. SIGMA 21 (2025), 014, 37 pages
1815-0659
2020 Mathematics Subject Classification: 35Q41; 22E45
arXiv:2308.16815
https://nasplib.isofts.kiev.ua/handle/123456789/212877
https://doi.org/10.3842/SIGMA.2025.014
In this paper, we prove Strichartz estimates for the (, )-generalized Laguerre operators ⁻¹(−||²⁻ᵃ Δₖ + ||ᵃ) which were introduced by Ben Saïd-Kobayashi-Ørsted, and for the operators ||²⁻ᵃ Δₖ. Here k denotes a non-negative multiplicity function for the Dunkl Laplacian Δₖ, and denotes a positive real number satisfying certain conditions. The cases = 1, 2 were studied previously. We consider more general cases here. The proof depends on symbol-type estimates of special functions and a discrete analog of the stationary phase theorem inspired by the work of Ionescu-Jerison.
KT was supported by JSPS KAKENHI Grant Number 23K13004, and HT was supported by JSPS KAKENHI Grant Numbers 20J00024 and 23K12947. HT is grateful to Toshiyuki Kobayashi for helpful comments. The authors would like to appreciate Hatem Mejjaoli for letting us know the papers [3, 24]. The authors would like to thank the anonymous referees for their valuable suggestions and improvements.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Strichartz Estimates for the (, )-Generalized Laguerre Operators
Article
published earlier
spellingShingle Strichartz Estimates for the (, )-Generalized Laguerre Operators
Taira, Kouichi
Tamori, Hiroyoshi
title Strichartz Estimates for the (, )-Generalized Laguerre Operators
title_full Strichartz Estimates for the (, )-Generalized Laguerre Operators
title_fullStr Strichartz Estimates for the (, )-Generalized Laguerre Operators
title_full_unstemmed Strichartz Estimates for the (, )-Generalized Laguerre Operators
title_short Strichartz Estimates for the (, )-Generalized Laguerre Operators
title_sort strichartz estimates for the (, )-generalized laguerre operators
url https://nasplib.isofts.kiev.ua/handle/123456789/212877
work_keys_str_mv AT tairakouichi strichartzestimatesforthegeneralizedlaguerreoperators
AT tamorihiroyoshi strichartzestimatesforthegeneralizedlaguerreoperators

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