$b$-coercive convolution equations in weighted function spaces and applications
We study the $b$-separability properties of elliptic convolution operators in weighted Besov spaces and establish sharp estimates for the resolvents of the convolution operators. As a result, it is shown that these operators are positive and, in addition, play the role of negative generators of anal...
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Institute of Mathematics, NAS of Ukraine
2017
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507653252317184 |
|---|---|
| author | Musaev, H. K. Shakhmurov, V. B. Мусаєв, Г. К. Шахмуров, В. Б. |
| author_facet | Musaev, H. K. Shakhmurov, V. B. Мусаєв, Г. К. Шахмуров, В. Б. |
| author_sort | Musaev, H. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:26:39Z |
| description | We study the $b$-separability properties of elliptic convolution operators in weighted Besov spaces and establish sharp
estimates for the resolvents of the convolution operators. As a result, it is shown that these operators are positive and, in
addition, play the role of negative generators of analytic semigroups. Moreover, the maximal $b$-regularity properties of
the Cauchy problem for a parabolic convolution equation are established. Finally, these results are applied to obtain the
maximal regularity properties for anisotropic integro-differential equations and the system of infinitely many convolution
equations. |
| first_indexed | 2026-03-24T02:12:44Z |
| format | Article |
| fulltext |
UDC 517.9
H. K. Musaev (Baku State Univ., Azerbaijan),
V. B. Shakhmurov (Okan Univ., Istanbul, Turkey)
\bfitB -COERCIVE CONVOLUTION EQUATIONS
IN WEIGHTED FUNCTION SPACES AND APPLICATIONS
\bfitB -КОЕРЦИТИВНI РIВНЯННЯ В ЗГОРТКАХ
У ВАГОВИХ ФУНКЦIОНАЛЬНИХ ПРОСТОРАХ ТА ЇХ ЗАСТОСУВАННЯ
We study the B -separability properties of elliptic convolution operators in weighted Besov spaces and establish sharp
estimates for the resolvents of the convolution operators. As a result, it is shown that these operators are positive and, in
addition, play the role of negative generators of analytic semigroups. Moreover, the maximal B -regularity properties of
the Cauchy problem for a parabolic convolution equation are established. Finally, these results are applied to obtain the
maximal regularity properties for anisotropic integro-differential equations and the system of infinitely many convolution
equations.
Вивчаються властивостi B -сепарабельностi елiптичних операторiв згортки у зважених просторах Бєсова. Встанов-
лено точнi оцiнки для резольвент операторiв згортки. В результатi показано, що цi оператори є додатними, а також
вiд’ємними генераторами аналiтичних напiвгруп. Крiм того, встановлено властивостi максимальної B -регулярностi
задачi Кошi для параболiчного рiвняння у згортках. Цi результати застосовано до отримання властивостей макси-
мальної регулярностi для анiзотропних iнтегро-диференцiальних рiвнянь та для систем нескiнченного числа рiвнянь
у згортках.
Introduction. In recent years, maximal regularity properties for differential operator equations
have been studied extensively, e.g., in [1 – 8, 12, 13, 18, 20, 22, 26, 28, 29]. Moreover, convolution-
differential equations (CDEs) have been investigated, e.g., in [10, 16, 19, 21, 22, 27] and the references
therein. However, convolution differential-operator equations (CDOEs) is relatively less investigated
subject. In [14, 18, 21, 23, 24] parabolic type CDEs with operator coefficient was investigated. In
[15, 21] regularity properties of degenerate CDOEs studed in weighted Lp spaces. In conrary to
these, the main aim of the present paper is to obtain separability property of the elliptic CDOE
(L+ \lambda )u =
\sum
| \alpha | \leq l
a\alpha \ast D\alpha u+A \ast u+ \lambda u = f(x) (1.1)
and the maximal regularity property of the Cauchy problem for the parabolic CDOE
\partial u
\partial t
+
\sum
| \alpha | \leq l
a\alpha \ast D\alpha u+A \ast u = f(t, x), u(0, x) = 0
in E -valued weighted Besov spaces, where E is a Banach space, A = A(x) is a linear operator in
E, a\alpha = a\alpha (x) are complex-valued functions and \lambda is a complex spectral parameter.
By using the Fourier multiplier theorems in weighted Banach valued Besov spaces Bs
p,q,\gamma (R
n;E),
in Section 2 we derive the coercive estimate of resolvent and particularly and we show that this
operator is positive. Namely, we prove that for all f \in Bs
p,q,\gamma (R
n;E) there is a unique solution
u \in Bl,s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
of the problem (1.1) and the following uniformly estimate holds:
c\bigcirc H. K. MUSAEV, V. B. SHAKHMUROV, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10 1385
1386 H. K. MUSAEV, V. B. SHAKHMUROV\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \| a\alpha \ast D\alpha u\| Bs
p,q,\gamma (R
n;E)+
+\| A \ast u\| Bs
p,q,\gamma (R
n;E) + | \lambda | \| u\| Bs
p,q,\gamma (R
n;E) \leq C\| f\| Bs
p,q,\gamma (R
n;E).
Particularly, this result implies that if f \in Bs
p,q,\gamma (R
n;E), then all terms of the equations (1.1) are
also from Bs
p,q,\gamma (R
n;E), (i.e., all terms are separated). This important effect allows to obtain some
spectral properties of the convolution operator Q.
Moreover, under some assumptions we conclude that the corresponding convolution operator Q
has a domain coinciding with the Besov space
Bl,s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
= Bl,s
p,q,\gamma (R
n;E) \cap Bs
p,q,\gamma (R
n;E(A))
and there are positive constants C1 and C2 such that
C1\| u\| Bl,s
p,q,\gamma
\bigl(
Rn;E(A),E
\bigr) \leq \| Qu\| Bs
p,q,\gamma (R
n;E) \leq C2\| u\| Bl,s
p,q,\gamma
\bigl(
Rn;E(A),E
\bigr)
for all u \in Bl,s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
.
By using the positivity properties of the convolution operator Q and the semigroup theory, in
Section 3 we conclude that the above Cauchy problem has a unique solution satisfying the coercive
estimate. In Sections 4 and 5, by putting concrete vector spaces instead of E and concrete linear
operators instead of A, the maximal regularity properties of convolution differential operators are
obtained in vector valued Besov spaces.
1. Notations and background. Let E be a Banach space and \gamma = \gamma (x), x = (x1, x2, . . . , xn),
be a positive measurable weighted function on a measurable subset \Omega \subset Rn. Let Lp,\gamma (\Omega ;E) denote
the space of strongly E -valued functions that are defined on \Omega with the norm
\| f\| Lp,\gamma (\Omega ;E) = \| f\| Lp(E;\gamma ) =
\int
\Omega
\Bigl( \bigm\| \bigm\| f(x)\bigm\| \bigm\| p
E
\gamma (x) dx
\Bigr) 1/p
, 1 \leq p <\infty .
For \gamma (x) \equiv 1, the space Lp,\gamma (\Omega ;E) will be denoted by Lp = Lp(\Omega ;E),
\| f\| L\infty ,\gamma (\Omega ;E) = \| f\| L\infty (E;\gamma ) = \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
x\in \Omega
\Bigl[
\gamma (x)
\bigm\| \bigm\| f(x)\bigm\| \bigm\|
E
\Bigr]
.
The weighted \gamma (x) we will consider satisfy an Ap condition; i.e., \gamma (x) \in Ap, 1 < p < \infty , if
there is a positive constant C such that\left( 1
| Q|
\int
Q
\gamma (x)dx
\right)
\left( 1
| Q|
\int
Q
\gamma
- 1
p - 1 (x) dx
\right)
p - 1
\leq C
for all compacts Q \subset Rn.
Let \BbbC be the set of complex numbers and
S\varphi =
\bigl\{
\lambda ;\lambda \in \BbbC , | \mathrm{a}\mathrm{r}\mathrm{g} \lambda | \leq \varphi
\bigr\}
\cup \{ 0\} , 0 \leq \varphi < \pi .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1387
Let E1 and E2 be two Banach spaces and L(E1, E2) denotes the spaces of bounded linear
operators acting from E1 to E2. For E1 = E2 = E it will be denoted by L(E).
A closed linear operator function A = A(x) is said to be uniformly \varphi -positive in Banach space
E, if D(A(x)) is dense in E and does not depend on x and there is a positive constant M so that\bigm\| \bigm\| \bigm\| \bigl( A(x) + \lambda I
\bigr) - 1
\bigm\| \bigm\| \bigm\|
L(E)
\leq M
\bigl(
1 + | \lambda |
\bigr) - 1
for every x \in Rn and \lambda \in S\varphi , \varphi \in [0, \pi ), where I is an identity operator in E. Sometimes instead
of A+ \lambda I we will write A+ \lambda and it will be denoted by A\lambda . It is known [25] (§ 1.15.1) that there
exists fractional powers A\theta of the positive operator A.
Let E
\bigl(
A\theta
\bigr)
denote the space D(A\theta ) with graphical norm
\| u\|
E
\bigl(
A\theta
\bigr) = \Bigl( \| u\| pE +
\bigm\| \bigm\| A\theta u
\bigm\| \bigm\| p
E
\Bigr) 1
p
, 1 \leq p <\infty , - \infty < \theta <\infty .
Let S = S(Rn;E), or simply S(E) denotes Schwarz class, i.e., the space of E -valued rapidly
decreasing smooth functions on Rn equipped with its usual topology generated by seminorms.
S(Rn;\BbbC ) denoted by just S.
Let S\prime (Rn;E) denote the space of all continuous linear operators, L : S - \rightarrow E, equipped with
the bounded convergence topology. Recall S(Rn;E) is norm dense in Bs
p,q,\gamma (R
n;E) when
1 \leq p <\infty , 1 \leq q <\infty , \gamma \in Ap.
We shall use Fourier analytic definition of weighted Besov spaces in this study. Therefore, we
need to consider some subsets \{ Jk\} \infty k=0 and \{ Ik\} \infty k=0 of Rn. Let \{ Jk\} \infty k=0 given by
J0 =
\bigl\{
t \in Rn : | t| \leq 1
\bigr\}
, Jk =
\bigl\{
t \in Rn : 2k - 1 \leq | t| \leq 2k
\bigr\}
for k \in \BbbN .
Enlarge each Jk to form a sequence \{ Ik\} \infty k=0 of overlapping subsets defined by
I0 =
\bigl\{
t \in Rn : | t| \leq 2
\bigr\}
, Ik =
\bigl\{
t \in Rn : 2k - 1 \leq | t| \leq 2k+1
\bigr\}
for k \in \BbbN .
Next, we define the unity \{ \varphi k\} k\in \BbbN 0 of functions from S(Rn;R), where \BbbN 0 = \{ 0\} \cup \BbbN , \BbbN =
= \{ 1, 2, . . .\} is the of natural numbers. Let \psi \in S(R,R) be nonnegative function with support in
[2 - 1, 2], which satisfies
\infty \sum
k= - \infty
\psi (2 - ks) = 1 for s \in R\setminus \{ 0\}
and
\varphi k(t) = \psi
\bigl(
2 - k| t|
\bigr)
, \varphi 0(t) = 1 -
\infty \sum
k=1
\varphi k(t) for k \in \BbbN , t \in Rn.
Let \varphi k \equiv 0 if k < 0. Later, we will need the following useful properties:
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\varphi k \subset Ik for each k \in \BbbN 0,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1388 H. K. MUSAEV, V. B. SHAKHMUROV
\infty \sum
k=0
\varphi k(s) = 1 for each s \in Rn,
Jm \cap \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\varphi k = \varnothing if | m - k| > 1,
\varphi k - 1(s) + \varphi k(s) + \varphi k+1(s) = 1 for each s \in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\varphi k, k \in \BbbN 0.
Let 1 \leq p \leq q \leq \infty and s \in R. The weighted Besov space is the set of all functions f \in S\prime (R;E)
for which
\| f\| Bs
p,q(E;\gamma ) = \| f\| Bp,q,\gamma (Rn;E) =
\bigm\| \bigm\| \bigm\| \Bigl\{ 2ks\bigl( \v \varphi k \ast f
\bigr) \Bigr\} \infty
k=0
\bigm\| \bigm\| \bigm\|
lq(Lp,\gamma (Rn;E))
\equiv
\equiv
\left\{
\Biggl[ \infty \sum
k=0
2ksq \| \v \varphi k \ast f\| qLp,\gamma (Rn;E)
\Biggr] 1
q
if q \not = \infty ,
\mathrm{s}\mathrm{u}\mathrm{p}
k\in \BbbN 0
\Bigl[
2ks \| \v \varphi k \ast f\| Lp,\gamma (Rn;E)
\Bigr]
if q = \infty
is finite; here p and s are main and smoothness indexes respectively.
Let \alpha = (\alpha 1, \alpha 2, . . . , \alpha n), | \alpha | =
\sum n
k=1
\alpha k, where \alpha k are integers and D\alpha = D\alpha 1
1 D\alpha 2
2 . . . D\alpha n
n .
An E -valued generalized function D\alpha f is called a generalized derivative in the sense of Schwarz
distributions of the function f \in S(Rn;E), equality
\langle D\alpha f, \varphi \rangle = ( - 1)| \alpha | \langle f,D\alpha \varphi \rangle
holds for all \varphi \in S.
Let F denote the Fourier transform. Through this section the Fourier transformation of a function
f will be denoted by \widehat f, Ff = \^f and F - 1f = \v f. It is known that
F (D\alpha
xf) = (i\xi 1)
\alpha 1 . . . (i\xi n)
\alpha n \widehat f, D\alpha
\xi (F (f)) = F
\Bigl[
( - ix1)\alpha 1 . . . ( - ixn)\alpha nf
\Bigr]
for all f \in S\prime (Rn;E), where \xi = (\xi 1, \xi 2, . . . , \xi n) \in Rn.
Let E0 and E be two Banach spaces and E0 is continuously and densely embedded to E. Let l
is a positive integer and Dl
k =
\partial l
\partial xlk
. Consider the E -valued function spaces defined by
Bl,s
p,q,\gamma (\Omega ;E0, E) =
\Biggl\{
u : u \in Bs
p,q,\gamma (\Omega ;E0), D
l
ku \in Bs
p,q,\gamma (\Omega ;E),
\| u\|
Bl,s
p,q,\gamma (\Omega ;E0,E)
= \| u\| Bs
p,q,\gamma (\Omega ;E0) +
n\sum
k=1
\bigm\| \bigm\| \bigm\| Dl
ku
\bigm\| \bigm\| \bigm\|
Bs
p,q,\gamma (\Omega ;E)
<\infty
\Biggr\}
.
A function u \in Bl,s
p,q,\gamma (Rn;E(A), E) satisfying the equation (1.1) a.e. on Rn is called a solution
of (1.1).
The CDOE (1.1) is said to be weighted B-separable (or Bs
p,q,\gamma (R
n;E)-separable) if for all
f \in Bs
p,q,\gamma (R
n;E) it has a unique solution u \in Bl,s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
and
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1389\sum
| \alpha | \leq l
\| a\alpha \ast D\alpha u\| Bs
p,q,\gamma (R
n;E) + \| A \ast u\| Bs
p,q,\gamma (R
n;E) \leq C\| f\| Bs
p,q,\gamma (R
n;E).
Let E1 and E2 be two Banach spaces. A function \Psi \in L\infty
\bigl(
Rn;L(E1, E2)
\bigr)
is called a multiplier
from Bs
p,q,\gamma (R
n;E1) to Bs
p,q,\gamma (R
n;E2) for p \in (1,\infty ) and q \in [1,\infty ] if the map u \rightarrow Ku =
= F - 1\Psi (\xi )Fu, u \in S(Rn;E1) are well defined and extends to a bounded linear operator
K : Bs
p,q,\gamma (R
n;E1) \rightarrow Bs
p,q,\gamma (R
n;E2).
M q
p (E1, E2, s) denotes the collection of multiplier from Bs
p,q(R
n;E1) to Bs
p,q(R
n;E2). Let
\Phi h =
\bigl\{
\Psi h \in M q
p (E1, E2, s), h \in M(h)
\bigr\}
.
We say that \Phi h is a collection of uniformly bounded multipliers (UBM) if there exists a positive
constant M independent on h \in M(h) such that\bigm\| \bigm\| F - 1\Psi hFu
\bigm\| \bigm\|
Bs
p,q(R
n;E2)
\leq M\| u\| Bs
p,q(R
n;E1)
for all h \in M(h) and u \in S(Rn;E1).
Definition 1.1. Let E be a Banach space and 1 \leq p \leq 2. Let E so that
\| Ff\| Lp\prime (R
n;E) \leq C\| f\| Lp(Rn;E) for each f \in S(Rn, E),
where
1
p
+
1
p\prime
= 1. Then the space E is said to be the Fourier type p.
Definition 1.2. Let A = A(x), x \in Rn, be a closed linear operator in E with domain definition
D(A) independent of x. Then, the Fourier transformation of A(x) is defined as\bigl\langle
\^Au,\varphi
\bigr\rangle
= \langle Au, \^\varphi \rangle , u \in D(A), \varphi \in S(Rn).
(For details see [3, p. 7].)
Let h \in R,m \in \BbbN and ek, k = 1, 2, . . . , n, be standart unit vectors of Rn. Let
\Delta k(h)f(x) = f
\bigl(
x+ hek
\bigr)
- f(x).
Definition 1.3. Let A = A(x) be a closed linear operator in E with the domain definition D(A)
independent of x. It is differentiable if\biggl(
\partial A
\partial xk
\biggr)
u = \mathrm{l}\mathrm{i}\mathrm{m}
h\rightarrow 0
\| \Delta k(h)A(x)u\| E
h
<\infty , u \in D(A).
Let A = A(x), x \in Rn, be a closed linear operator in E with domain definition D(A) indepen-
dent of x and u \in S \shortmid \bigl( Rn;E(A)
\bigr)
. We define the convolution A \ast u in the distribution sense (see,
e.g., [3]).
The space C(m)(\Omega ;E) will denote the spaces of E -valued uniformly bounded, m-times conti-
nuously differentiable functions on \Omega .
Let us first recall an important fact [11] (Corollary 4.11) that will used in this section.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1390 H. K. MUSAEV, V. B. SHAKHMUROV
Theorem 1.1. Let p, q \in [1,\infty ]. If \Psi \in C l
\bigl(
Rn, L(E1, E2)
\bigr)
satisfies for some positive constant
K,
\mathrm{s}\mathrm{u}\mathrm{p}
x\in Rn
\bigm\| \bigm\| \bigm\| (1 + | x| )| \alpha | D\alpha \Psi (x)
\bigm\| \bigm\| \bigm\|
L(E1,E2)
\leq K
for each multiindex \alpha with | \alpha | \leq l, then \Psi is Fourier multiplier from Bs
p,q(R
n, E1) to Bs
p,q(R
n, E2)
provided one of the following conditions hold:
(a) E1 and E2 are arbitrary Banach spaces and l = n+ 1;
(b) E1 and E2 are uniformly convex Banach spaces and l = n;
(c) E1 and E2 have Fourier type p and l =
\biggl\lceil
n
p
\biggr\rceil
+ 1.
2. Convolution-elliptic operator equations. In this section we present the separability properties
of the CDOE
(L+ \lambda )u =
\sum
| \alpha | \leq l
a\alpha \ast D\alpha u+A \ast u+ \lambda u = f(x),
where A = A(x) is a linear operator in a Banach space E, a\alpha = a\alpha (x) are complex valued functions
and \lambda is a complex parameter.
We fined the sufficient conditions under which the problem is separable in Bs
p,q,\gamma (R
n;E).
Condition 2.1. Let a\alpha \in L\infty (Rn) such that the following conditions satisfied:
L(\xi ) =
\sum
| \alpha | \leq l
\^a\alpha (\xi )(i\xi )
\alpha \in S\varphi 1 ,
\bigm| \bigm| L(\xi )\bigm| \bigm| \geq C
\bigm| \bigm| a0(\xi )\bigm| \bigm| | \xi | l,
| a0(\xi )| = \mathrm{m}\mathrm{a}\mathrm{x}
\alpha
\bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| .
For proving the main result of this section let at first, show the following lemmas.
Lemma 2.1. Suppose the Condition 2.1 holds. Assume \^A(\xi ) is an uniformly \varphi -positive operator
in a Banach space E with 0 < \varphi < \pi - \varphi 1. Then operator functions
\sigma 0(\xi , \lambda ) = \lambda G(\xi , \lambda ), \sigma 1(\xi , \lambda ) = \^A(\xi )G(\xi , \lambda ),
\sigma 2(\xi , \lambda ) =
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \^a\alpha (\xi )(i\xi )
\alpha G(\xi , \lambda )
are uniformly bounded, where
G(\xi , \lambda ) =
\Bigl[
\^A(\xi ) + \lambda + L(\xi )
\Bigr] - 1
.
Proof. Let us note that for the sake of simplicity we shall note change constants in every step.
By virtue of [9] (Lemma 2.3), for L(\xi ) \in S\varphi 1 , \lambda \in S\varphi and \varphi 1 + \varphi < \pi there is a positive constant
C so that \bigm| \bigm| \lambda + L(\xi )
\bigm| \bigm| \geq C
\Bigl(
| \lambda | +
\bigm| \bigm| L(\xi )\bigm| \bigm| \Bigr) . (2.1)
By using the resolvent properties of positive operators and by (2.1) we obtain
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1391
\bigm\| \bigm\| \sigma 0(\xi , \lambda )\bigm\| \bigm\| L(E)
\leq M | \lambda |
\Bigl[
1 + | \lambda | +
\bigm| \bigm| L(\xi )\bigm| \bigm| \Bigr] - 1
\leq M0,\bigm\| \bigm\| \sigma 1(\xi , \lambda )\bigm\| \bigm\| L(E)
=
\bigm\| \bigm\| \bigm\| I - \bigl( \lambda + L(\xi )
\bigr)
G(\xi , \lambda )
\bigm\| \bigm\| \bigm\|
L(E)
\leq
\leq 1 +
\bigm| \bigm| \lambda + L(\xi )
\bigm| \bigm| \bigm\| \bigm\| G(\xi , \lambda )\bigm\| \bigm\|
L(E)
\leq 1 +M
\bigm| \bigm| \lambda + L(\xi )
\bigm| \bigm| \bigl( 1 + \bigm| \bigm| \lambda + L(\xi )
\bigm| \bigm| \bigr) - 1 \leq M1.
Next, let us consider \sigma 2. It is clear to see that
\bigm\| \bigm\| \sigma 2(\xi , \lambda )\bigm\| \bigm\| L(E)
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \^a\alpha (\xi )(i\xi )
\alpha G(\xi , \lambda )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
L(E)
\leq
\leq C
\sum
| \alpha | \leq l
| \lambda |
\bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| \Bigl[ | \xi | | \lambda | - 1
l
\Bigr] | \alpha | \bigm\| \bigm\| G(\xi , \lambda )\bigm\| \bigm\|
L(E)
\leq
\leq C
\sum
| \alpha | \leq l
| \lambda |
\bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| n\prod
k=1
\Bigl[
| \xi k| | \lambda | -
1
l
\Bigr] \alpha k \bigm\| \bigm\| G(\xi , \lambda )\bigm\| \bigm\|
L(E)
.
Therefore, \sigma 2(\xi , \lambda ) is bounded if
\sum
| \alpha | \leq l
| \lambda |
\bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| n\prod
k=1
\Bigl[
| \xi k| | \lambda | -
1
l
\Bigr] \alpha k \bigm\| \bigm\| G(\xi , \lambda )\bigm\| \bigm\|
L(E)
\leq C.
Since \^A is uniformly positive and L(\xi ) \in S\varphi 1 , by using the well known inequalities
y\alpha 1
1 y\alpha 2
2 . . . y\alpha n
n \leq C
\Biggl(
1 +
n\sum
k=1
ylk
\Biggr)
, yk \geq 0, | \alpha | \leq l, (2.2)
for yk =
\Bigl(
| \lambda | -
1
l | \xi k|
\Bigr) \alpha k
we obtain
\sum
| \alpha | \leq l
| \lambda |
\bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| n\prod
k=1
\Bigl[
| \xi k| | \lambda | -
1
l
\Bigr] \alpha k \bigm\| \bigm\| G(\xi , \lambda )\bigm\| \bigm\|
L(E)
\leq
\leq C
\sum
| \alpha | \leq l
| \lambda |
\bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| \Biggl[ 1 + n\sum
k=1
| \xi k| l| \lambda | - 1
\Biggr] \bigl[
1 + | \lambda + L(\xi )|
\bigr] - 1
.
Taking into account Condition 2.1 and by (2.2) we get
\bigm\| \bigm\| \sigma 2(\xi , \lambda )\bigm\| \bigm\| L(E)
\leq C
\Biggl[
| \lambda | +
n\sum
k=1
| \xi k| l
\Biggr] \bigl[
1 + | \lambda | +
\bigm| \bigm| L(\xi )\bigm| \bigm| \bigr] - 1 \leq
\leq C
\Biggl[
| \lambda | +
n\sum
k=1
| \xi k| l
\Biggr] \left[ 1 + | \lambda | +
\sum
| \alpha | \leq l
\bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| | \xi \alpha |
\right] - 1
\leq C.
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1392 H. K. MUSAEV, V. B. SHAKHMUROV
Lemma 2.2. Let all conditions of Lemma 2.1 hold. Let \^a\alpha \in C(n)(Rn),
\bigl[
D\beta \^A(\xi )
\bigr]
\^A - 1(\xi ) \in
\in C
\bigl(
Rn;L(E)
\bigr)
for | \beta | \leq n+ 1 and\bigm\| \bigm\| \bigm\| \bigl[ D\beta \^A(\xi )
\bigr]
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\|
L(E)
\leq C1. (2.3)
Then operator functions D\beta \sigma j(\xi , \lambda ), j = 0, 1, 2, are uniformly bounded.
Proof. Let us first estimate
\partial \sigma 1
\partial \xi i
. It is easy to see that
\bigm| \bigm| D\beta \^a\alpha (\xi )
\bigm| \bigm| \leq C2, | \beta | \leq n+ 1. (2.4)
Really, \bigm\| \bigm\| \bigm\| \bigm\| \partial \sigma 1\partial \xi i
\bigm\| \bigm\| \bigm\| \bigm\|
L(E)
\leq \| I1\| L(E) +
\bigm\| \bigm\| I2\bigm\| \bigm\| L(E)
+
\bigm\| \bigm\| I3\bigm\| \bigm\| L(E)
,
where
I1 =
\Biggl[
\partial \^A(\xi )
\partial \xi i
\Biggr]
G(\xi , \lambda ), I2 = \^A(\xi )
\Biggl[
\partial \^A(\xi )
\partial \xi i
\Biggr] \bigl[
G(\xi , \lambda )
\bigr] 2
and
I3 = \^A(\xi )
\biggl[
\partial L(\xi )
\partial \xi i
\biggr] \bigl[
G(\xi , \lambda )
\bigr] 2
.
By using (2.3) we get
\| I1\| L(E) =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\Biggl[
\partial \^A(\xi )
\partial \xi i
\Biggr]
\^A - 1(\xi ) \^A(\xi )G(\xi , \lambda )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
L(E)
\leq
\leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\Biggl[
\partial \^A(\xi )
\partial \xi i
\Biggr]
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
L(E)
\bigm\| \bigm\| \sigma 1(\xi , \lambda )\bigm\| \bigm\| L(E)
\leq C.
Due to resolvent properties of \^A and by using (2.3) we obtain
\bigm\| \bigm\| I2\bigm\| \bigm\| L(E)
\leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\Biggl[
\partial \^A(\xi )
\partial \xi i
\Biggr]
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
L(E)
\bigm\| \bigm\| \sigma 1(\xi , \lambda )\bigm\| \bigm\| 2L(E)
\leq C.
By using (2.3), (2.4) for \lambda \in S\varphi and L(\xi ) \in S\varphi 1 with \varphi 1 + \varphi < \pi we have
\bigm\| \bigm\| I3\bigm\| \bigm\| L(E)
\leq
\bigm| \bigm| \bigm| \bigm| \partial L(\xi )\partial \xi i
\bigm| \bigm| \bigm| \bigm| \bigm\| \bigm\| G(\xi , \lambda )\bigm\| \bigm\| L(E)
\| \sigma 1(\xi , \lambda )\| L(E) \leq
\leq C
\sum
| \alpha | \leq l
\biggl[ \bigm| \bigm| \bigm| \bigm| \partial \^a\alpha (\xi )\partial \xi i
\bigm| \bigm| \bigm| \bigm| | \xi \alpha | + \bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| \bigm| \bigm| \bigm| \xi \alpha 1
1 . . . \xi \alpha i - 1
i . . . \xi \alpha n
n
\bigm| \bigm| \bigm| \biggr] [1 + | \lambda + L(\xi )| ] - 1 \leq C.
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B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1393
Then by using (2.2) we get from above \| I3\| L(E) \leq C. In a similar way the uniformly boundedness
of \sigma 0(\xi , \lambda ) is proved. Next we shall prove
\partial
\partial \xi i
\sigma 2(\xi , \lambda ) is uniformly bounded. Similarly,
\bigm\| \bigm\| \bigm\| \bigm\| \partial
\partial \xi i
\sigma 2(\xi , \lambda )
\bigm\| \bigm\| \bigm\| \bigm\|
L(E)
\leq \| J1\| L(E) + \| J2\| L(E) + \| J3\| L(E) + \| J4\| L(E) ,
where
J1 =
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l
\biggl[
\partial \^a\alpha (\xi )
\partial \xi i
\biggr]
(i\xi )\alpha G(\xi , \lambda ),
J2 =
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \^a\alpha (\xi )i\alpha i\xi
- 1
i (i\xi )\alpha G(\xi , \lambda ),
J3 =
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \^a\alpha (\xi )(i\xi )
\alpha
\biggl[
\partial L(\xi )
\partial \xi i
\biggr] \bigl[
G(\xi , \lambda )
\bigr] 2
and
J4 =
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \^a\alpha (\xi )(i\xi )
\alpha
\biggl[
\partial
\partial \xi i
\^A(\xi )
\biggr] \bigl[
G(\xi , \lambda )
\bigr] 2
.
Let us first show J1 is uniformly bounded. Since,
\| J1\| L(E) \leq
\sum
| \alpha | \leq l
\bigm| \bigm| \bigm| \bigm| \partial \^a\alpha (\xi )\partial \xi i
\bigm| \bigm| \bigm| \bigm| \bigm\| \bigm\| \bigm\| | \lambda | 1 - | \alpha |
l (i\xi )\alpha G(\xi , \lambda )
\bigm\| \bigm\| \bigm\|
L(E)
.
By resolvent properties of \^A, by virtue of (2.1), (2.2) and (2.4) we obtain \| J1\| L(E) \leq C. In a similar
way by using (2.1), (2.2) and (2.4) we get
\| Jk\| L(E) \leq C, k = 2, 3, 4.
Hence, operator functions
\partial \sigma j
\partial \xi i
, j = 0, 1, 2, are uniformly bounded. Now, it remains to show
D\beta \sigma j(\xi , \lambda ) are uniformly bounded for | \beta | \leq n+ 1. It is clear to see that\bigm\| \bigm\| \bigm\| \bigm\| \partial 2\sigma 1\partial \xi 2i
\bigm\| \bigm\| \bigm\| \bigm\|
L(E)
\leq \| I4\| L(E) + \| I5\| L(E) + \| I6\| l(E),
where
I4 =
\partial 2 \^A(\xi )
\partial \xi 2i
G(\xi , \lambda ) -
\Biggl[
\partial \^A(\xi )
\partial \xi i
\Biggr] 2 \bigl[
G(\xi , \lambda )
\bigr] 2 - \partial \^A(\xi )
\partial \xi i
\partial L(\xi )
\partial \xi i
\bigl[
G(\xi , \lambda )
\bigr] 2
,
I5 =
\Biggl[
\partial \^A(\xi )
\partial \xi i
\Biggr] 2 \bigl[
G(\xi , \lambda )
\bigr] 2
+
\partial \^A(\xi )
\partial \xi i
\partial 2 \^A(\xi )
\partial \xi 2i
\bigl[
G(\xi , \lambda )
\bigr] 2 -
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1394 H. K. MUSAEV, V. B. SHAKHMUROV
- 2 \^A(\xi )
\Biggl[
\partial \^A(\xi )
\partial \xi i
\^A(\xi )
\Biggr] \Biggl[
\partial \^A(\xi )
\partial \xi i
+
\partial L(\xi )
\partial \xi i
\Biggr] \bigl[
G(\xi , \lambda )
\bigr] 3
+
+ \^A(\xi )
\partial L(\xi )
\partial \xi i
\Biggl[
\partial \^A(\xi )
\partial \xi i
+
\partial L(\xi )
\partial \xi i
\Biggr] \bigl[
G(\xi , \lambda )
\bigr] 3
,
I6 =
\partial \^A(\xi )
\partial \xi i
\partial L(\xi )
\partial \xi i
\bigl[
G(\xi , \lambda )
\bigr] 2
+ \^A(\xi )
\partial 2L(\xi )
\partial \xi 2i
\bigl[
G(\xi , \lambda )
\bigr] 2
.
By using resolvent properties of positive operator \^A(\xi ) and conditions of lemma we have
\| I4\| L(E) \leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \partial 2 \^A(\xi )\partial \xi 2i
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \| \sigma 1\| +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \partial \^A(\xi )
\partial \xi i
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
\| \sigma 1\| 2+
+2 \| \sigma 1\|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \partial \^A(\xi )
\partial \xi i
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\Biggl[
\partial \^A(\xi )
\partial \xi i
+
\partial L
\partial \xi i
\Biggr]
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \| \sigma 1\| 2 .
Then by using (2.1) – (2.4) we obtain from above \| I4\| L(E) \leq M. By using the same arguments
we get
\| Ik\| L(E) \leq M, k = 5, 6.
From the representations of \sigma j(\xi , \lambda ), j = 0, 1, 2, it easy to see that operator functions D\beta \sigma j(\xi , \lambda )
contain the similar terms as Ik; namely, the functions D\beta \sigma j(\xi , \lambda ) will be represented as the combi-
nations of principal terms\Bigl[
Dm \^A(\xi ) + \xi \sigma D\gamma \^a\alpha (\xi )
\Bigr] \Bigl[
\^A(\xi ) + \lambda + L(\xi )
\Bigr] - | \beta |
, (2.5)
where | m| \leq | \beta | and | \sigma | + | \gamma | \leq | \beta | . So, by using the similar arguments as above we obtain\bigm\| \bigm\| D\beta \sigma j(\xi , \lambda )
\bigm\| \bigm\| \leq C, j = 0, 1, 2.
Hence, we get operator functions D\beta \sigma j(\xi , \lambda ) are uniformly bounded for each multiindex | \beta | \leq
\leq n+ 1.
Lemma 2.3. Let all conditions of Lemma 2.2 are satisfied and
| \xi | \eta
\bigm| \bigm| D\beta \^a\alpha (\xi )
\bigm| \bigm| \leq C1, | \xi | \eta
\bigm\| \bigm\| \bigm\| \bigl[ D\beta \^A(\xi )
\bigr]
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\|
L(E)
\leq C2. (2.6)
Then the estimates hold
| \xi | \eta
\bigm\| \bigm\| D\beta \sigma j(\xi , \lambda )
\bigm\| \bigm\|
L(E))
\leq M, j = 0, 1, 2,
for all \eta \leq | \beta | \leq n+ 1 and \xi \in Rn.
Proof. Since, D\beta \sigma j(\xi , \lambda ) is in the form of (2.5), by reasoning as in Lemma 2.2, by (2.6) we
have \bigm\| \bigm\| \bigm\| | \xi i| \eta D\beta \sigma j(\xi , \lambda )
\bigm\| \bigm\| \bigm\|
L(E))
\leq C, j = 0, 1, 2, i = 1, 2, . . . , n,
that in its turn implies assertion of Lemma 2.3.
From Lemma 2.3 we obtain the following corollary.
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B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1395
Corollary 2.1. Let all conditions of Lemma 2.3 are satisfied, p, q \in [1,\infty ). Then operator-
functions \sigma j(\xi , \lambda ), j = 0, 1, 2, are UBM in Bs
p,q,\gamma (R
n;E).
Proof. To prove \sigma j(\xi , \lambda ) are UBM in Bs
p,q,\gamma (R
n;E), we need the following estimates:\bigm\| \bigm\| \bigm\| (1 + | \xi | )\eta D\beta \sigma j(\xi , \lambda )
\bigm\| \bigm\| \bigm\|
L\infty (Rn,L(E))
\leq K, K > 0,
for \xi \in Rn\setminus 0, | \beta | \leq n+ 1. From Lemma 2.3 it follows \sigma i \in C(n)(Rn\setminus 0, L(E)) and\bigm\| \bigm\| D\beta \sigma j(\xi , \lambda )
\bigm\| \bigm\|
L\infty (L(E)))
\leq K1, | \xi | \eta
\bigm\| \bigm\| D\beta \sigma j(\xi , \lambda )
\bigm\| \bigm\|
L\infty (L(E)))
\leq K2, \eta \leq | \beta | \leq n+ 1.
Hence, operator functions \sigma i(\xi , \lambda ) are Fourier multipliers in Bs
p,q,\gamma (R
n, E). Let we denote
Bs
p,q,\gamma (R
n, E) by X.
Now we are ready to state the main result of the present section.
Condition 2.2. Suppose the following conditions be satisfied:
(1) L(\xi ) =
\sum
| \alpha | \leq l
\^a\alpha (\xi )(i\xi )
\alpha \in S\varphi 1 ,
\bigm| \bigm| L(\xi )\bigm| \bigm| \geq C
\bigm| \bigm| a0(\xi )\bigm| \bigm| | \xi | l, \bigm| \bigm| a0(\xi )\bigm| \bigm| = \mathrm{m}\mathrm{a}\mathrm{x}\alpha
\bigm| \bigm| \^a\alpha (\xi )\bigm| \bigm| , \xi =
=
\bigl(
\xi 1, \xi 2, . . . , \xi n
\bigr)
\in Rn;
(2) E is a Banach space;
(3) \^A(\xi ) is an uniformly \varphi -positive operator in E, with 0 < \varphi < \pi - \varphi 1 and\bigl[
D\beta \^A(\xi )
\bigr]
\^A - 1(\xi ) \in C
\bigl(
Rn;L(E)
\bigr)
,
\^a\alpha \in C(Rn), | \xi | k
\bigm| \bigm| \bigm| D\beta \^a\alpha (\xi )
\bigm| \bigm| \bigm| \leq C1, k \leq | \beta | \leq n+ 1,
| \xi | k
\bigm\| \bigm\| \bigm\| \bigl[ D\beta \^A(\xi )
\bigr]
\^A - 1(\xi )
\bigm\| \bigm\| \bigm\|
L(E)
\leq C2, k \leq | \beta | \leq n+ 1.
Theorem 2.1. Suppose the Condition 2.2 is satisfied, then for f \in Bs
p,q,\gamma (R
n;E), p, q \in [1,\infty ),
the equation (1.1) has a unique solution u \in Bl,s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
and the following coercive uniform
estimate holds:
| \lambda | \| u\| X +
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l
\bigm\| \bigm\| a\alpha \ast D\alpha u
\bigm\| \bigm\|
X
+ \| A \ast u\| X \leq C\| f\| X (2.7)
for \lambda \in S\varphi with | \lambda | \geq \lambda 0 > 0.
Proof. By applying the Fourier transform to equation (1.1), we obtain\bigl[
\^A(\xi ) + L(\xi ) + \lambda
\bigr]
u\^(\xi ) = f\^(\xi ).
Since L(\xi ) \in S\varphi 1 for \xi \in Rn and \^A is a positive operator in E, we get
\bigl[
\^A(\xi ) +L(\xi ) + \lambda
\bigr] - 1 \in
\in L(E). So we obtain that the solution of the equation (1.1) can be represented in the form
u(x) = F - 1
\bigl[
\^A(\xi ) + \lambda + L(\xi )
\bigr] - 1
f\^.
There are positive constants C1 and C2 such that
C1| \lambda | \| u\| X \leq
\bigm\| \bigm\| F - 1 [\sigma 0(\xi , \lambda )f\^]
\bigm\| \bigm\|
X
\leq C2| \lambda | \| u\| X ,
C1\| A \ast u\| X \leq
\bigm\| \bigm\| F - 1 [\sigma 1(\xi , \lambda )f\^]
\bigm\| \bigm\|
X
\leq C2\| A \ast u\| X , (2.8)
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1396 H. K. MUSAEV, V. B. SHAKHMUROV
C1
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \| a\alpha \ast D\alpha u\| X \leq
\bigm\| \bigm\| F - 1 [\sigma 2(\xi , \lambda )f\^]
\bigm\| \bigm\|
X
\leq
\leq C2
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \| a\alpha \ast D\alpha u\| X ,
where
\sigma 0(\xi , \lambda ) = \lambda
\bigl[
\^A(\xi ) + (\lambda + L(\xi )
\bigr] - 1
, \sigma 1(\xi , \lambda ) = \^A(\xi )
\bigl[
\^A(\xi ) + \lambda + L(\xi )
\bigr] - 1
,
\sigma 2(\xi , \lambda ) =
\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \^a\alpha (\xi )(i\xi )
\alpha
\bigl[
\^A(\xi ) + \lambda + L(\xi )
\bigr] - 1
.
To show the estimate (2.7) it is enough to prove that operator functions \sigma j(\xi , \lambda ), j = 0, 1, 2, are
UBM in X. This fact is obtained from the Lemma 2.3. That is we obtain the assertion.
Let Q be the operator in X = Bs
p,q,\gamma (R
n;E) generated by problem (1.1) for \lambda = 0, i.e.,
D(Q) \subset X0 = Bl,s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
, Qu =
\sum
| \alpha | \leq l
a\alpha \ast D\alpha u+A \ast u.
Result 2.1. Assume all conditions of Theorem 2.1 hold. Then for all \lambda \in S\varphi , | \lambda | \geq \lambda 0 > 0,
there exist the resolvent of operator Q and the following estimate holds:\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l
\bigm\| \bigm\| \bigm\| a\alpha \ast
\bigl[
D\alpha (Q+ \lambda ) - 1
\bigr] \bigm\| \bigm\| \bigm\|
L(X)
+
+| \lambda |
\bigm\| \bigm\| (Q+ \lambda ) - 1
\bigm\| \bigm\|
L(X)
+
\bigm\| \bigm\| A \ast (Q+ \lambda ) - 1
\bigm\| \bigm\|
L(X)
\leq C.
Condition 2.3. Let
D(A) = D( \^A) = D
\bigl(
\^A(\xi 0)
\bigr)
, \xi 0 \in Rn.
Moreover, there are positive constants C1 and C2 so that for u \in D(A), x \in Rn,
C1
\bigm\| \bigm\| \^A(\xi 0)u\bigm\| \bigm\| \leq \| A(x)u\| \leq C2
\bigm\| \bigm\| \^A(\xi 0)u\bigm\| \bigm\| .
Remark 2.1. The Condition 2.2 is checked for regular elliptic operators with sufficiently smooth
coefficients. Really, by setting E = Lp1(\Omega ), p1 \in (1,\infty ), for bounded domain \Omega \subset Rm with
enough smooth boundary \partial \Omega and by putting regular elliptic operators instead of A(x) and \^A(\xi ) one
can get it. So, by virtue of [17, 19] we obtain that the operators A(x), \^A(\xi ) are positive and the
above estimates hold.
Theorem 2.2. Assume all conditions of Theorem 2.1 and Condition 2.3 are satisfied. Then the
problem (1.1) has a unique solution u \in X0 and the coercive uniform estimate holds\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \| D\alpha u\| X + \| Au\| X \leq C\| f\| X
for all f \in X, p, q \in [1,\infty ) and \lambda \in S\varphi with | \lambda | \geq \lambda 0 > 0.
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B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1397
Proof. By applying the Fourier transform to equation (1.1), we have \^u(\xi ) = D(\xi , \lambda )f\^(\xi ),
where
D(\xi , \lambda ) =
\bigl[
\^A(\xi ) + L(\xi ) + \lambda )
\bigr] - 1
.
So we obtain that the solution of equation (1.1) can be represented as u(x) = F - 1D(\xi , \lambda )f\^.
Moreover, by Condition 2.2 we get\bigm\| \bigm\| AF - 1D(\xi , \lambda )f\^
\bigm\| \bigm\|
X
\leq M
\bigm\| \bigm\| \bigm\| \^A(\xi 0)F - 1D(\xi , \lambda )f\^
\bigm\| \bigm\| \bigm\|
X
.
Hence, by using (2.8) it sufficient to show that the operator functions\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \xi \alpha D(\xi , \lambda ) and \^A(\xi 0)D(\xi , \lambda )
are UBMs in X. Really, in view of (3) part of Condition 2.2 these fact are proved by reasoning as in
Lemma 2.3.
Condition 2.4. There are positive constants C1 and C2 such that
C1
n\sum
k=1
\bigm| \bigm| ak\xi k\bigm| \bigm| l \leq \bigm| \bigm| L(\xi )\bigm| \bigm| \leq C2
n\sum
k=1
\bigm| \bigm| ak\xi k\bigm| \bigm| l, \xi \not = 0,
and
D(A) = D( \^A) = D
\bigl(
A(x0)
\bigr)
, \^A(\xi )A - 1(x0) \in L\infty
\bigl(
Rn;B(E)
\bigr)
, \xi , x0 \in Rn,
C1
\bigm\| \bigm\| A(x0)u\bigm\| \bigm\| \leq
\bigm\| \bigm\| A(x)u\bigm\| \bigm\| \leq C2
\bigm\| \bigm\| A(x0)u\bigm\| \bigm\| , u \in D(A), x \in Rn.
Theorem 2.3. Let all conditions of Theorem 2.2 and Condition 2.4 hold. Then for u \in X0 there
are positive constants M1 and M2 so that
M1\| u\| X0 \leq
\sum
| \alpha | \leq l
\bigm\| \bigm\| a\alpha \ast D\alpha u
\bigm\| \bigm\|
X
+ \| A \ast u\| X \leq M2\| u\| X0 .
Proof. The left part of the above inequality is obtained from Theorem 2.2. So, it remains to
prove the right-hand side of the estimate. Really, from Condition 2.3 we have
\| A \ast u\| X \leq M
\bigm\| \bigm\| F - 1 \^A\^u
\bigm\| \bigm\|
X
\leq C
\bigm\| \bigm\| \bigm\| F - 1 \^AA - 1(x0)A(x0)\^u
\bigm\| \bigm\| \bigm\|
X
\leq
\leq C
\bigm\| \bigm\| F - 1A(x0)\^u
\bigm\| \bigm\|
X
\leq C\| Au\| X , u \in X0.
Hence, applying the Fourier transform to equation (1.1) and by reasoning as Theorem 2.2, it is
sufficient to prove that the function
\sum
| \alpha | \leq l
\^a\alpha \xi
\alpha
\Biggl[
n\sum
k=1
\xi lkk
\Biggr] - 1
is a multiplier in X. Then by using Condition 2.3 and proof of Lemma 2.3 we get the desired result.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1398 H. K. MUSAEV, V. B. SHAKHMUROV
Result 2.2. Theorem 2.3 implies the estimate
C1\| u\| X0 \leq \| Qu\| X \leq C2\| u\| X0
for u \in X0.
Result 2.3. Result 2.1 particularly implies that the operator Q + a is positive in X for a > 0,
i.e., if \^A is uniformly positive for \varphi \in
\Bigl( \pi
2
, \pi
\Bigr)
, then it is clear to see that the operator Q + a is a
generator of an analytic semigroup in X.
3. The Cauchy problem for parabolic CDOE. In this section we derive the maximal regularity
properties of parabolic CDOE.
Let E0 and E be two Banach spaces, where E0 is continuously and densely embedded into
E. Let X = Bs
p,q,\gamma (R
n;E), Y = Bs
p,q,\gamma (R+;X) and X0 = Bl,s
p,q,\gamma
\bigl(
Rn;E0, E
\bigr)
. \~Bl,1,s
p,q,\gamma
\bigl(
Rn
+;E0, E
\bigr)
denotes the space of all functions u \in \~Bl,1,s
p,q,\gamma
\bigl(
Rn
+;E0, E
\bigr)
that possess the generalized derivatives
Dtu,D
\alpha
xu \in Bs
p,q,\gamma (R+;X)
with the norm
\| u\|
Bl,1,s
p,q,\gamma
\bigl(
Rn
+;E0,E
\bigr) = \| u\| Bs
p,q,\gamma (R+;X) + \| Dtu\| Bs
p,q,\gamma (R+;X) +
\bigm\| \bigm\| D\alpha u
\bigm\| \bigm\|
Bs
p,q,\gamma (R+;X)
.
Consider the Cauchy problem for parabolic CDOE
\partial u
\partial t
+
\sum
| \alpha | \leq l
a\alpha \ast D\alpha u+A \ast u = f(t, x), u(0, x) = 0, (3.1)
where A = A(x) is a linear operator in E, a\alpha = a\alpha (x) are complex-valued functions.
Theorem 3.1. Assume Condition 2.2 holds for \varphi \in
\biggl(
\pi
2
, \pi
\biggr)
, s > 0. Suppose \gamma \in Ap and E is
a Banach space satisfies the B-weighted multiplier condition. Then for f \in Y the problem (3.1) has
a unique solution u \in \~Bl,1,s
p,q
\bigl(
Rn+1
+ ;E(A), E
\bigr)
satisfying the estimate\bigm\| \bigm\| \bigm\| \bigm\| \partial u\partial t
\bigm\| \bigm\| \bigm\| \bigm\|
Y
+
\sum
| \alpha | \leq l
\| a\alpha \ast D\alpha u\| Y + \| A \ast u\| Y \leq C\| f\| Y . (3.2)
Proof. It is clear to see that
\~Bl,1,s
p,q
\bigl(
Rn+1
+ ;E(A), E
\bigr)
= B1,s
p,q
\bigl(
R+;D(Q), X
\bigr)
.
Fom the Result 2.1 that the operator Q is positive in X for \varphi >
\pi
2
. Then it is known that the operator
Q generated an analytic semigroup in X. It is easy to see that the problem (3.1) can be expressed as
the following problem:
\partial u
\partial t
+Qu(t) = f(t), u(0) = 0, t \in R+. (3.3)
In view of resolvent properties of Q, \varphi \in
\biggl(
\pi
2
, \pi
\biggr)
, by virtue of [3] (Proposition 8.10), [15,
19] and by Result 2.2 we obtain that, for f \in Y the equation (3.3) has a unique solution u \in
\in B1,s
p,q
\bigl(
R+;D(Q), X
\bigr)
satisfying \bigm\| \bigm\| \bigm\| \bigm\| \partial u\partial t
\bigm\| \bigm\| \bigm\| \bigm\|
X
+ \| Qu\| X \leq C\| f\| X . (3.4)
By Theorem 2.1 and estimate (3.4) we obtain (3.2).
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B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1399
4. Degenerate convolution elliptic equations. Consider the E -valued weighted Besov spaces
B
[l],s
p,q,\gamma (\Omega ;E0, E) defined as
B[l],s
p,q,\gamma (\Omega ;E0, E) =
\Bigl\{
u; u \in Bs
p,q,\gamma (\Omega ;E0), D
[l]
xk
u \in Bs
p,q,\gamma (\Omega ;E)
\Bigr\}
,
\| u\|
B
[l],s
p,q,\gamma (\Omega ;E0,E)
= \| u\| Bs
p,q,\gamma (\Omega ;E0) +
n\sum
k=1
\bigm\| \bigm\| \bigm\| D[l]
xk
u
\bigm\| \bigm\| \bigm\|
B,s
p,q,\gamma (\Omega ;E)
<\infty .
Let us consider the following degenerate elliptic CDOE:
(L+ \lambda )u =
\sum
| \alpha | \leq l
a\alpha \ast D[\alpha ]u+A \ast u+ \lambda u = f. (4.1)
We shall show that for all f \in Bs
p,q,\gamma (R
n;E) and sufficiently large | \lambda | , \lambda \in S\varphi the equation (4.1)
has a unique solution u \in B
[l],s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
and the coercive uniform estimate\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l
\bigm\| \bigm\| \bigm\| a\alpha \ast D[\alpha ]u
\bigm\| \bigm\| \bigm\|
Bs
p,q,\gamma (R
n;E)
+ \| A \ast u\| Bs
p,q,\gamma (R
n;E) + | \lambda | \| u\| Bs
p,q,\gamma (R
n;E) \leq
\leq C\| f\| Bs
p,q,\gamma (R
n;E) (4.2)
holds.
Recall that \~\gamma k(x), k = 1, 2, . . . , n, are positive measurable functions in R and
D[\alpha ] = D
[\alpha 1]
1 D
[\alpha 2]
2 . . . D[\alpha n]
n , D
[ak]
k =
\biggl(
\~\gamma k(xk)
\partial
\partial xk
\biggr) ak
, \~\gamma =
\bigl(
\~\gamma 1, \~\gamma 2, . . . , \~\gamma n
\bigr)
.
It is clear that under the substitution
zk =
xk\int
0
\~\gamma - 1
k (y) dy (4.3)
spaces Bs
p,q(R
n;E) and B
[l],s
p,q
\bigl(
Rn;E(A), E
\bigr)
are mapped isomorphically onto the weighted spaces
Bs
p,q,\gamma (R
n;E), Bl,s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
respectively, where
\gamma =
n\prod
k=1
\~\gamma k
\bigl(
xk(zk)
\bigr)
.
Moreover, under substitution (4.3) the degenerate problem (4.1) is redused to the following nonde-
generate problem:
(L+ \lambda )u =
\sum
| \alpha | \leq l
a\alpha \ast D\alpha u+A \ast u+ \lambda u = f (4.4)
considered in weighted space Bs
p,q,\gamma (R
n;E), where A is a linear operator in Banach space E and
a\alpha are complex numbers.
Now, we consider Cauchy problem the degenerate parabolic convolution equation
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1400 H. K. MUSAEV, V. B. SHAKHMUROV
\partial u
\partial t
+
\sum
| \alpha | \leq l
a\alpha \ast D[\alpha ]u+A \ast u = f(t, x),
u(0, x) = 0, t \in R+, x \in R.
(4.5)
In a similar way, under the substitution (4.3) the degenerate Cauchy problem (4.5) considered in
Bs
p,q(R
n;E) is transformed into undegenerate Cauchy problem (3.1) considered in the weighted
space Bs
p,q,\gamma (R
n;E).
Let H be the operator generated by problem (4.1), i.e.,
D(H) = B[l],s
p,q,\gamma
\bigl(
Rn;E(A), E
\bigr)
, Hu =
\sum
| \alpha | \leq l
a\alpha \ast D[\alpha ]u+A \ast u,
and we denote Bs
p,q,\gamma (R
n;E) by X.
From Theorems 2.1 and 3.1 we obtain the following results.
Result 4.1. Under conditions of Theorem 2.1 and (4.3) the equation (4.1) has a unique solution
u(x) that belongs to space B[l],s
p,q,\gamma (Rn;E(A), E) and the coercive uniform estimate\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l
\bigm\| \bigm\| a\alpha \ast D[\alpha ]u
\bigm\| \bigm\|
X
+ \| A \ast u\| X + | \lambda | \| u\| X \leq C\| f\| X
holds for all f \in Bs
p,q,\gamma (R
n;E) and for sufficiently large \lambda \in S\varphi .
Moreover, for \lambda \in S\varphi there exist the resolvent of operator H and has the estimate\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l
\bigm\| \bigm\| a\alpha \ast D[\alpha ](H + \lambda ) - 1
\bigm\| \bigm\|
L(X)
+
+
\bigm\| \bigm\| A \ast (H + \lambda ) - 1
\bigm\| \bigm\|
L(X)
+
\bigm\| \bigm\| \lambda (H + \lambda ) - 1
\bigm\| \bigm\|
L(X)
\leq C.
Result 4.2. For all f \in Bs
p,q,\gamma (R+;X) there is unique solution u(t, x) of problem (4.1) satisfying
the following coercive estimate:\bigm\| \bigm\| \bigm\| \bigm\| \partial u\partial t
\bigm\| \bigm\| \bigm\| \bigm\|
Y
+
\sum
| \alpha | \leq l
\bigm\| \bigm\| \bigm\| a\alpha \ast D[\alpha ]u
\bigm\| \bigm\| \bigm\|
Y
+ \| A \ast u\| Y \leq C\| f\| Y .
5. Boundary-value problems for CDEs. In this section the boundary-value problems (BVPs)
for the anisotropic type integro-differential equations is studied. The maximal regularity properties of
this problem in weighted mixed Bs
\bfp ,q,\gamma norm is obtained. In this direction it can be mention, e.g., the
works [3, 10, 11, 16, 19]. Let \~\Omega = Rn \times \Omega , where \Omega \subset R\mu is an open connected set with compact
C2m-boundary \partial \Omega . Consider the BVP for CDE
(L+ \lambda )u =
\sum
| \alpha | \leq l
a\alpha \ast D\alpha u+
\sum
| \alpha | \leq 2m
\bigl(
b\alpha c\alpha D
\alpha
y
\bigr)
\ast u+ \lambda u = f(x, y), (5.1)
x \in Rn, y \in \Omega \subset R\mu ,
Bju =
\sum
| \beta | \leq mj
bj\beta (y)D
\beta
yu(x, y) = 0, y \in \partial \Omega , (5.2)
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B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1401
where
Dj = - i \partial
\partial yj
, y = (y1, . . . , y\mu ), a\alpha = a\alpha (x), b\alpha = b\alpha (x), c\alpha = c\alpha (y),
\alpha = (\alpha 1, \alpha 2, . . . , \alpha n), u = u(x, y), j = 1, 2, . . . ,m.
Let \~\Omega = Rn\times \Omega , \bfp =(p1, p), and \gamma (x) = | x| \alpha , L\bfp ,\gamma
\bigl(
\~\Omega
\bigr)
will be denote the space of all \bfp -summable
scalar-valued functions with weighted mixed norm (see, e.g., [7], § 1), i.e., the space of all measurable
functions f defined on \~\Omega , for which
\| f\|
L\bfp ,\gamma
\bigl(
\~\Omega
\bigr) =
\left( \int
Rn
\left( \int
\Omega
| f(x, y)| p1 \gamma (x)dx
\right)
p
p1
dy
\right)
1
p
<\infty .
Analogously Bs
\bfp ,q,\gamma
\bigl(
\~\Omega
\bigr)
denotes the Besov space with corresponding weighted mixed norm [7] (§ 18)
and let
\~Bs
\bfp ,q,\gamma
\bigl(
\~\Omega
\bigr)
= Bs
p,q,\gamma
\bigl(
Rn;Bs
p1,q,\gamma (\Omega )
\bigr)
,
\~Bl,2m,s
\bfp ,q,\gamma (\~\Omega ) = Bl,s
p,q,\gamma
\Bigl(
Rn;B2m,s
p1,q,\gamma (\Omega ), B
s
p1,q,\gamma (\Omega )
\Bigr)
.
Let Q denote the operator generated by BVP (5.1), (5.2).
In general, l \not = 2m so equation (5.1) is anisotropic. For l = 2m we get isotropic equation.
Theorem 5.1. Let the following conditions be satisfied:
(1) c\alpha \in C
\bigl(
\=\Omega
\bigr)
for each | \alpha | = 2m and c\alpha \in L\infty (\Omega ) + Lrk(\Omega ) for each | \alpha | = k < 2m with
rk \geq p1, p1 \in (1,\infty ) and 2m - k >
l
rk
, - 1 < \alpha < p - 1, k = 1, 2, . . . , n;
(2) bj\beta \in C2m - mj (\partial \Omega ) for each j, \beta , mj < 2m, p, q \in (1,\infty );
(3) for y \in \=\Omega , \xi \in R\mu , \lambda \in S\varphi 0 , \varphi 0 \in
\biggl(
0,
\pi
2
\biggr)
, | \xi | + | \lambda | \not = 0, let \lambda +
\sum
| \alpha | =2m
c\alpha (y)\xi
\alpha \not = 0;
(4) for each y0 \in \partial \Omega local BVP in local coordinates corresponding to y0
\lambda +
\sum
| \alpha | =2m
c\alpha (y0)D
\alpha g(y) = 0,
Bj0g =
\sum
| \beta | =mj
bj\beta (y0)D
\beta g(y) = hj , j = 1, 2, . . . ,m,
has a unique solution g(y) \in C0 (R+) for all h = (h1, h2, . . . , hm) \in Rm and for \xi \prime \in R\mu - 1 with
| \xi \prime | + | \lambda | \not = 0;
(5) the (1) part of Condition 2.2 satisfied; \^a\alpha ,\^b\alpha \in C(n)(Rn) and there are positive constants
C1 and C2, so that
| \xi | k
\bigm| \bigm| D\beta \^a\alpha (\xi )
\bigm| \bigm| \leq C1, | \xi | k
\bigm| \bigm| D\beta \^b\alpha (\xi )
\bigm| \bigm| \leq C2
for all k \leq | \beta | \leq n+ 1 and \xi \in Rn\setminus \{ 0\} .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
1402 H. K. MUSAEV, V. B. SHAKHMUROV
Then for all f \in \~Bs
\bfp ,q,\gamma (
\~\Omega ) problem (5.1), (5.2) has a unique solution u \in \~Bl,2m,s
\bfp ,q,\gamma
\bigl(
\~\Omega
\bigr)
and the
following coercive uniform estimate holds:\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \| a\alpha \ast D\alpha u\| \~Bs
\bfp ,q,\gamma
\bigl(
\~\Omega
\bigr) + | \lambda | \| u\| \~Bs
\bfp ,q,\gamma (\~\Omega )
+
+
\sum
| \alpha | \leq 2m
\| b\alpha c\alpha D\alpha \ast u\| \~Bs
\bfp ,q,\gamma
\bigl(
\~\Omega
\bigr) \leq C\| f\| \~Bs
\bfp ,q,\gamma
\bigl(
\~\Omega
\bigr)
for \lambda \in S\varphi and \varphi \in [0, \pi ).
Proof. Let X = Bs
p1,q,\gamma (\Omega ). Consider the operator A defined by the following equalities:
D(A) = B2m,s
p1,q (\Omega ;Bju = 0) , A(x)u =
\sum
| \alpha | \leq 2m
b\alpha (x)c\alpha (y)D
\alpha u(y). (5.3)
The problem (5.1), (5.2) can be rewritten in the form of (1.1), where u(x) = u(x, \cdot ), f(x) =
= f(x, \cdot ) are functions with values in X = Bs
p1,q,\gamma (\Omega ). It is easy to see that \^A(\xi ) and D\beta \^A(\xi ) are
operators in X defined by
D( \^A) = D(D\beta \^A) = B2m,s
p1,q (\Omega ;Bju = 0),
\^A(\xi )u =
\sum
| \alpha | \leq 2m
\^b\alpha (\xi )c\alpha (y)D
\alpha u(y), | \beta | \leq n, (5.4)
D\beta
\xi
\^A(\xi )u =
\sum
| \alpha | \leq 2m
D\beta
\xi
\^b\alpha (\xi )c\alpha (y)D
\alpha u(y).
In view of conditions (1) – (5) and by virtue of [17, 19] the operators \^A(\xi ) + \lambda and D\beta \^A(\xi ) + \lambda
for sufficiently large \lambda > 0 are uniformly positive in X. Moreover, following problems for f \in X
and for arg \lambda \in S\varphi 0 , | \lambda | \rightarrow \infty :
\lambda u(y) +
\sum
| \alpha | \leq 2m
\^b\alpha (\xi )c\alpha (y)D
\alpha u(y) = f(y),
Bju =
\sum
| \beta | \leq mj
bj\beta (y)D
\beta u(y) = 0, j = 1, 2, . . . ,m,
(5.5)
\lambda u(y) +
\sum
\alpha \leq 2m
D\beta \^b\alpha (\xi )c\alpha (y)D
\alpha u(y) = f(y),
Bju =
\sum
| \beta | \leq mj
bj\beta (y)D
\beta u(y) = 0, j = 1, 2, . . . ,m,
(5.6)
has unique solutions belong to B2m,s
p1,q,\gamma (\Omega ) and the coercive estimates hold
\| u\|
B2m,s
p1,q,\gamma
(\Omega )
\leq C
\bigm\| \bigm\| ( \^A(\xi ) + \lambda )u
\bigm\| \bigm\|
X
, \| u\|
B2m,s
p1,q,\gamma
(\Omega )
\leq C
\bigm\| \bigm\| (D\beta \^A(\xi ) + \lambda )u
\bigm\| \bigm\|
X
(5.7)
for solutions of problems (5.5) and (5.6) respectively. Then by (5.4) in view of (5) condition and by
virtue of embedding theorems [7] (§ 18.4) and [21] we obtain
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
B -COERCIVE CONVOLUTION EQUATIONS IN WEIGHTED FUNCTION SPACES AND APPLICATIONS 1403\bigm\| \bigm\| ( \^A(\xi ) + \lambda )u
\bigm\| \bigm\|
X
\leq C\| u\|
B2m,s
p1,q,\gamma
(\Omega )
\leq C
\bigm\| \bigm\| ( \^A(\xi ) + \lambda )u
\bigm\| \bigm\|
X
,\bigm\| \bigm\| (D\beta \^A(\xi ) + \lambda )u
\bigm\| \bigm\|
X
\leq C\| u\|
B2m,s
p1,q,\gamma
(\Omega )
\leq C
\bigm\| \bigm\| (D\beta \^A(\xi ) + \lambda )u
\bigm\| \bigm\|
X
. (5.8)
Morever by using condition (5), for u \in B2m,s
p1,q,\gamma (\Omega ) we have
| \xi | k
\bigm\| \bigm\| (D\beta \^A(\xi ) + \lambda )u
\bigm\| \bigm\|
X
\leq C
\bigm\| \bigm\| ( \^A(\xi ) + \lambda )u
\bigm\| \bigm\|
X
,
i.e., all conditions of Theorem 2.1 hold and we obtain the assertion.
6. The system of infinite many integro-differential equations. Consider the following infinity
system of convolution equation:
\sum
| \alpha | \leq l
a\alpha \ast D\alpha um +
\infty \sum
j=1
dj \ast uj(x) + \lambda u = fm(x), x \in Rn, m = 1, 2, . . . ,\infty . (6.1)
Condition 6.1. There are positive constants C1 and C2 so that for
\bigl\{
dj(x)
\bigr\} \infty
1
\in lr for all x \in Rn
and some x0 \in Rn,
C1
\bigm| \bigm| dj(x0)\bigm| \bigm| \leq | dj(x)| \leq C2
\bigm| \bigm| dj(x0)\bigm| \bigm| .
Let
D(x) = \{ dm(x)\} , dm > 0, u = \{ um\} , D \ast u = \{ dm \ast um\} , m = 1, 2, . . . ,\infty ,
lr(D) =
\left\{ u : u \in lr, \| u\| lr(D) = \| D \ast u\| lr =
\Biggl( \infty \sum
m=1
| dm(x0) \ast um| r
\Biggr) 1
r
<\infty
\right\} , 1 < r <\infty .
Let Q be a differential operator in X = Bs
p,q,\gamma
\bigl(
Rn; lr
\bigr)
generated by problem (6.1) and B =
= L
\bigl(
Bs
p,q,\gamma (R
n; lr)
\bigr)
. Here \gamma (x) = | x| \alpha , - 1 < \alpha < p - 1.
Theorem 6.1. Suppose the first part of Conditions 2.1 and 6.1 hold, \^a\alpha , \^dm \in C(n)(Rn) and
there are positive constants C1 and C2 so that
| \xi | k
\bigm| \bigm| D\beta \^a\alpha (\xi )
\bigm| \bigm| \leq C1, | \xi | k
\bigm| \bigm| D\beta \^dm(\xi )
\bigm| \bigm| \leq C2
for all k \leq | \beta | \leq n+ 1 and \xi \in Rn\setminus \{ 0\} .
Then:
(a) for all f(x) = \{ fm(x)\} \infty 1 \in Bs
p,q,\gamma
\bigl(
Rn; lr(D)
\bigr)
for \lambda \in S\varphi , \varphi \in [0, \pi ), problem (6.1) has a
unique solution u = \{ um(x)\} \infty 1 that belongs to space Bl,s
p,q,\gamma
\bigl(
Rn; lr(D), lr
\bigr)
and the coercive uniform
estimate \sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l \| a\alpha \ast D\alpha u\| X + \| D \ast u\| X + | \lambda | \| u\| X \leq C\| f\| X (6.2)
holds for the solution of the problem (6.1);
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1404 H. K. MUSAEV, V. B. SHAKHMUROV
(b) for \lambda \in S\varphi there exists a resolvent (Q+ \lambda ) - 1 of operator Q and\sum
| \alpha | \leq l
| \lambda | 1 -
| \alpha |
l
\bigm\| \bigm\| a\alpha \ast
\bigl[
D\alpha (Q+ \lambda ) - 1
\bigr] \bigm\| \bigm\|
B
+
+
\bigm\| \bigm\| D \ast (Q+ \lambda ) - 1
\bigm\| \bigm\|
B
+
\bigm\| \bigm\| \lambda (Q+ \lambda ) - 1
\bigm\| \bigm\|
B
\leq C. (6.3)
Proof. Really, let E = lr, A be infinite matrices, such that
A =
\bigl[
dm(x)\delta jm
\bigr]
, m, j = 1, 2, . . . ,\infty .
Then
\^A(\xi ) =
\bigl[
\^dm(\xi )\delta jm
\bigr]
, D\beta \^A(\xi ) =
\bigl[
D\beta \^dm(\xi )\delta jm
\bigr]
, m, j = 1, 2, . . . ,\infty .
It is clear to see that \^A and D\beta \^A(\xi ) are uniformly positive in lr. Therefore, by virtue of
Theorem 2.1 and Result 2.1 we obtain that the problem (6.1) for all f \in X and \lambda \in S\varphi has a unique
solution u \in Bl,s
p,q,\gamma (Rn; lr(D), lr) and estimates (6.2), (6.3) are satisfied.
Remark 6.1. There are a lot of positive operators in concrete Banach spaces. Therefore, putting
concrete Banach spaces instead of E and concrete positive differential, pseudodifferential operators,
or finite, infinite matrices, etc. instead of operator A on (1.1) or (3.1) we can obtain the maximal
regularity properties of different class of convolution equations and Cauchy problems for parabolic
CDOEs or system of equations by virtue of Theorems 2.1 and 3.1.
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Received 05.04.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 10
|
| id | umjimathkievua-article-1789 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:44Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b8/b133528638ca53316aeeefe5895aa5b8.pdf |
| spelling | umjimathkievua-article-17892019-12-05T09:26:39Z $b$-coercive convolution equations in weighted function spaces and applications $b$ -коерцитивнi рiвняння в згортках у вагових функцiональних просторах та їх застосування Musaev, H. K. Shakhmurov, V. B. Мусаєв, Г. К. Шахмуров, В. Б. We study the $b$-separability properties of elliptic convolution operators in weighted Besov spaces and establish sharp estimates for the resolvents of the convolution operators. As a result, it is shown that these operators are positive and, in addition, play the role of negative generators of analytic semigroups. Moreover, the maximal $b$-regularity properties of the Cauchy problem for a parabolic convolution equation are established. Finally, these results are applied to obtain the maximal regularity properties for anisotropic integro-differential equations and the system of infinitely many convolution equations. Вивчаються властивостi $b$-сепарабельностi елiптичних операторiв згортки у зважених просторах Бєсова. Встановлено точнi оцiнки для резольвент операторiв згортки. В результатi показано, що цi оператори є додатними, а також вiд’ємними генераторами аналiтичних напiвгруп. Крiм того, встановлено властивостi максимальної $b$-регулярностi адачi Кошi для параболiчного рiвняння у згортках. Цi результати застосовано до отримання властивостей максимальної регулярностi для анiзотропних iнтегро-диференцiальних рiвнянь та для систем нескiнченного числа рiвнянь у згортках. Institute of Mathematics, NAS of Ukraine 2017-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1789 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 10 (2017); 1385-1405 Український математичний журнал; Том 69 № 10 (2017); 1385-1405 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1789/771 Copyright (c) 2017 Musaev H. K.; Shakhmurov V. B. |
| spellingShingle | Musaev, H. K. Shakhmurov, V. B. Мусаєв, Г. К. Шахмуров, В. Б. $b$-coercive convolution equations in weighted function spaces and applications |
| title | $b$-coercive convolution equations in weighted function spaces
and applications |
| title_alt | $b$ -коерцитивнi рiвняння в згортках
у вагових функцiональних просторах та їх застосування |
| title_full | $b$-coercive convolution equations in weighted function spaces
and applications |
| title_fullStr | $b$-coercive convolution equations in weighted function spaces
and applications |
| title_full_unstemmed | $b$-coercive convolution equations in weighted function spaces
and applications |
| title_short | $b$-coercive convolution equations in weighted function spaces
and applications |
| title_sort | $b$-coercive convolution equations in weighted function spaces
and applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1789 |
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