The space $\Omega^p_m(R^d)$ and some properties
Let $m$ be a $v$-moderate function defined on $R^d$ and let $g \in L^2(R^d)$. In this work, we define $\Omega ^p_m(R^d)$ to be the vector space of $f \in L^2_n(R^d)$ such that the Gabor transform $V_gf$ belongs to $L^p(R^{2d})$, where $1 \leq p < \infty$. We endowe it with a norm and show tha...
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509534460575744 |
|---|---|
| author | Gürkanli, A. T. Sandikçi, A. Гурканлі, А. Т. Сандикчи, А. |
| author_facet | Gürkanli, A. T. Sandikçi, A. Гурканлі, А. Т. Сандикчи, А. |
| author_sort | Gürkanli, A. T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:54:30Z |
| description | Let $m$ be a $v$-moderate function defined on $R^d$ and let $g \in L^2(R^d)$. In this work, we define $\Omega ^p_m(R^d)$ to be the vector space of $f \in L^2_n(R^d)$ such that the Gabor transform $V_gf$ belongs to $L^p(R^{2d})$, where $1 \leq p < \infty$. We endowe it with a norm and show that it is a Banach space with this norm.
We also study some preliminary properties of $\Omega ^p_m(R^d)$.
Later we discuss inclusion properties and obtain the dual space of $\Omega ^p_m(R^d)$. At the end of this work, we study multipliers from $L_w^1 (R^d)$
into $\Omega ^p_w(R^d)$ and from $\Omega ^p_w(R^d)$ into $L^{\infty}_{w^{-1}}(R^d)$, where $w$
is Beurling's weight function. |
| first_indexed | 2026-03-24T02:42:38Z |
| format | Article |
| fulltext |
UDC 517.5
A. Sandikçi, A. T. Gürkanli (Ondokuz Mayis Univ., Turkey)
THE SPACE Ωp
m
(
Rd
)
AND SOME PROPERTIES
PROSTIR Ωp
m
(
Rd
)
TA DEQKI VLASTYVOSTI
Let m be a v-moderate function defined on Rd and let g ∈ L2(Rd). In this work, we define Ωp
m(Rd) to be
the vector space of f ∈ L2
m(Rd) such that the Gabor transform Vgf belongs to Lp(R2d), where 1 ≤ p < ∞.
We endowe it with a norm and show that it is a Banach space with this norm. We also study some preliminary
properties of Ωp
m(Rd). Later we discuss inclusion properties and obtain the dual space of Ωp
m(Rd). At the end
of this work, we study multipliers from L1
w(Rd) into Ωp
w(Rd) and from Ωp
w(Rd) into L∞
w−1 (Rd), where w
is Beurling’s weight function.
Nexaj m [ v-pomirnog funkci[g, wo vyznaçena na Rd, i g ∈ L2(Rd). U danij roboti Ωp
m(Rd) vyz-
naçeno qk vektornyj prostir elementiv f ∈ L2
m(Rd) takyx, wo peretvorennq Habora Vgf naleΩyt\
do Lp(R2d), de 1 ≤ p < ∞. Cej prostir osnaweno normog i pokazano, wo vin [ banaxovym iz ci[g
normog. TakoΩ vyvçeno deqki poperedni vlastyvosti Ωp
m(Rd). Rozhlqnuto vlastyvosti vklgçennq,
oderΩano dual\nyj do Ωp
m(Rd) prostir. Nasamkinec\ vyvçeno mul\typlikatory z L1
w(Rd) do Ωp
w(Rd)
ta z Ωp
w(Rd) do L∞
w−1 (Rd), de w [ vahovog funkci[g Berlinha.
1. Introduction. Throughout this paper,Cc(Rd) andC0(Rd) denote the space of complex-
valued continuous functions onRd with compact support and the space of complex-valued
continuous functions on Rd vanishing at infinity, respectively. For 1 ≤ p ≤ ∞, we
consider the Lebesgue spaces
(
Lp(Rd), ‖ · ‖p
)
. For any function f : Rd → C, the
translation and modulation operator are defined as Txf (t) = f (t− x) and Mwf (t) =
= e2πiwtf (t) for x,w ∈ Rd, respectively. It is easy to see that TxMt = e−2πixtMtTx
and ‖TxMtf‖p = ‖f‖p [1]. A weight is a positive locally integrable function m : Rd →
→ (0,∞). A weight v is called submultiplicative if v (x+ y) ≤ v (x) v (y) for all
x, y ∈ Rd. A weight w is right moderate (or simply v-moderate) if there exists a submul-
tiplicative function v such that w (x+ y) ≤ w (x) v (y) for all x, y ∈ Rd. Especially any
continuous submultiplicative function satisfying w (x) ≥ 1 is called Beurling’s weight
function. For 1 ≤ p < ∞, we set
Lp
w(Rd) =
{
f | fw ∈ Lp(Rd)
}
,
‖f‖p,w =
∫
Rd
|f (x)|p wp (x) dx
1
p
.
This is a Banach space with the norm.
Particularly, L1
w(Rd) is a Banach convolution algebra. It is called a Beurling algebra.
Let L∞
w−1(Rd) be the algebra of all measurable functions f on Rd for which
‖f‖∞,w−1 = ess sup
x∈Rd
∣∣∣∣ f (x)
w (x)
∣∣∣∣ < ∞.
Under the norm ‖ · ‖∞,w−1 , L∞
w−1(Rd) is a Banach algebra, which is the dual space of
L1
w(Rd) [2]. It is also known that if
1
p
+
1
q
= 1, then the dual of Lp
w(Rd) is the space
Lq
w−1(Rd) [2 – 4].
c© A. SANDIKÇI, A. T. GÜRKANLI, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1 139
140 A. SANDIKÇI, A. T. GÜRKANLI
Let w1 and w2 be two weight functions. We say that w2 < w1 if and only if there
exists c > 0 such that w2 (x) < cw1 (x) for all x ∈ Rd. Two weights w1 and w2 are
equivalent, denoted w1 ≈ w2, if there exist contants A,B > 0 such that Aw1 (x) ≤
≤ w2 (x) ≤ Bw1 (x) .
Let 〈x, t〉 =
∑d
i=1
xiti be the usual scalar product on Rd. For f ∈ L1(Rd), the
Fourier transform
∧
f (or Ff ) is given by the relation
∧
f (t) =
∫
Rd
f (x) e−2πi〈x,t〉dx.
It is known that
∧
f ∈ C0(Rd).
In engineering, t is a frequency and
∧
f (t) is the amplitude of the frequency t. In the
physics, t is the momentum variable. To obtain information about local properties of f
and about some local frequency spectrum, we restrict f to an interval and take the Fourier
transform. Therefore, given any fixed function g
= 0 (called the window function), the
Short-Time Fourier transform (STFT) or Gabor transform, of a function f with respect to
g is defined by
Vgf (x,w) =
∫
Rd
f (t) g (t− x)e−2πitwdt
for x,w ∈ Rd. It is known that if f, g ∈ L2(Rd), then Vgf ∈ L2
(
Rd ×Rd
)
and Vgf is
uniformly continuous. Moreover,
Vg (TuMηf) (x,w) = e−2πiuwVgf (x− u,w − η)
for all x,w, u, η ∈ Rd [1]. A very important inequality for STFT was proved by E. Lieb
[5]. That is if f, g ∈ L2(Rd) and 2 ≤ p < ∞, then∫∫
R2d
|Vgf (x,w)|p dxdw ≤
(
2
p
)d
(‖f‖2 ‖g‖2)
p
.
If 1 ≤ p ≤ 2 and f, g ∈ L2(Rd), then∫∫
R2d
|Vgf (x,w)|p dxdw ≥
(
2
p
)d
(‖f‖2 ‖g‖2)
p
.
The equality holds if and only if p > 1 and f, g are certain Gaussians.
For two Banach modules B1 and B2 over a Banach algebra A, we write MA (B1,B2)
or HomA (B1, B2) for the space of all bounded linear operators satisfying T (ab) =
= aT (b) for all a ∈ A, b ∈ B1. This operators are called multiplier (right) or module
homomorphism from B1 into B2.
2. The space Ωp
m(Rd).
Definition 1. Let v be a weight and m be a v-moderate function on Rd. For 1 ≤
≤ p < ∞ and g ∈ L2(Rd), define
Ωp
m(Rd) =
{
f ∈ L2
m(Rd) : Vgf ∈ Lp
(
R2d
)}
.
It is easy to see that ‖f‖Ω = ‖f‖2,m + ‖Vgf‖p is a norm on the vector space Ωp
m(Rd).
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
THE SPACE Ωp
m
(
Rd
)
AND SOME PROPERTIES 141
Theorem 1. Let 1 ≤ p < ∞. Then the following assertions are true:
a)
(
Ωp
m(Rd), ‖ · ‖Ω
)
is a Banach space;
b) if v(z) ≥ 1 is a submultiplicative function, then Ωp
m(Rd) is a translation invariant
and the function z → Tzf is continuous from Rd into Ωp
m(Rd);
Proof. a) Suppose that (fn)n∈N is a Cauchy sequence in Ωp
m(Rd). Clearly, (fn)n∈N
and (Vgfn)n∈N are Cauchy sequences in L2
m(Rd) and Lp
(
R2d
)
, respectively. Since
L2
m(Rd) and Lp
(
R2d
)
are Banach spaces, there exists f ∈ L2
m(Rd) and h ∈ Lp
(
R2d
)
such that ‖fn − f‖2,m → 0, ‖Vgfn − h‖p → 0. Moreover, using the subsequence prop-
erty, we obtain Vgf = h. Thus, ‖fn − f‖Ω → 0 and f ∈ Ωp
m(Rd). Hence, Ωp
m(Rd) is a
Banach space.
b) Let f ∈ Ωp
m(Rd) be given. Then we write f ∈ L2
m(Rd) and Vgf ∈ Lp
(
R2d
)
. It is
easy to see that ‖Tzf‖2,m ≤ v(z)‖f‖2,m and Tzf ∈ L2
m(Rd) for all z ∈ Rd. Using the
properties of Gabor transform, we obtain
Vg (Tzf) (x,w) = Vgf (x− z, w) (1)
and
‖Vg (Tzf)‖p = ‖Vgf‖p .
Thus, we have
‖Tzf‖Ω ≤ v(z)‖f‖Ω < ∞
and Tzf ∈ Ωp
m(Rd). This means that Ωp
m(Rd) is a translation invariant. From equality
(1) we have
|Vg (Tzf) (x,w)| = |Vgf (x− z, w)|
and
‖Vg (Tzf) − Vgf‖p =
∥∥T(z,0) (Vgf) − Vgf
∥∥
p
.
It is known that the function z → Tzf and (z, u) → T(z,u)f are continuous from Rd into
L2
m(Rd) and from R2d into Lp
(
R2d
)
, respectively, by Lemma 1.6 in [6]. By using these
properties, the proof is completed.
Theorem 2. Ωp
m(Rd) is an essential Banach module over L1
v(Rd).
Proof. It is known that Ωp
m(Rd) is a Banach space by Theorem 1. Let f ∈ Ωp
m(Rd)
and h ∈ L1
v(Rd). Since Lp
m(Rd) is a Banach module over L1
v(Rd), we have f ∗ h ∈
∈ L2
m(Rd) and ‖f ∗ h‖2,m ≤ ‖f‖2,m ‖h‖1,v [7]. Moreover, using the equality Vgf(x,
w) = e−2πixw (f ∗Mwg
∗) (x), we obtain
‖Vg (f ∗ h)‖p =
∥∥e−2πixw ((f ∗ h) ∗Mwg
∗)
∥∥
p
≤
≤ ‖h‖1 ‖f ∗Mwg
∗‖p ≤ ‖h‖1,v ‖Vgf‖p < ∞. (2)
Thus, Vg (f ∗ h) ∈ Lp
(
R2d
)
. From (2) we write
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
142 A. SANDIKÇI, A. T. GÜRKANLI
‖f ∗ h‖Ω = ‖f ∗ h‖2,m + ‖Vg (f ∗ h)‖p ≤
≤ ‖f‖2,m ‖h‖1,v + ‖Vgf‖p ‖h‖1,v = ‖h‖1,v ‖f‖Ω.
Hence, Ωp
m(Rd) is a Banach module over L1
v(Rd).
It is known that L1
v(Rd) has a bounded approximate identity [8]. To show that
Ωp
m(Rd) is an essential module in L1
v(Rd), it suffices to prove that L1
v(Rd) ∗ Ωp
m(Rd)
is dense in Ωp
m(Rd) by Module Factorization Theorem. Take any h ∈ Ωp
m(Rd). Since
the map z → Tzh is contiuonus from Rd into Ωp
m(Rd) by Theorem 1, for any given
ε > 0 there exists a compact neighbourhood U of the unit element of Rd such that
‖Tzh− h‖Ω < ε for all z ∈ U. Let f be a continuous function on Rd for which f ≥ 0,∫
U
f(x)dx = 1 and the support of f is contained in U. Then
‖f ∗ h− f‖Ω =
∥∥∥∥∥∥
∫
Rd
f(z)h (y − z) dz −
∫
Rd
f(z)h (y) dz
∥∥∥∥∥∥
Ω
=
=
∥∥∥∥∥∥
∫
U
f(z) (h (y − z) − h (y)) dz
∥∥∥∥∥∥
Ω
≤
≤
∫
U
f(z) ‖Tzh− h‖Ω dz = ‖Tzh− h‖Ω
∫
U
f(z)dz =
= ‖Tzh− h‖Ω < ε.
Thus, L1
v(Rd) ∗ Ωp
m(Rd) is dense in Ωp
m(Rd) and the proof is completed.
Corollary 1. Let (eα)α∈I be a bounded approximate identity in L1
v(Rd). Since
Ωp
m(Rd) is an essential Banach module over L1
v(Rd), we have lim
α
eα ∗ f = f for all
f ∈ Ωp
m(Rd) by Corollary 15.3 in [9].
Proposition 1. If 2 ≤ p < ∞, then the spaces Ωp
m(Rd) and L2
m(Rd) are al-
gebrically isomorphic and topologically homeomorphic.
Proof. Take any f ∈ Ωp
m(Rd). Then we write f ∈ L2
m(Rd) and ‖f‖2 ≤ ‖f‖2,m <
< ∞. Conversely, let f ∈ L2
m(Rd). By the Lieb Uncertainty Principle, we have
‖Vgf‖p ≤
(
2
p
)d
p
‖f‖2 ‖g‖2 < ∞.
Thus, f ∈ Ωp
m(Rd) and consequently, we obtain Ωp
m(Rd) = L2
m(Rd). Moreover, it is
easy to see that the norms ‖ · ‖2,m , ‖ · ‖Ω are equivalent.
Proposition 2. Let 2 ≤ p < ∞. Then Cc(Rd) is dense in Ωp
m(Rd).
Proof. It is easy to see the inclusion Cc(Rd) ⊂ Ωp
m(Rd). Take any f ∈ Ωp
m(Rd). Let
C0 = max
{(
2
p
)d
p
‖g‖2 , 1
}
. Since Cc(Rd) is dense in L2
m(Rd), for any given ε > 0
there exists h ∈ Cc(Rd) such that
‖f − h‖2 ≤ ‖f − h‖2,m <
ε
2C0
<
ε
2
. (3)
By the Lieb Uncertainty Principle and (3), we write
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
THE SPACE Ωp
m
(
Rd
)
AND SOME PROPERTIES 143
‖f − h‖Ω = ‖f − h‖2,m + ‖Vg (f − h)‖p ≤
≤ ‖f − h‖2,m +
(
2
p
)d
p
‖g‖2 ‖f − h‖2 <
<
ε
2C0
+
(
2
p
)d
p
‖g‖2
ε
2C0
<
ε
2
+ C0
ε
2C0
<
ε
2
+
ε
2
= ε.
This completes the proof.
Lemma 1. Let w be Beurling’s weight function. Then for every f ∈ Ωp
w(Rd),
f
= 0, there exists c (f) > 0 such that
c (f)w(z) ≤ ‖Tzf‖Ω ≤ w(z)‖f‖Ω.
Proof. Let f ∈ Ωp
w(Rd). By Theorem 1.9 in [6], there exists c(f) > 0 such that
c (f)w(z) ≤ ‖Tzf‖2,w ≤ w(z)‖f‖2,w.
Moreover, it is known that ‖Vg (Tzf)‖p = ‖Vgf‖p by Theorem 1. Hence,
c (f)w(z) ≤ ‖Tzf‖2,w + ‖Vg (Tzf)‖p ≤ w(z)‖f‖2,w + ‖Vgf‖p ≤
≤ w(z)‖f‖2,w + w(z) ‖Vgf‖p =
= w(z)
(
‖f‖2,w + ‖Vgf‖p
)
= w(z)‖f‖Ω
for all f ∈ Ωp
w(Rd). Consequently, we obtain
c (f)w(z) ≤ ‖Tzf‖Ω ≤ w(z)‖f‖Ω.
It is easy to prove the following lemma.
Lemma 2. Letw1 andw2 be Beurling’s weight functions and Ωp
w1
(Rd) ⊂ Ωp
w2
(Rd).
Then Ωp
w1
(Rd) is a Banach space under the norm ‖f‖Ωp
w
= ‖f‖Ωp
w1
+ ‖f‖Ωp
w2
.
Theorem 3. Ifw1 andw2 are Beurling’s weight functions, then Ωp
w1
(Rd) ⊂ Ωp
w2
(Rd)
if and only if w2 < w1.
Proof. Suppose w2 < w1. Then there exists c > 0 such that w2(z) ≤ cw1(z) for
all z ∈ Rd. Let f ∈ Ωp
w1
(Rd). Then we write ‖fw2‖2 ≤ c ‖fw1‖2 . Furthermore, since
‖Vgf‖p < ∞, we have
‖f‖Ωp
w2
= ‖f‖2,w2 + ‖Vgf‖p ≤ c‖f‖2,w1 + c ‖Vgf‖p = c‖f‖Ωp
w1
< ∞
and Ωp
w1
(Rd) ⊂ Ωp
w2
(Rd).
Conversely, assume that Ωp
w1
(Rd) ⊂ Ωp
w2
(Rd). For given f ∈ Ωp
w1
(Rd) we have
f ∈ Ωp
w2
(Rd). By the Lemma 1, the function z → ‖Tzf‖Ωp
w1
is equivalent to the weight
function w1 and the function z → ‖Tzf‖Ωp
w2
is equivalent to the weight function w2.
Hence, there are costants c1, c2, c3, c4 > 0 such that
c1w1(z) ≤ ‖Tzf‖Ωp
w1
≤ c2w1(z), (4)
c3w2(z) ≤ ‖Tzf‖Ωp
w2
≤ c4w2(z) (5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
144 A. SANDIKÇI, A. T. GÜRKANLI
for every z ∈ Rd. By Lemma 2, the space Ωp
w1
(Rd) is a Banach space under the norm
‖f‖Ωp
w
= ‖f‖Ωp
w1
+‖f‖Ωp
w2
, f ∈ Ωp
w1
(Rd). Thus, by closed graph mapping theorem the
norms ‖ · ‖Ωp
w1
and ‖ · ‖Ωp
w
are equivalent. Hence, there exists c > 0 such that
‖f‖Ωp
w2
≤ c‖f‖Ωp
w1
(6)
for all f ∈ Ωp
w1
(Rd). Moreover, we also have Tzf ∈ Ωp
w2
(Rd) and
‖Tzf‖Ωp
w2
≤ c ‖Tzf‖Ωp
w1
.
If we combine (4), (5) and (6) find w2(z) ≤
cc2
c3
w1(z). If we take k =
cc2
c3
, then we have
w2(z) ≤ kw1(z) for all z ∈ Rd.
The theorem is proved.
Let Φp : Ωp
m(Rd) → L2
m−1(Rd) × Lq
(
R2d
)
, Φp (f) = (f, Vgf) be a function and
H = Φp
(
Ωp
m(Rd)
)
. Then
‖Φp (f)‖ = ‖(f, Vgf)‖ = ‖f‖2,m + ‖Vgf‖p
is a norm on H and Φp is an isometry.
Theorem 4. If
1
p
+
1
q
= 1, then the dual of the space Ωp
m(Rd) is the space
L2
m−1(Rd) × Lq
(
R2d
)
/K,
where
K =
(Φ,Ψ) ∈ L2
m−1(Rd) × Lq
(
R2d
) ∣∣∣∣ ∫
Rd
f (x) Φ (x) dx+
+
∫∫
R2d
Vgf (y, w) Ψ (y, w) dydw = 0, (f, Vgf) ∈ H
.
Proof. Φp is an isometry. Since Ωp
m(Rd) is a Banach space, H = Φp
(
Ωp
m(Rd)
)
is
closed. By the Duality Theorem in [10], we have
H∗ ∼= L2
m−1(Rd) × Lq
(
R2d
)
/K, (7)
where H∗ is the dual of H . Also, since Φp is an isometry, from (7) we obtain(
Ωp
m(Rd)
)∗ ∼= L2
m−1(Rd) × Lq
(
R2d
)
/K.
3. Multiplier from L1
w(Rd) into Ωp
w(Rd) and from Ωp
w(Rd) into L∞
w−1(Rd).
Let w be Beurling’s weight function on Rd and (eα)α∈I be bounded approximate iden-
tity in the weighted space L1
w(Rd). The relative completion Ω̃p
w(Rd) of Ωp
w(Rd) is defi-
ned by
Ω̃p
w(Rd) =
{
f ∈ L1
w(Rd)
∣∣∣ f ∗ eα ∈ Ωp
w(Rd)
for all α ∈ I and sup
α∈I
‖f ∗ eα‖Ω < ∞
}
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
It is known that Ω̃p
w(Rd) is a Banach space with the norm
‖f‖Ω̃ = sup
α∈I
‖f ∗ eα‖Ω .
It is also known that this does not depend on the approximate identity (eα)α∈I [11].
Theorem 5. If g ∈ L2(Rd), then the space M
(
L1
w(Rd),Ωp
w(Rd)
)
and Ω̃p
w(Rd)
are algebrically isomorphic and homeomorphic.
Proof. It is known that Ωp
w(Rd) is an essential Banach module over L1
w(Rd) by
Theorem 3. Let (eα)α∈I be a bounded approximate identity of L1
w(Rd). Hence,
‖f ∗ eα − f‖Ω → 0
for all f ∈ Ωp
w(Rd) by Corollary 1. We also have ‖f‖2,w ≤ ‖f‖Ω. Thus, by Theorem 3.8
in [11], we have
M
(
L1
w(Rd), Ωp
w(Rd)
) ∼= Ω̃p
w(Rd).
Theorem 6. If g ∈ L2(Rd), then the space HomL1
w
(
Ωp
w(Rd), L∞
w−1(Rd)
)
and
L2
w−1(Rd) × Lq
(
R2d
)
/K are algebrically isomorphic and homeomorphic.
Proof. It is known that Ωp
w(Rd) is an essential Banach module over L1
w(Rd) by
Theorem 2 and
(
Ωp
w(Rd)
)∗ ∼= L2
w−1(Rd) × Lq
(
R2d
)
/K by Theorem 4. If we use
Corollary 2.13 in [4], we obtain
HomL1
w
(
Ωp
w(Rd), L∞
w−1(Rd)
)
= HomL1
w
(
Ωp
w(Rd),
(
L1
w(Rd)
)∗) =
=
(
Ωp
w(Rd) ∗ L1
w(Rd)
)∗ =
=
(
Ωp
w(Rd)
)∗ ∼= L2
w−1(Rd) × Lq
(
R2d
)
/K.
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– P. 71 – 82.
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42, # 3. – P. 395 – 409.
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Received 16.05.2005
|
| id | umjimathkievua-article-3441 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:42:38Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d7/c07795d62c424ef441bbe17e37ecd4d7.pdf |
| spelling | umjimathkievua-article-34412020-03-18T19:54:30Z The space $\Omega^p_m(R^d)$ and some properties Простір $\Omega^p_m(R^d)$ тa деякі властивості Gürkanli, A. T. Sandikçi, A. Гурканлі, А. Т. Сандикчи, А. Let $m$ be a $v$-moderate function defined on $R^d$ and let $g \in L^2(R^d)$. In this work, we define $\Omega ^p_m(R^d)$ to be the vector space of $f \in L^2_n(R^d)$ such that the Gabor transform $V_gf$ belongs to $L^p(R^{2d})$, where $1 \leq p < \infty$. We endowe it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of $\Omega ^p_m(R^d)$. Later we discuss inclusion properties and obtain the dual space of $\Omega ^p_m(R^d)$. At the end of this work, we study multipliers from $L_w^1 (R^d)$ into $\Omega ^p_w(R^d)$ and from $\Omega ^p_w(R^d)$ into $L^{\infty}_{w^{-1}}(R^d)$, where $w$ is Beurling's weight function. Нехай $m$ є $v$-помірною функцією, що визначена на $R^d$, i $g \in L^2(R^d)$. У даній роботі $\Omega ^p_m(R^d)$ визначено як векторний простір елементів $f \in L^2_n(R^d)$ таких, що перетворення Габора $V_gf$ належить до $L^p(R^{2d})$, де $1 \leq p < \infty$. Цей простір оснащено нормою і показано, що він є банаховим із цією нормою. Також вивчено деякі попередні властивості $\Omega ^p_m(R^d)$. Розглянуто властивості включення, одержано дуальний до $\Omega ^p_m(R^d)$ простір. Насамкінець вивчено мультиплікатори з $L_w^1 (R^d)$ до $\Omega ^p_w(R^d)$ та з $\Omega ^p_w(R^d)$ до $L^{\infty}_{w^{-1}}(R^d)$, де $w$ є ваговою функцією Берлінга. Institute of Mathematics, NAS of Ukraine 2006-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3441 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 1 (2006); 139-145 Український математичний журнал; Том 58 № 1 (2006); 139-145 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3441/3621 https://umj.imath.kiev.ua/index.php/umj/article/view/3441/3622 Copyright (c) 2006 Gürkanli A. T.; Sandikçi A. |
| spellingShingle | Gürkanli, A. T. Sandikçi, A. Гурканлі, А. Т. Сандикчи, А. The space $\Omega^p_m(R^d)$ and some properties |
| title | The space $\Omega^p_m(R^d)$ and some properties |
| title_alt | Простір $\Omega^p_m(R^d)$ тa деякі властивості |
| title_full | The space $\Omega^p_m(R^d)$ and some properties |
| title_fullStr | The space $\Omega^p_m(R^d)$ and some properties |
| title_full_unstemmed | The space $\Omega^p_m(R^d)$ and some properties |
| title_short | The space $\Omega^p_m(R^d)$ and some properties |
| title_sort | space $\omega^p_m(r^d)$ and some properties |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3441 |
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