The space Ωpm(Rd) and some properties

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Дата:	2006
Автори:	Sandikçi, A., Gürkanli, A.T.
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Опубліковано:	Інститут математики НАН України 2006
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Теми:	Короткі повідомлення
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Цитувати:	The space Ωpm(Rd) and some properties / A.T. Gürkanli, Sandikçi A. // Український математичний журнал. — 2006. — Т. 58, № 1. — С. 139–145. — Бібліогр.: 11 назв. — англ.

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spelling	irk-123456789-1642512020-02-09T01:26:30Z The space Ωpm(Rd) and some properties Sandikçi, A. Gürkanli, A.T. Короткі повідомлення 2006 Article The space Ωpm(Rd) and some properties / A.T. Gürkanli, Sandikçi A. // Український математичний журнал. — 2006. — Т. 58, № 1. — С. 139–145. — Бібліогр.: 11 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164251 517.5 en Український математичний журнал Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Короткі повідомлення Короткі повідомлення
spellingShingle	Короткі повідомлення Короткі повідомлення Sandikçi, A. Gürkanli, A.T. The space Ωpm(Rd) and some properties Український математичний журнал
format	Article
author	Sandikçi, A. Gürkanli, A.T.
author_facet	Sandikçi, A. Gürkanli, A.T.
author_sort	Sandikçi, A.
title	The space Ωpm(Rd) and some properties
title_short	The space Ωpm(Rd) and some properties
title_full	The space Ωpm(Rd) and some properties
title_fullStr	The space Ωpm(Rd) and some properties
title_full_unstemmed	The space Ωpm(Rd) and some properties
title_sort	space ωpm(rd) and some properties
publisher	Інститут математики НАН України
publishDate	2006
topic_facet	Короткі повідомлення
url	http://dspace.nbuv.gov.ua/handle/123456789/164251
citation_txt	The space Ωpm(Rd) and some properties / A.T. Gürkanli, Sandikçi A. // Український математичний журнал. — 2006. — Т. 58, № 1. — С. 139–145. — Бібліогр.: 11 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT sandikcia thespaceōpmrdandsomeproperties AT gurkanliat thespaceōpmrdandsomeproperties AT sandikcia spaceōpmrdandsomeproperties AT gurkanliat spaceōpmrdandsomeproperties
first_indexed	2025-07-14T16:45:16Z
last_indexed	2025-07-14T16:45:16Z
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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. 1. Gröchenig K. Foundations of time-freuquency analysis. – Boston: Birkhauser, 2001. 2. Rieter H. Classical harmonic analysis and local compact groups. – Oxford: Clarendan Press, 1968. 3. Murthy G. N. K., Unni K. R. Multipliers on weighted spaces, functional analysis and its applications // Int. Conf. Modras. Lect. Notes Math. – 1973. – 399. 4. Rieffel M. A. Induced Banach representation of Banach algebras and locally compact groups // J. Funct. Anal. – 1967. – P. 443 – 491. 5. Lieb E. H. Integral bounds for radar ambiguity functions and Wigner distributions // J. Math. Phys. – 1990. – 31, # 3. – P. 594 – 599. 6. Fischer R. H., Gürkanli A. T., Liu T. S. On family of weighted spaces // Math. Slovaca. – 1996. – 46, # 1. – P. 71 – 82. 7. Feichtinger H. Generalized amalgams, with applications to Fourier transform // Can. J. Math. – 1990. – 42, # 3. – P. 395 – 409. 8. Gaudry G. I. Multipliers of weighted Lebesgue and measure spaces // Proc. London Math. Soc. – 1969. – 3, # 19. 9. Doran R. S., Wichman J. Approximate identities and factorization in Banach modules // Lect. Notes Math. – 1979. – 768. 10. Liu T., Van Rooij A. Sums and intersections of normed linear spaces // Math. Nachr. – 1969. – 42. – S. 29 – 42. 11. Duyar C., Gürkanli A. T. Multipliers and relative completion in weighted Lorentz spaces // Acta Math. Sci. Ser. B. – 2003. – 23, # 4. – P. 467 – 476. Received 16.05.2005

The space Ωpm(Rd) and some properties

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