ON A FAMILY OF WIENER TYPE SPACES

Research on Wiener type spaces was initiated by N.Wiener in [15]. A number of authors worked on these spaces or some special cases of these spaces. A kind of generalization of the Wiener's definition was given by H.Feichtinger in [2] as a Banach spaces of functions (or measures, distributions) on locally compact groups that are defined by means of the global behaviour of certain local properties of their elements. In the present paper we discussed Wiener type spaces using the spaces Aw,ωp,q(G) and Fw,ωp,q(G) (c. f. [8]) as a local component, and Lνr(G) as a global component, where w and ν are Beurling weights on G and ω is a Beurling weight on Gˆ (c. f. [13]).

Let Q be a fixed open and relatively compact subset of G.We define where S O (G)--{ f lf=ELynfn,YneG,fneAQ,n>_l and Ell fnllA<=}.AQ-={ heA (G) supp (h)   (l) and L denotes the translation operator.Any Representation of f in y the form (1) is called an admissible representation.Endowed with the norm ----inf{llf I I f--z, f admissible I I fllSo n A' Yn n S (G) is the smallest strongly character invariant Segal algebra on G O It is well-known that the Fourier transforms induce isomorphisms bet- () The generalized Fourier transform then wean the spaces S o(G) and S o is defined by <o,f>---<o,> for feS O() oeS' (G) 0 It is easy to see that above mentioned qeneralized Fourier trans- form coincides with the Fourier transform in the sense of tempered distributions for the special case of G=IR m.
Throughout this work, we also will use Beurling weights, i.e.real valued, measurable and locally bounded functions w on a locally com- pact abelian group G which satisfy l<_w(x), w(x+y)<_w(x)w(y) for x,yeG.

F=
where F is the generalized Fourier transform, w and m are Beurling weights on G and respectively.
The main parts of this work deal with certain Wiener-type spaces in the sense [2]: the definition is the following: Let B be a Banach space.Assume that there exists a homogeneous Banach space continuously embedded into (Cb(G),ll.ll.), which is a Banach algebra under pointwise multiplication and is stable under complex conjugati- on, such that (B ll'II B is continuously embedded into A' (G)=(A(G)nC (S))' c c and also is a Banach module under pointwise multiplication.Here C (G) c is the space of continuous functions with compact support, Ac(G)=A(G) NCc(G) is given the (locally convex) inductive limit topology of its subspaces (AK(G) ,II "II A) KcmG compact, and A'c (G) is the topological dual of Ac(G).Let now Bioc be the space of all feA(G) such that hfeB for heA c(G) this is a locally convex vector space whose topology is de- fined by the seminorms fllhfllB,heAc(G).Fix am open, relatively com- pact set and define, for fEB and xeG, the "restriction of f over x/Q" to be [11111 hf-" hg for IcA (t;) xQ For fEBloc, set Ff(x):llfllB(x.O If now c is a solid, translation tn... variant BFspace on G, one defines the Wiener-type space W(B,C) by W(B,C)--{fEBIoclFfEC} and IlfllwB,C)--tlll )- Lastly recall that a Banach convolution triple (BCT) is a triple 1 B2 3 (B ,B of BF-spaces such that convolution, given by fl f2 i f2 x (x)= ff (x-y) (y)dy G for fieBiNC (G) (i:=i,2), extends to a continuous bilinear nap BIxB 2/ B 3 c (of norm I).
I. THE WIENER-TYPE SPACES W(AwP'q(G) ,Lr) The construction of the Wiener-type spaces mentioned in the sec- tion title requires some preliminary considerations, notably Theorem 1.5 below.First of all we introduce the Banach spaces.
u where u is an arbitrary weight function on G with the norm !llluIglll, u and F is the classical Fourier transform We set Au(G)=A u(G)NC c(G) and c equip it with the inductive limit topology T u of the subspaces AK(G)= =AU(G),CK(G), KeG compact, equipped with their ll.llu-nOrms.Since it is obvious that T u then is finer than the norm-topology of Ac(G), it is Hausdorff and hence the dual space Au(G) '----(Au(G) ,T u) separates the p- ints of AU(G) Note also that the subspaces A K c Indeed, if (hn)CA(G) converges to hEAU(G) with respect to II'II u, the sequence also converges uniformly to h and so supp(h)cK.The same re- u (G) Consequently, G) ,Tu) also is complete since then it is a w2 (shortly (BD) i.e E <, taG), then A (G)cA (G) and A (G) tlsfied and A (G) is everywhere dense in A 1 with respect to the Also since w 2 satisfies (BD) then there is an approximate norm II wl identity (ee)eiCAw K (G), where wK2={flfLw 1 (G),supp compact}.Take cW 2 w any eA I(G) and e>0.There exists a function feA 2(G) such that R. H. FISCHER, A. T. 0RKANLI AND T.  (G).This proves our lemma.c conver-COROLLARY 1.2.Let w I and w 2 be satisfied the hypothesis in lem- ma I.I.We endow the space AWl(G) with the induced inductive limit fo- In order to obtain all the properties of AU(G), etc., required for the construction of Wiener-type spaces in the sense e.g. of Feich- tinger, cf.
[2], we assume henceforth that the weight function u on G satisfies (BD) and symmetric ands<u, where is the second weight func- tion in AP'(G) (For example one can take.u=m(x) /(-x)) w First of all, AU(G) now satisfies the requirements of [2]: It is clear that AU(G) is continuously embedded into C b(G) in fact, the em- bedding map has norm<l.Moreover, since u satisfies (BD), Au(G) is known to be a Wiener space, cf.[13], hence is reqular.It is a Banach algebra under pointwise multiplication and also is translation-invari-L I( 6) is character-invariant and that the maps M are Isometries; ant: u x moreover, for geLl(G),x / Mxg is continuous [8].By taking Fourier transforms, we now conclude that AU(G) is translation-invariant, trans- lation maps are isometries and that x Lxh is continuous from G into AU(G) for each beAU(G).In other words: AU(G) is homogeneous Banach space.Lastly, the symmetry of u implies that AU(G) is closed under complex conjugation.
Secondly A p'q (G) is a pointwise Banach module over A(G), [8].Since <_u w then AP'q(G) is also a Banach module over Au(G).Consequently one shws w that AwP'q(G) is continuously embedded into (AU(G) ) by the Proposi- c tlon I. 4. With this, Feichtinger's general hypotheses are satisfied and the construction of the Wiener-type spaces w(AP'q(G),C), C a solid BF-space w} proceeds in the standard manner.
Using the arguments in Theorem I.
in [2] and doing some small modification, t|'e proof of the following theorem is completed.
(V) W(A'q(G),L r , u(G)) is a Banach module over :(A (G) ,L (G)) with respect to the pointwise multiplication.
PROPOSITION 1 6 Let p>l Then w(AP'q(G) L(G)) is a Banach modu- We le over W(AI'q(G) LI(G)) with respect to convolution.
It is easy to show that every locally compact lian grip is a group (i.e. a group having a compact neighbourhood of identity is Invariant under inner automorphisms) It is known that AP'q(G) s a W) left(rlght) convolutlon module over Al'q() 8].Hence since W (A'q(GI,AP'q(G) ,A'q(G)) and (L(G) ,LP(G) ,LP(G)) is a Banach convolution triples on G and the inequality IIflW(A'q(G) LP(G))11 < W l,q) (G) ,L(G) and Is Satisfied "[ the Theorem 3. in [2] for el].feW(Aw, 4,.P,q(G), (G)).One can easily show the algebraic conditions which are needed to be a module.
PROPOSITION 1.8.The Wiener type space W(AwP'q(G), ,Lu(G)r is a Ba- nach convolution module (left and right because G is an Abelian grou over some Beurllng algebra L 1 (G).
W o in Proposition 1.7 that W(A.P.'(G),L,r.(G)) is a PROOF.We proved BF-space, thus W(AwP'q(G) Lr(G)) 1 , + Lloc(G).By the Theorem 1.5., this space is left invarlant and translation operator in W(AP'q(G),Lr(G)is W continuous. NOW a simple application of vector valued inteqral shows that W(AP'q(G),Lr(G)) is a Banach module over L 1 (G), where is a weight satisfying v(x)>Wo(X) for all xSG,where w (x) is defined as in Proposition 1.8.o REMARK.By the Theorem 1.5 (iii), one writes IlL x III < III L x III p' q" IlL x III w, r,u' (l) where III-III III.III p'q and III .III., are operator norms on W(AwP:q(G),L(G)), W) <v(x) AP'q(G) and Lr(G) respectively.It is also known that W and" IIILxlIIP'q<w(x) [8].Then we have for all xeG.Since W,v are weight functions, then the function w.9 is locally bounded.Using the inequality(2) it is easily seen that x-IIILIII i so .oooune.
Given a weighted space LP(G) the associated weighted sequence space W is denoted by P and defined w r_{ (a) %eq r w i ie (aiW(i) iele r} where the discrete weioht W given by W(i)--W(x i) for iel It is r known that is a Banach space with respect to the norm w 1 I[zll r w---( E laiw(i)Ir) il where z----(a i) ieI" LEMMA I.i0.For any Zw r, z0 the function z-+ llLpZllr,w is equi- valent to the weight function w, i.e there is a constant c>0 such that one has -Iw <c.W (p) c (P) <ll LpZll r, w- PROOF.Result can be obtained by a slight modifications of the proof of Lemma 2.2. in [7].
It is also easy to prove the followi Imam using the closed graph theonaa.respectively.Also assume that Wl,W2,91,u 2 weights on ; i,2 weights on G and l_<p,q,rl,r2 <.If UI~U2 r 2 4_r I u 1 < u 2 and then A p'q (G) = A p'q (O) c strict inductive limit of Banach spaces LEMMA I.I.If wl<w 2 and w satisfies Beurling Domar condition logw 2 (tn[ w 2 original topo]ogv on A (G) which is induced topology bv A (G) c COROLLARY 1.3.Again let w.,w be satisfied the hypothesis in lem-4.If l_<p<_ and the weight function on G satis- fies (BD) condition then AP'q(G) is continuously embedded into o(A(G)',-w,{ A c (S)).PROOF.It is known that AP'g(G) is continuously embedded into w O(A(G) ,Ac(G))[8].If one uses the above embedding and the Corollary 1.3., easily proves the Proposition.
12. w c w if and only if w2<wI.