WEIGHTED VARIABLE EXPONENT AMALGAM SPACES. İsmail Aydin and A. Turan Gürkanli Sinop University and Ondokuz Mayıs University, Turkey

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1 GLASNIK MATEMATIČKI Vol 47(67(202, WEIGHTED VARIABLE EXPONENT AMALGAM SPACES W(L p(x,l q İsmail Aydin and A Turan Gürkanli Sinop University and Ondokuz Mayıs University, Turkey Abstract In the present paper a ne family of Wiener amalgam spaces W(L p(x,l q is defined, ith local component hich is a variable exponent Lebesgue space L p(x (R n and the global component is a eighted Lebesgue space L q (R n We proceed to sho that these Wiener amalgam spaces are Banach function spaces We also present ne Höldertype inequalities and embeddings for these spaces At the end of this paper e sho that under some conditions the Hardy-Littleood maximal function is not mapping the space W(L p(x,l q into itself Introduction A number of authors orked on amalgam spaces or some special cases of these spaces The first appearance of amalgam spaces can be traced to N Wiener ([22] But the first systematic study of these spaces as undertaken by F Holland ([7,8] The amalgam of L p and l q on the real line is the space (L p,l q (R (or shortly (L p,l q consisting of functions f hich are locally in L p and have l q behavior at infinity in the sense that the norms over [n,n+] form an l q -sequence For p,q the norm q q f p,q = n= n+ n f (x p dx p < makes (L p,l q into a Banach space If p = q then (L p,l q reduces to L p A generalization of Wiener s definition as given by H G Feichtinger in [9], 200 Mathematics Subject Classification 42B25,42B35 Key ords and phrases Variable exponent Lebesgue space, Hardy-Littleood maximal function, Wiener amalgam space 65

2 66 İ AYDIN AND A TURAN GÜRKANLI describing certain Banach spaces of functions (or measures, distributions on locally compact groups by global behaviour of certain local properties of their elements C Heil in [6] gave a good summary of results concerning amalgam spaces ith global components being eighted L q (R spaces For a historical background of amalgams see [5] Letp : R n [, beameasurablefunction(calledthevariable exponent on R n We put p = ess inf x R n p(x, p = ess sup x R n p(x Thevariable exponentlebesgue space (orgeneralized Lebesgue spacel p(x (R n is defined to be the space of measurable functions (equivalence classes f such that ρ p (λf = λf(x p(x dx < R n for some λ = λ(f > 0 The function ρ p is called modular of the space L p(x (R n Then f L p(x = inf{λ > 0 : ρ p (f/λ } defines a norm (Luxemburg norm This makes L p(x (R n a Banach space If p(x = p is a constant function, then the variable exponent Lebesgue space L p(x (R n coincides ith the classical Lebesgue space L p (R n, see [9] Also there are recent many interesting and important papers appeared in variable exponent Lebesgue spaces (see [3 5,7,8] In this paper e ill assume that p < The space L loc (Rn consists of all (classes of measurable functions f on R n such that fχ K L (R n forany compactsubset K R n, hereχ K is the characteristic function of K It is a topological vector space ith the family of seminorms f fχ K L A Banach function space (shortly BF-space on R n is a Banach space (B, B of measurable functions hich is continuously embedded into L loc (Rn, that is for any compact subset K R n there exists some constant C K > 0 such that fχ K L C K f B for all f B and to functions equal almost everyhere are identified as usual We denote it by B L loc (Rn Obviously L p(x (R n L loc (Rn and the space L p(x (R n is a solid space, that is, if f L p(x (R n is given and g L loc (Rn satisfies g(x f(x ae, then g L p(x (R n and g L p(x f L p(x by [, Lemma ] A positive, measurable and locally integrable function ϑ : R n (0, is called a eight function We say that a eight function ϑ is submultiplicative if ϑ(x+y ϑ(xϑ(y

3 WEIGHTED VARIABLE EXPONENT AMALGAM SPACES W(L p(x,l q 67 for any x,y R n A eight function is moderate ith respect to a submultiplicative function ϑ (or ϑ-moderate if (x+y (xϑ(y for any x,y R n If the eight is moderate than / is also moderate We say that if there exists a constant C > 0 such that C (x (x for all x R n To eight functions are called equivalent and ritten, if and The space L q (R n (eighted L q (R n is the space of all complex-valued measurable functions on R n for hich f L q (R n Obviously ( L q (R n, L q is a Banach space ith the norm or f L q = f L q = f(x(x q dx R n q, q <, f L = f L = esssup x R n f(x (x, q = Also the dual of the space L q (R n is the space L s (R n, here q <, q + s = (see [2,4,6] Given a discrete family X = (x i i I in R n and a eightedspace L q (Rn, the associated eighted sequence space over X is the appropriateeighted l q - spacel q, the discrete being givenby (i = (x i fori I (see [, Lemma 35] 2 The Wiener Amalgam Space W ( L p(x,l q Let C b (R n be the the regular Banach algebra (ith respect to pointise multiplication of complex-valued bounded, continuous functions on R n Also let C 0 (R n, C c (R n be the spaces of complex-valued continuous function R n vanishing at infinity and the space of complex-valued continuous functions ith compact support defined on R n endoed ith its natural inductive limit topology respectively It is knon that (C 0, (C b, and the dual space of C c (R n (ith respect to its natural inductive limit topology is M (R n, the space of regularborel measures For everyh C c (R n e define the semi-norm q h on M (R n by q h (h = h (h The locally convex topology on M (R n defined by the family (q h h Cc(R n of seminorms is called the topology σ(m (R n,c c (R n or eak -topology, also called vague topology We define L p(x loc (Rn = { } σ M (R n : φσ L p(x (R n for all φ C c (R n L p(x loc (Rn isatopologicalvectorspaceithrespecttothefamilyofseminorms given by σ φ = φσ L p(x, φ C c (R n

4 68 İ AYDIN AND A TURAN GÜRKANLI It isknonby[,lemma]that L p(x (R n iscontinuouslyembedded into L loc (Rn Hence it is easily shon that L p(x loc (Rn is continuously embedded into L loc (Rn It is also obvious that L loc (Rn is continuously embedded into M (R n ith the eak -topology Therefore L p(x loc (Rn is continuously embedded into M (R n Since the general hypotheses for the Wiener amalgam space denoted by W ( L p(x (R n,l q (Rn (shortly W ( L p(x,l q are satisfied, it is defined as follos as in [9] Let fix an open set Q R n ith compact closure The Wiener amalgam space W ( L p(x,l q p(x consists of all elements f L loc (Rn such that F f (z = f L p(x (z+q belongs to Lq (R n ; the norm of W ( L p(x,l q is f W(L p(x,l q = F f L q In this definition f L p(x (z+q that is denotes the restriction norm of f to z + Q, f L p(x (z+q = inf { g L p(x: } g L p(x (R n,g coincides ith f on z +Q, ie, hf = hg for all h C C (R n ith supp(h z +Q By the solidity of the BF-space the assumptions imply f L p(x (z+q = fχ z+q L p(x The folloing theorem, based on [9, Theorem ], describes the basic properties of W ( L p(x,l q Theorem 2 i W ( L p(x,l q is a Banach space ith norm W(L p(x,l q ii W ( L p(x,l q p(x is continuously embedded into L loc (Rn iii The space { } Λ 0 = f L p(x (R n : supp(f is compact is continuously embedded into W ( L p(x,l q iv W ( L p(x,l q does not depend on the particular choice of Q, ie different choices of Q define the same space ith equivalent norms By iii and [, Lemma 4] it is easy to see that C c (R n is continuously embedded into W ( L p(x,l q By using the techniques in [3], e prove the folloing proposition Proposition 22 W ( L p(x,l q is a solid BF-space on R n

5 WEIGHTED VARIABLE EXPONENT AMALGAM SPACES W(L p(x,l q 69 Proof Let K R n be a compact subset Since C 0 (R n is a regular Banach algebra ith respect to pointise multiplication one may choose a function h 0 C c (R n ith 0 h 0 (x and h 0 (x = for all x K Let supp(h 0 = K 0 Then χ K (x h 0 (x and hence χ K (x f(x h 0 (x f(x for all x R n Since L p(x (R n L loc (Rn, there exists D K0 > 0 such that (2 h 0 (xf(x dx D K0 h 0 f L p(x K 0 Since K K 0 (22 K f(x dx K 0 h 0 (xf(x dx On the other hand by Theorem 2 ii, W ( L p(x,l q p(x L loc (Rn Hence for this h 0 C c (R n there exists a constant number D h0 > 0 such that (23 p h0 (f = h 0 f L p(x D h0 f W(L p(x,l q for all f W ( L p(x,l q Combining (2, (22 and (23 e obtain f(x dx h 0 (xf(x dx D K0 h 0 f Lp(x K K 0 D K0 D h0 f W(L p(x,l q = C K f W(L p(x,l q It is easy to sho that W ( L p(x,l q is solid Proposition 23 Let, and 3 be eight functions Suppose that there exist constants C,C 2 > 0 such that and h L p(x (R n, k L p2(x (R n, hk L p 3 (x C h L p (x k L p 2 (x u L q (R n, v L q2 (R n, uv L q 3 3 C 2 u L q v L q 2 2 Then there exists C > 0 such that for all f W ( L p(x,l q and g W ( L p2(x,l q2 e have In other ords fg W(L p 3 (x,l q 3 3 C f W(L p (x,l q g W(L p 2 (x,l q 2 2 W ( L p(x,l q W ( ( L p2(x,l q2 W L p3(x,l q3 3

6 70 İ AYDIN AND A TURAN GÜRKANLI Proof Let f W ( ( L p(x,l q and g W L p 2(x,L q2 Then fg W(L p 3 (x,l q 3 3 = fgχ z+q L p 3 (x q L 3 3 (24 = (fχz+q (gχ z+q L p 3 (x q L 3 3 C fχz+q L p (x gχ z+q L p 2 (x If e put by (24 e obtain F f (z = fχ z+q L p (x and F g (z = gχ z+q L p 2 (x, q L 3 3 fg W(L p 3 (x,l q 3 3 C F f F g L q 3 3 C C 2 F f L q F g L q 2 2 = C C 2 f W(L p (x,l q g W(L p 2 (x,l q 2 2 = C f W(L p (x,l q g W(L p 2 (x,l q 2 2 Corollary 24 Define k(x by p(x + r(x = k(x and suppose k <, q + s = Then there exists a constant C > 0 such that fg W(L k(x,l C f W(L p(x,l q g W(L r(x,l s for all f W ( ( L p(x,l q and g W L r(x,l s Thus ( ( W L p(x,l q W (L r(x,l s W L k(x,l Proof Let f W ( L p(x,l q and g W ( L r(x,l s Then fχz+q L p(x (R n and gχ z+q L r(x (R n Thus there exists C(z > 0 such that (25 fgχ z+q L k(x C(z fχ z+q L p(x gχ z+q L r(x by [20, Lemma 28] Also it is knon by [20, Lemma 28]that C(z 2k = C < Since L (R n is solid, then by (25 fg W(L (26 k(x,l 2k fχ z+q L p(x gχ z+q L r(x L = C fχz+q L p(x gχ z+q L r(x L Finally since fχ z+q L p(x L q (Rn, gχ z+q L r(x L s (R n, by the Hölder inequality and (26 e obtain fg W(L k(x,l C f W(L p(x,l q g W(L r(x,l s Proposition 25 W a If p (x p 2 (x, q 2 q and, then ( ( L p2(x,l q2 W L p(x,l q

7 WEIGHTED VARIABLE EXPONENT AMALGAM SPACES W(L p(x,l q 7 b If p (x p 2 (x, q 2 q and, then ( ( W L p(x L p2(x,l q2 W L p(x,l q c If, then L p (R n W (L p(x,l p and W ( L p(x,l p L p (R n Proof a Let f W ( L p2(x,l q2 be given Since p (x p 2 (x, then L p2(x (z +Q L p(x (z +Q and (27 fχ z+q L p (x (µ(z +Q+ fχ z+q L p 2 (x = (µ(q+ fχ z+q L p 2 (x for all z R n by [9, Theorem 28], here µ is the Lebesgue measure Hence by (27 and the solidity of L q2 (R n e have ( ( W L p2(x,l q2 W L p(x,l q2 It is knon by [, Proposition 37], that ( ( W L p(x,l q2 W L p(x,l q if and only if l q2 l q, here l q2 and l q are the associated sequence spaces of L q2 (R n and L q (R n respectively Since q 2 q and, then l q2 l q ([3] This completes the proof b The proof of this part is easy by a c By using a and [6, Proposition 52], e have L p (R n = W Since, then l p l p Similarly e can prove (L p,l p W ([2] Hence L p (R n W W (L p(x,l p ( L p(x,l p L p (R n (L p(x,l p The folloing lemma follos directly from the closed graph theorem Lemma 26 If p,p 2 <, then L p(x (R n L p2(x (R n if and only if there exists a constant C > 0 such that f L p 2 (x C f L p (x for all f L p(x (R n Proposition 27 Let B be any solid space If q 2 q and, then e have W ( ( B,L q L q2 = W B,L q 2

8 72 İ AYDIN AND A TURAN GÜRKANLI Proof It is easy to see that the associated sequence space of L q (R n L q2 (R n isl q l q2 Sinceq 2 q and, thusthe associatedsequence space of L q (R n L q2 (R n is l q2 Then by [, Proposition 37] W ( B,L q L q2 = W ( B,L q 2 Corollary 28 a If p,p 2 <, L p(x (R n L p2(x (R n, q 2 q, q 4 q 3, q 4 q 2,, 3 4 and 4, then ( ( ( W L p(x,l q3 3 L q4 4 = W L p(x,l q4 4 W L p2(x,l q L q2 ( = W L p2(x,l q2 b If p (x p 2 (x, q q 2 and, then ( ( W L p(x L p2(x,l q W L p2(x,l q2 A general interpolation theorem in Wiener Amalgam space has been given by H Feichtinger (see [0, Theorem 22] We ill give a similar theorem for W ( L p(x,l q next: Proposition 29 If p 0 (x and p (x are variable exponents ith < p j, p j <, j = 0, Then, for θ (0,, e have [ ( ( ] ( [ W L p0(x,l q0 0,W L p(x,l q = W L p0(x,l p(x],l q θ [θ] [θ] ( = W L pθ(x,l q θ here p θ (x = θ p 0(x + θ p (x, q θ = θ q 0 + θ q, = θ 0 θ Proof By [0, Theorem 22] the interpolation space [ ( ( ] W L p0(x,l q0 0,W L p(x,l q for ( W ( ( ( L p0(x,l q0 0,W L p [L (x,l q isw p 0(x,L p(x],[ L q0 [θ] 0,L q ][θ] We kno that [ ] L q0 0,L q = L q θ [θ], [2] and by [6, Corollary A2] that [ L p 0(x,L p(x] [θ] = Lp θ(x [θ] 3 The Hardy-Littleood maximal function on W ( L p(x,l q (R n We use the notation B r (x to denote the open ball centered at x of radius r For a locally integrable function f on R, e define the (centered Hardy- Littleood maximal function Mf of f by (3 Mf(x = sup f(y dy µ(b r (x r>0 B r(x

9 WEIGHTED VARIABLE EXPONENT AMALGAM SPACES W(L p(x,l q 73 here the supremum is taken over all balls B r (x and µ(b r (x denotes the Lebesgue measure of B r (x Although the local Hardy-Littleood maximal function has been shon to be a bounded mapping on L p(x over a bounded domain, it is not bounded on many of the amalgam spaces We have the folloing result Proposition 3 Let p : R [,, q and is a eight function If Ls (R and q + s = then the Hardy-Littleood maximal function M is not bounded on W ( L p(x (R,L q (R Proof Since Ls (R and q + s = then Lq (R L (R Hence ( ( (32 W L p(x (R,L q (R W L p(x (R,L (R L (R Take the indicator function χ [,] It obvious by Theorem 2 iii that χ [,] W ( L p(x (R,L q (R By [2, Theorem ] the Hardy-Littleood maximal function f M (f is not bounded on L (R Also if f L (R is not identically zero then M (f is never integrable on R This implies that the Hardy-Littleood maximal function M ( χ [,] is not in L (R Hence M ( χ [,] / W ( L p(x (R,L q (R This completes the proof Acknoledgements The authors ould like to thank to H G Feichtinger for his various comments on earlier versions of this paper References [] I Aydın and A T Gürkanlı, On some properties of the spaces A p(x (R n, Proc Jangjeon Math Soc 2 (2009, 4 55 [2] J Bergh and J Löfström, Interpolation spaces An introduction, Springer-Verlag, Berlin, Heidelberg, Ne York, 976 [3] D Cruz Uribe and A Fiorenza, LlogL results for the maximal operator in variable L p spaces, Trans Amer Math Soc 36, (2009, [4] D Cruz Uribe, A Fiorenza, J M Martell and C Perez Moreno, The boundedness of classical operators on variable L p spaces, Ann Acad Sci Fenn Math 3 (2006, [5] L Diening, Maximal function on generalized Lebesgue spaces L p(, Math Inequal Appl 7 (2004, [6] L Diening, P Hästö and A Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, In FSDONA04 Proc (Milovy, Czech Republic, 2004, [7] L Diening, P Hästö, and S Roudenko, Function spaces of variable smoothness and integrability, J Funct Anal 256 (2009, [8] D Edmunds, J Lang, and A Nekvinda, On L p(x norms, R Soc Lond Proc Ser A Math Phys Eng Sci 455 (999, [9] H G Feichtinger, Banach convolution algebras of Wiener type, in Proc Conf functions, series, operators (Budapest, 980, Colloq Math Soc János Bolyai, North- Holland, 983,

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