WEIGHTED VARIABLE EXPONENT AMALGAM SPACES ˙Ismail ...

ˇ GLASNIK MATEMATICKI Vol. 47(67)(2012), 165 – 174

WEIGHTED VARIABLE EXPONENT AMALGAM SPACES W (Lp(x) , Lqw ) ˙ ¨rkanli Ismail Aydin and A. Turan Gu Sinop University and Ondokuz Mayıs University, Turkey

Abstract. In the present paper a new family of Wiener amalgam spaces W (Lp(x) , Lqw ) is defined, with local component which is a variable exponent Lebesgue space Lp(x) (Rn ) and the global component is a weighted Lebesgue space Lqw (Rn ) . We proceed to show that these Wiener amalgam spaces are Banach function spaces. We also present new H¨ oldertype inequalities and embeddings for these spaces. At the end of this paper we show that under some conditions the Hardy-Littlewood maximal function is not mapping the space W (Lp(x) , Lqw ) into itself.

1. Introduction A number of authors worked 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 was undertaken by F. Holland ([17, 18]). The amalgam of Lp and lq on the real line is the space (Lp , lq ) (R) (or shortly (Lp , lq ) ) consisting of functions f which are locally in Lp and have lq behavior at infinity in the sense that the norms over [n, n + 1] form an lq -sequence. For 1 ≤ p, q ≤ ∞ the norm   n+1  pq  1q Z ∞  X   p kf kp,q =  |f (x)| dx  < ∞ n=−∞

p

n

q

makes (L , l ) into a Banach space. If p = q then (Lp , lq ) reduces to Lp . A generalization of Wiener’s definition was given by H. G. Feichtinger in [9], 2010 Mathematics Subject Classification. 42B25,42B35. Key words and phrases. Variable exponent Lebesgue space, Hardy-Littlewood maximal function, Wiener amalgam space. 165

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˙ AYDIN AND A. TURAN GURKANLI ¨ I.

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 [16] gave a good summary of results concerning amalgam spaces with global components being weighted Lq (R) spaces. For a historical background of amalgams see [15]. Let p : Rn → [1, ∞) be a measurable function (called the variable exponent on Rn ). We put p∗ = ess inf p(x), n x∈R

p∗ = ess sup p(x). x∈Rn

The variable exponent Lebesgue space (or generalized Lebesgue space) Lp(x) (Rn ) is defined to be the space of measurable functions (equivalence classes) f such that Z p(x) dx < ∞ ρp (λf ) = |λf (x)| Rn

for some λ = λ (f ) > 0. The function ρp is called modular of the space Lp(x) (Rn ). Then kf kLp(x) = inf {λ > 0 : ρp (f /λ) ≤ 1} defines a norm (Luxemburg norm). This makes Lp(x) (Rn ) a Banach space. If p(x) = p is a constant function, then the variable exponent Lebesgue space Lp(x) (Rn ) coincides with the classical Lebesgue space Lp (Rn ), see [19]. Also there are recent many interesting and important papers appeared in variable exponent Lebesgue spaces (see [3–5, 7, 8]). In this paper we will assume that p∗ < ∞. The space L1loc (Rn ) consists of all (classes of ) measurable functions f on n R such that f χK ∈ L1 (Rn ) for any compact subset K ⊂ Rn , where χK is the characteristic function of K. It is a topological vector space with the family of seminorms f 7→ kf χK kL1 . A Banach function space (shortly BF-space) on Rn is a Banach space (B, k.kB ) of measurable functions which is continuously embedded into L1loc (Rn ), that is for any compact subset K ⊂ Rn there exists some constant CK > 0 such that kf χK kL1 ≤ CK kf kB for all f ∈ B and two functions equal almost everywhere are identified as usual. We denote it by B ֒→ L1loc (Rn ) . Obviously Lp(x) (Rn ) ֒→ L1loc (Rn ) and the space Lp(x) (Rn ) is a solid space, that is , if f ∈ Lp(x) (Rn ) is given and g ∈ L1loc (Rn ) satisfies |g(x)| ≤ |f (x)| a.e., then g ∈ Lp(x) (Rn ) and kgkLp(x) ≤ kf kLp(x) by [1, Lemma 1]. A positive, measurable and locally integrable function ϑ : Rn → (0, ∞) is called a weight function. We say that a weight function ϑ is submultiplicative if ϑ(x + y) ≤ ϑ(x)ϑ(y).

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for any x, y ∈ Rn . A weight function w is moderate with respect to a submultiplicative function ϑ (or ϑ-moderate) if w(x + y) ≤ w(x)ϑ(y) n

for any x, y ∈ R . If the weight w is moderate than 1/w is also moderate. We say that w1 ≺ w2 if there exists a constant C > 0 such that Cw1 (x) ≤ w2 (x) for all x ∈ Rn . Two weight functions are called equivalent and written w1 ≈ w2 , if w1 ≺ w2 and w2 ≺ w1 . The space Lqw (Rn ) (weighted Lq (Rn )) is the space of all complex-valued measurable functions on Rn for which f w ∈
q n q n L (R ). Obviously Lw (R ) , k.kLqw is a Banach space with the norm kf kLqw = kf wkLq =

 Z 

Rn

or

 1q  q , 1 ≤ q < ∞, |f (x)w(x)| dx 

kf kL∞ = kf wkL∞ = ess sup |f (x)| w(x), q = ∞. w

x∈Rn

Lqw

n

Also the dual of the space (R ) is the space Lsw−1 (Rn ) , where 1 ≤ q < ∞, 1 1 q + s = 1 (see [12, 14, 16]). Given a discrete family X = (xi )i∈I in Rn and a weighted space Lqw (Rn ) , the associated weighted sequence space over X is the appropriate weighted ℓq space ℓqw , the discrete w being given by w(i) = w(xi ) for i ∈ I (see [11, Lemma 3.5]). 2. The Wiener Amalgam Space W Lp(x) , Lqw




Let Cb (Rn ) be the the regular Banach algebra (with respect to pointwise multiplication) of complex-valued bounded, continuous functions on Rn . Also let C0 (Rn ) , Cc (Rn ) be the spaces of complex-valued continuous function Rn vanishing at infinity and the space of complex-valued continuous functions with compact support defined on Rn endowed with its natural inductive limit topology respectively. It is known that (C0 , k.k∞ ) ֒→ (Cb , k.k∞ ) and the dual space of Cc (Rn ) (with respect to its natural inductive limit topology) is M (Rn ) , the space of regular Borel measures. For every h ∈ Cc (Rn ) we define the semi-norm qh on M (Rn ) by qh (hp ) = hp (h) . The locally convex topology on M (Rn ) defined by the family (qh )h∈Cc (Rn ) of seminorms is called the topology σ (M (Rn ) , Cc (Rn )) or weak ∗ -topology, also called vague topology. We define n o p(x) Lloc (Rn ) = σ ∈ M (Rn ) : φσ ∈ Lp(x) (Rn ) for all φ ∈ Cc (Rn ) . p(x)

Lloc (Rn ) is a topological vector space with respect to the family of seminorms given by kσkφ = kφσkLp(x) , φ ∈ Cc (Rn ).

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It is known by [1, Lemma 1] that Lp(x) (Rn ) is continuously embedded into p(x) L1loc (Rn ). Hence it is easily shown that Lloc (Rn ) is continuously embedded into L1loc (Rn ). It is also obvious that L1loc (Rn ) is continuously embedded p(x) into M (Rn ) with the weak ∗ -topology. Therefore Lloc (Rn ) is continuously n embedded into M (R ). Since the general hypotheses for the Wiener
amalgam space denoted by
W Lp(x) (Rn ) , Lqw (Rn ) (shortly W Lp(x) , Lqw ) are satisfied, it is defined as follows as in [9]. Let fix an open set Q ⊂ Rn with compact closure. The Wiener amalgam
p(x) space W Lp(x) , Lqw consists of all elements f ∈ Lloc (Rn ) such that Ff (z) =
kf kLp(x) (z+Q) belongs to Lqw (Rn ); the norm of W Lp(x) , Lqw is kf kW (Lp(x) ,Lqw ) = kFf kLqw .

In this definition kf kLp(x) (z+Q) denotes the restriction norm of f to z + Q, that is kf kLp(x) (z+Q)  = inf kgkLp(x) :

g ∈ Lp(x) (Rn ) , g coincides with f on z + Q, i.e., hf = hg for all h ∈ CC (Rn ) with supp (h) ⊂ z + Q



.

By the solidity of the BF-space the assumptions imply kf kLp(x) (z+Q) = kf χz+Q kLp(x) . The following theorem,
based on [9, Theorem 1], describes the basic properties of W Lp(x) , Lqw . Theorem 2.1.


i) W Lp(x) , Lqw is a Banach space with norm k.kW (Lp(x) ,Lqw ) .
p(x) ii) W Lp(x) , Lqw is continuously embedded into Lloc (Rn ). iii) The space n o Λ0 = f ∈ Lp(x) (Rn ) : supp (f ) is compact
is continuously embedded into W Lp(x) , Lqw .
iv) W Lp(x) , Lqw does not depend on the particular choice of Q, i.e. different choices of Q define the same space with equivalent norms. n By iii) and [1, Lemma 4]
it is easy to see that Cc (R ) is continuously p(x) q embedded into W L , Lw . By using the techniques in [13], we prove the following proposition.
Proposition 2.2. W Lp(x) , Lqw is a solid BF-space on Rn .

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Proof. Let K ⊂ Rn be a compact subset. Since C0 (Rn ) is a regular Banach algebra with respect to pointwise multiplication one may choose a function h0 ∈ Cc (Rn ) with 0 ≤ h0 (x) ≤ 1 and h0 (x) = 1 for all x ∈ K. Let supp(h0 ) = K0 . Then χK (x) ≤ h0 (x) and hence χK (x) |f (x)| ≤ h0 (x) |f (x)| for all x ∈ Rn . Since Lp(x) (Rn ) ֒→ L1loc (Rn ), there exists DK0 > 0 such that Z |h0 (x)f (x)| dx ≤ DK0 kh0 f kLp(x) . (2.1) K0

Since K ⊂ K0 Z

(2.2)

|f (x)| dx ≤

K

Z

|h0 (x)f (x)| dx.

K0


p(x) On the other hand by Theorem 2.1 ii), W Lp(x) , Lqw ֒→ Lloc (Rn ). Hence for this h0 ∈ Cc (Rn ) there exists a constant number Dh0 > 0 such that ph0 (f ) = kh0 f kLp(x) ≤ Dh0 kf kW (Lp(x) ,Lqw )

(2.3)


for all f ∈ W Lp(x) , Lqw . Combining (2.1), (2.2) and (2.3) we obtain Z Z |h0 (x)f (x)| dx ≤ DK0 kh0 f kLp(x) |f (x)| dx ≤ K

K0

≤ DK0 Dh0 kf kW (Lp(x) ,Lqw ) = CK kf kW (Lp(x) ,Lqw ) .
It is easy to show that W Lp(x) , Lqw is solid. Proposition 2.3. Let w1 , w2 and w3 be weight functions. Suppose that there exist constants C1 , C2 > 0 such that ∀h ∈ Lp1 (x) (Rn ) , ∀k ∈ Lp2 (x) (Rn ) ,

khkkLp3 (x) ≤ C1 khkLp1 (x) kkkLp2 (x)

and ∀u ∈ Lqw11 (Rn ) , ∀v ∈ Lqw22 (Rn ) ,

kuvkLqw3 ≤ C2 kukLqw1 kvkLqw2 .

Then there exists C > 0 such that for all f ∈ W L
W Lp2 (x) , Lqw22 we have

2

1

3

p1 (x)

, Lqw11




kf gkW (Lp3 (x) ,Lqw3 ) ≤ C kf kW (Lp1 (x) ,Lqw1 ) kgkW (Lp2 (x) ,Lqw2 ) . 3

1

2

In other words       W Lp1 (x) , Lqw11 .W Lp2 (x) , Lqw22 ⊂ W Lp3 (x) , Lqw33 .

and g ∈

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Proof. Let f ∈ W Lp1 (x) , Lqw11 and g ∈ W Lp2 (x) , Lqw22 . Then

kf gkW (Lp3 (x) ,Lqw3 ) = kf gχz+Q kLp3 (x) Lq3 w3 3



= k(f χ ) (gχ )k z+Q z+Q Lp3 (x) Lq3 (2.4) w3

≤ C1 kf χz+Q kLp1 (x) kgχz+Q kLp2 (x) Lq3 . w3

If we put

Ff (z) = kf χz+Q kLp1 (x) and Fg (z) = kgχz+Q kLp2 (x) , by (2.4) we obtain kf gkW (Lp3 (x) ,Lqw3 ) ≤ C1 kFf Fg kLqw3 ≤ C1 C2 kFf kLqw1 kFg kLqw2 3

3

2

1

= C1 C2 kf kW (Lp1 (x) ,Lqw1 ) kgkW (Lp2 (x) ,Lqw2 ) 2

1

= C kf kW (Lp1 (x) ,Lqw1 ) kgkW (Lp2 (x) ,Lqw2 ) . 2

1

1 1 1 Corollary 2.4. Define k (x) by p(x) + r(x) = k(x) ≤ 1 and suppose 1 1 ∗ k < ∞, q + s = 1. Then there exists a constant C > 0 such that

kf gkW (Lk(x) ,L1 ) ≤ C kf kW (Lp(x) ,Lqw ) kgkW (Lr(x) ,Ls ) w−1

r(x) s p(x) q , Lw−1 . Thus for all f ∈ W L , Lw and g ∈ W L       W Lp(x) , Lqw W Lr(x) , Lsw−1 ⊂ W Lk(x) , L1 .

Proof. Let f ∈ W Lp(x) , Lqw and g ∈ W Lr(x) , Lsw−1 . Then f χz+Q ∈ Lp(x) (Rn ) and gχz+Q ∈ Lr(x) (Rn ). Thus there exists C(z) > 0 such that (2.5)

kf gχz+Q kLk(x) ≤ C(z) kf χz+Q kLp(x) kgχz+Q kLr(x)

by [20, Lemma 2.18]. Also it is known by [20, Lemma 2.18] that C(z) ≤ 2k ∗ = C < ∞. Since L1 (Rn ) is solid, then by (2.5)

kf gk ≤ 2k ∗ kf χz+Q kLp(x) kgχz+Q kLr(x) L1 W (Lk(x) ,L1 ) (2.6)

= C kf χz+Q kLp(x) kgχz+Q kLr(x) L1 .

Finally since kf χz+Q kLp(x) ∈ Lqw (Rn ), kgχz+Q kLr(x) ∈ Lsw−1 (Rn ), by the H¨older inequality and (2.6) we obtain kf gkW (Lk(x) ,L1 ) ≤ C kf kW (Lp(x) ,Lqw ) kgkW (Lr(x) ,Ls

w−1

Proposition 2.5.

).

a) If p1 (x) ≤ p2 (x), q2 ≤ q1 and w1 ≺ w2 , then    W Lp2 (x) , Lqw22 ⊂ W Lp1 (x) , Lqw11 . 

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b) If p1 (x) ≤ p2 (x), q2 ≤ q1 and w1 ≺ w2 , then     W Lp1 (x) ∩ Lp2 (x) , Lqw22 ⊂ W Lp1 (x) , Lqw11 .

c) If w1 ≺ w2 , then     ∗ ∗ Lpw2 (Rn ) ⊂ W Lp(x) , Lpw1 and W Lp(x) , Lpw∗2 ⊂ Lpw∗1 (Rn ) .
Proof. a) Let f ∈ W Lp2 (x) , Lqw2 be given. Since p1 (x) ≤ p2 (x), then Lp2 (x) (z + Q) ֒→ Lp1 (x) (z + Q) and (2.7)

kf χz+Q kLp1 (x) ≤ (µ (z + Q) + 1) kf χz+Q kLp2 (x) =

(µ (Q) + 1) kf χz+Q kLp2 (x)

for all z ∈ Rn by [19, Theorem 2.8], where µ is the Lebesgue measure. Hence by (2.7) and the solidity of Lqw22 (Rn ) we have     W Lp2 (x) , Lqw22 ⊂ W Lp1 (x) , Lqw22 . It is known by [11, Proposition 3.7], that     W Lp1 (x) , Lqw22 ⊂ W Lp1 (x) , Lqw11

if and only if ℓqw22 ⊂ ℓqw11 , where ℓqw22 and ℓqw11 are the associated sequence spaces of Lqw22 (Rn ) and Lqw11 (Rn ) respectively. Since q2 ≤ q1 and w1 ≺ w2 , then ℓqw22 ⊂ ℓqw11 ([13]). This completes the proof. b) The proof of this part is easy by a). c) By using a) and [16, Proposition 11.5.2], we have  ∗    ∗ ∗ ∗ Lpw2 (Rn ) = W Lp , Lpw2 ⊂ W Lp(x) , Lpw2 . ∗

∗

Since w1 ≺ w2 , then ℓpw2 ⊂ ℓpw1 ([12]). Hence   ∗ ∗ Lpw2 (Rn ) ⊂ W Lp(x) , Lpw1 .

Similarly we can prove

  W Lp(x) , Lpw∗2 ⊂ Lpw∗1 (Rn ) . The following lemma follows directly from the closed graph theorem. Lemma 2.6. If p∗1 , p∗2 < ∞, then Lp1 (x) (Rn ) ⊂ Lp2 (x) (Rn ) if and only if there exists a constant C > 0 such that kf kLp2 (x) ≤ C kf kLp1 (x) for all f ∈ Lp1 (x) (Rn ). Proposition 2.7. Let B be any solid space. If q2 ≤ q1 and w1 ≺ w2 , then we have

W B, Lqw11 ∩ Lqw22 = W B, Lqw22 .

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Proof. It is easy to see that the associated sequence space of Lqw11 (Rn ) ∩ (Rn ) is ℓqw11 ∩ℓqw22 . Since q2 ≤ q1 and w1 ≺ w2 , thus the associated sequence space of Lqw11 (Rn ) ∩ Lqw22 (Rn ) is ℓqw22 . Then by [11, Proposition 3.7]

W B, Lqw11 ∩ Lqw22 = W B, Lqw22 . Lqw22

Corollary 2.8. a) If p∗1 , p∗2 < ∞, Lp1 (x) (Rn ) ⊂ Lp2 (x) (Rn ), q2 ≤ q1 , q4 ≤ q3 , q4 ≤ q2 , w1 ≺ w2 , w3 ≺ w4 and w2 ≺ w4 , then       W Lp1 (x) , Lqw33 ∩ Lqw44 = W Lp1 (x) , Lqw44 ⊂ W Lp2 (x) , Lqw11 ∩ Lqw22   = W Lp2 (x) , Lqw22 . b) If p1 (x) ≤ p2 (x), q1 ≤ q2 and w2 ≺ w1 , then     1 W Lp1 (x) ∩ Lp2 (x) , Lqw1 ⊂ W Lp2 (x) , Lqw22 .

A general interpolation theorem in Wiener Amalgam space has been given by H. Feichtinger (see [10, Theorem 2.2]). We will give a similar theorem for
W Lp(x) , Lqw next:

Proposition 2.9. If p0 (x) and p1 (x) are variable exponents with 1 < pj,∗ ≤ p∗j < ∞, j = 0, 1. Then, for θ ∈ (0, 1) , we have  h i i   h  qθ p0 (x) p1 (x) p1 (x) q1 p0 (x) q0 , Lw =W L ,L , Lw1 W L , Lw0 , W L [θ] [θ]   = W Lpθ (x) , Lqwθ where

1 pθ (x)

=

1−θ p0 (x)

+

θ 1 p1 (x) , qθ

=

1−θ q0

+

θ q1 ,

w = w01−θ w1θ .

Proof. By [10, Theorem 2.2] the interpolation space i   h  W Lp0 (x) , Lqw00 , W Lp1 (x) , Lqw11 [θ] 


    for W Lp0 (x) , Lqw00 , W Lp1 (x) , Lqw11 is W Lp0 (x) , Lp1 (x) [θ] , Lqw00 , Lqw11 [θ] .   We know that Lqw00 , Lqw11 [θ] = Lqwθ , [2] and by [6, Corollary A.2] that  p (x) p (x)  L 0 ,L 1 = Lpθ (x) . [θ]
3. The Hardy-Littlewood maximal function on W Lp(x) , Lqw (Rn )

We use the notation Br (x) to denote the open ball centered at x of radius r. For a locally integrable function f on R, we define the (centered) HardyLittlewood maximal function M f of f by Z 1 (3.1) M f (x) = sup |f (y)| dy. r>0 µ (Br (x)) Br (x)

WEIGHTED VARIABLE EXPONENT AMALGAM SPACES W (Lp(x) , Lqw )

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where the supremum is taken over all balls Br (x) and µ (Br (x)) denotes the Lebesgue measure of Br (x). Although the local Hardy-Littlewood maximal function has been shown to be a bounded mapping on Lp(x) over a bounded domain, it is not bounded on many of the amalgam spaces. We have the following result. Proposition 3.1. Let p : R → [1, ∞), 1 ≤ q ≤ ∞ and w is a weight function. If w1 ∈ Ls (R) and 1q + 1s = 1 then the Hardy-Littlewood maximal
function M is not bounded on W Lp(x) (R) , Lqw (R) .

Proof. Since w1 ∈ Ls (R) and 1q + 1s = 1 then Lqw (R) ⊂ L1 (R) . Hence     (3.2) W Lp(x) (R) , Lqw (R) ⊂ W Lp(x) (R) , L1 (R) ⊂ L1 (R) .

Take the indicator function χ[−1,1] . It obvious by Theorem 2.1 iii) that
χ[−1,1] ∈ W Lp(x) (R) , Lqw (R) . By [21, Theorem 1] the Hardy-Littlewood maximal function f → M (f ) is not bounded on L1 (R) . Also if f ∈ L1 (R) is not identically zero then M (f ) is never integrable
on R. This implies that the Hardy-Littlewood maximal function M χ[−1,1] is not in L1 (R) . Hence

M χ[−1,1] ∈ / W Lp(x) (R) , Lqw (R) . This completes the proof. Acknowledgements. The authors would like to thank to H. G. Feichtinger for his various comments on earlier versions of this paper. References p(x)

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