VECTOR-VALUED INEQUALITIES ON HERZ SPACES AND CHARACTERIZATIONS OF HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT. Mitsuo Izuki Hokkaido University, Japan

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1 GLASNIK MATEMATIČKI Vol 45(65)(2010), VECTOR-VALUED INEQUALITIES ON HERZ SPACES AND CHARACTERIZATIONS OF HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT Mitsuo Izuki Hokkaido University, Japan Abstract Our first aim in this paper is to prove the vector-valued inequalities for some sublinear operators on Herz spaces with variable exponent As an application, we obtain some equivalent norms and wavelet characterization of Herz Sobolev spaces with variable exponent 1 Introduction The origin of Herz spaces is the study of characterization of functions and multipliers on the classical Hardy spaces ([1, 8]) By virtue of many authors works Herz spaces have became one of the remarkable classes of function spaces in harmonic analysis now One of the important problems on the spaces is boundedness of sublinear operators satisfying proper conditions Hernández, Li, Lu and Yang ([7,17,19]) have proved that if a sublinear operator T is bounded on L p (R n ) and satisfies the size condition Tf(x) C x y n f(y) dy R n for all f L 1 (R n ) with compact support and ae x / suppf, then T is bounded on both of the homogeneous Herz space K p (R n ) and the nonhomogeneous Herz space Kp (R n ) This result is extended to the weighted case by Lu, Yabuta and Yang ([18]), and to the vector-valued case by Tang and Yang ([25]) As an application of [18], noting that the Hardy Littlewood maximal operator M is a sublinear operator satisfying the assumption above, 2010 Mathematics Subject Classification 42B20, 42B35, 42C40, 46B15 Key words and phrases Herz Sobolev space with variable exponent, wavelet, vectorvalued inequality 475

2 476 M IZUKI Nakai, Tomita and Yabuta ([22]) have proved the density of the set of all infinitely differentiable functions with compact support in weighted Herz spaces On the other hand, Lu and Yang ([20]) have initially introduced Herz-type Sobolev spaces and Bessel potential spaces, and proved the equivalence of them using boundedness of sublinear operators on Herz spaces Later Xu and Yang ([27, 28]) generalize the result on Herz-type Sobolev spaces to the case of Herz-type Triebel Lizorkin spaces Many results on wavelet characterization of various function spaces including Herz spaces are well-known now (cf [5, 6, 9, 10, 12, 14, 21]) The first wavelet characterization of Herz spaces is proved by Hernández, Weiss and Yang ([6]) The author and Tachizawa ([12]) have obtained the characterizations of weighted Herz spaces applying the boundedness of sublinear operators ([18]) Recently the author ([9]) and Kopaliani ([14]) have independently proved wavelet characterization of Lebesgue spaces with variable exponent L p( ) (R n ) by virtue of the extrapolation theorem due to Cruz-Uribe, Fiorenza, Martell and Pérez ([2]) Herz spaces with variable exponent are initially defined by the author in the paper [10] where he uses the Haar functions to obtain wavelet characterization of those spaces by virtue of the result on L p( ) (R n ) ([9,14]) He also gives wavelet characterization of non-homogeneous Herz Sobolev spaces with variable exponent in terms of wavelets with proper smoothness and compact support ([11]) But the homogeneous case is not considered in [10, 11] where he follows an argument applicable only to the non-homogeneous case using local properties of wavelets In the present paper we will prove the vector-valued inequalities for sublinear operators on Herz spaces with variable exponent K and K Additionally we apply the result to obtain Littlewood-Paley type and wavelet characterization of Herz Sobolev spaces with variable exponent K p( ) W s (R n ) and K p( ) W s (R n ) We note that our method is applicable to both the homogeneous and the non-homogeneous cases Let us explain the outline of this article We first define function spaces with variable exponent in Section 2 We will state properties of variable exponent in Section 3 In Section 4 we prove the vector-valued inequalities for sublinear operators on K and K Applying the results, we give Littlewood Paley-type characterization of K p( ) W s (R n ) and K p( ) W s (R n ) in Section 5 We additionally prove wavelet characterization of them in terms of the Meyer scaling function and the Meyer wavelets in Section 6 Throughout this paper S denotes the Lebesgue measure and χ S means the characteristic function for a measurable set S R n The set of all nonnegative integers is denoted by N 0 A symbol C always means a positive constant independent of the main parameters and may change from one occurrence to another The Fourier transform of a function f on R n is denoted

3 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 477 by Ff(ξ) = ˆf(ξ) := (2π) n/2 R n f(x)e ix ξ dx 2 Definition of function spaces with variable exponent In this section we define some function spaces with variable exponent Let E be a measurable set in R n with E > 0 by L 1 loc Definition 21 Let p( ) : E [1, ) be a measurable function 1 The Lebesgue space with variable exponent L p( ) (E) is defined by L p( ) (E) := f is measurable : ρ p (f/λ) < for some constant λ > 0, where ρ p (f) := E f(x) p(x) dx 2 The space L p( ) loc (E) is defined by L p( ) loc (E) := f : f L p( ) (K) for all compact subsets K E The Lebesgue space L p( ) (E) is a Banach space with the norm defined f L p( ) (E) := inf λ > 0 : ρ p (f/λ) 1 Now we define two classes of exponent functions Given a function f (E), the Hardy Littlewood maximal operator M is defined by Mf(x) := sup r n f(y) dy (x E), r>0 B(x,r) E where B(x, r) := y R n : x y < r We also use the following notation (21) (22) p := ess infp(x) : x E, Definition 22 P(E) := B(E) := p + := esssupp(x) : x E p( ) : E [1, ) : p > 1 and p + < p( ) P(E) : M is bounded on L p( ) (E) Later we will state some properties of variable exponent Now we define Herz and Herz Sobolev spaces with variable exponent We use the following notation in order to define them Let l Z (23) (24) (25) (26) B l := x R n : x 2 l R l := B l \ B l 1 χ l := χ Rl χ l := χ Rl if l 1 and χ 0 := χ B0 We first define Herz spaces with variable exponent by analogy with the definition of the usual Herz spaces (cf [6,7,17,19])

4 478 M IZUKI Definition 23 Let α R, 0 < q < and p( ) P(R n ) 1 The homogeneous Herz space K is defined by where K := f K := f L p( ) loc (Rn \ 0) : f K <, = 2 αl f χ l L p( ) (R n ) ( l= l= l q (Z) 2 αlq f χ l q L p( ) (R n )) 1/q 2 The non-homogeneous Herz space K is defined by K := f L p( ) loc (Rn ) : f K <, where f K := = 2 αl f χ l L p( ) (R n ) l=0 l q (N 0) ( 1/q 2 αlq f χ l q L p( ) (R )) n l=0 The Herz spaces K and K are quasi-norm spaces with K and K respectively Next we define Herz Sobolev spaces with variable exponent Definition 24 Let α R, 0 < q <, p( ) P(R n ) and s N 0 K 1 The homogeneous Herz Sobolev space p( ) W s (R n ) is the space of all functions f K satisfying weak derivatives D γ f K for all γ N n 0 with γ s 2 The non-homogeneous Herz Sobolev space K p( ) W s (R n ) is the space of all functions f K satisfying weak derivatives D γ f K for all γ N n 0 with γ s The Herz Sobolev spaces for constant p are initially defined by Lu and Yang ([20]) Both of K p( ) W s (R n ) and K p( ) W s (R n ) are also quasi-norm spaces with respectively f K p( ) W s (R n ) := γ s f K p( ) W s (R n ) := γ s D γ f K, D γ f K

5 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT Properties of variable exponent Cruz-Uribe, Fiorenza and Neugebauer ([3]) and Nekvinda ([23]) proved the following sufficient conditions independently We remark that Nekvinda ([23]) gave a more general condition in place of (32) (31) (32) Proposition 31 Suppose that E is an open set If p( ) P(E) satisfies p(x) p(y) p(x) p(y) C log( x y ) C log(e + x ) if x y 1/2, if y x, where C > 0 is a constant independent of x and y, then we have p( ) B(E) Below p ( ) means the conjugate exponent of p( ), namely 1/p(x) + 1/p (x) = 1 holds Then Y denotes all families of disjoint and open cubes in R n The following propositions are due to Diening ([4, Theorem 81 and Lemma 55]) We remark that Diening has proved general results on Musielak Orlicz spaces We describe them for Lebesgue spaces with variable exponent Proposition 32 Suppose p( ) P(R n ) Then the following conditions are equivalent (I) There exists a constant C > 0 such that for all Y Y and all f L p( ) (R n ), (33) f Q χ Q C f L p( ) (R ) n Q Y L p( ) (R n ) (II) p( ) B(R n ) (III) p ( ) B(R n ) (IV) There exists a constant p 0 (1, p ) such that p 1 0 p( ) B(Rn ) Proposition 33 Let p( ) P(R n ) If p( ) satisfies condition (I) in Proposition 32, then there exist two positive constants δ and C such that for all Y Y, all non-negative numbers t Q and all f L 1 loc (Rn ) with f Q := 1 Q (34) Q f(x)dx 0 (Q Y ), f δ t Q f Q χ Q Q Y L p( ) (R n ) C t Q χ Q Q Y Propositions 32 and 33 lead the following corollary L p( ) (R n ) Corollary 34 Let p( ) P(R n ) If p( ) satisfies condition (I) in Proposition 32, then there exists a positive constant C such that for all balls

6 480 M IZUKI B in R n and all measurable subsets S B, (35) (36) χ B L p( ) (R n ) χ S L p( ) (R n ) χ S L p( ) (R n ) χ B L p( ) (R n ) C B S, ( ) δ S C, B where δ is the constant appearing in (34) Proof Take a ball B and a measurable subset S B arbitrarily We first show (35) By Proposition 32, p( ) belongs to B(R n ) Thus M satisfies the weak (p( ), p( )) inequality, ie, for all f L p( ) (R n ) and all λ > 0 we have λ χ Mf(x)>λ L p( ) (R n ) C f L p( ) (R n ) If we take λ (0, S / B ) arbitrarily, then we get namely λ χ B L p( ) (R n ) λ χ M(χS)(x)>λ L p( ) (R n ) C χ S L p( ) (R n ), χ B L p( ) (R n ) Cλ 1 χ S L p( ) (R n ) Since λ is arbitrary, we obtain (35) Next we prove (36) We can take a open cube Q B so that B Q B nb Putting f = χ S and Y = Q B in (34), we get ( ) χ S δ L p( ) (R n ) S C χ QB L p( ) (R n ) Q B By virtue of B Q B nb and (35), we see that χ S L p( ) (R n ) χ B L p( ) (R n ) = χ S L p( ) (R n ) χ Q B L p( ) (R n ) χ QB L p( ) (R n ) χ B L p( ) (R n ) χ S L p( ) (R n ) χ L nb p( ) (R n ) χ QB L p( ) (R n ) ( S C Q B ) δ nb B χ B L p( ) (R n ) ( ) δ S = C C Q B ( ) δ S B The next lemma describes the generalized Hölder inequality and the duality of L p( ) (E) The proof is found in [15] Lemma 35 Suppose p( ) P(E) Then the following hold

7 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT For all f L p( ) (E) and all g L p ( ) (E) we have (37) f(x)g(x) dx r p f L p( (E) g ) L p ( ) (E), E where r p := 1 + 1/p 1/p + 2 For all f L p( ) (E) we have (38) f L p( ) (E) sup f(x)g(x) dx : g L p ( ) (E) 1 E In particular, L p( ) (E) coincides with the dual space of L p ( ) (E) and the norm f L p( ) (E) is equivalent to the value sup f(x)g(x) dx : g L p ( ) (E) 1 E Remark 36 Below we write u r ( ) := r 1 p( ) for positive constant r > 0 1 If p( ) B(R n ), then Corollary 34 implies that there exists a positive constant δ 1 such that for all balls B in R n and all measurable subsets S B, (39) χ S L p( ) (R n ) χ B L p( ) (R n ) C ( ) δ1 S B (310) Thus we have that for all r > 0, χ S L ur( ) (R n ) χ B L ur( ) (R n ) = χ S r 1 L p( ) (R n ) C χ B r 1 L p( ) (R n ) ( ) r S 1 δ 1 B 2 If p( ) P(R n ) satisfies (31) and (32) in Proposition 31, then p ( ), u r ( ) and u r ( ) also belong to P(Rn ) and satisfy (31) and (32) for all 0 < r < p Because they are in B(R n ), we can take constants δ 2, δ(r) > 0 such that for all balls B in R n and all measurable subsets S B, (311) (312) χ S L p ( ) (R n ) χ B L p ( ) (R n ) χ S L u r ( ) (R n ) χ B L u r ( ) (R n ) ( ) δ2 S C, B ( ) δ(r) S C B 3 If p( ) B(R n ), then Proposition 32 implies that p ( ), u p0 B(R n ) Thus (311) holds In addition (310) and (312) are true with r = p 0 Applying Lemma 35 we obtain the following

8 482 M IZUKI Lemma 37 Let p( ) P(R n ) If p( ) satisfies condition (I) in Proposition 32, then there exists a constant C > 0 such that for all balls B in R n, 1 B χ B L p( ) (R n ) χ B L p ( ) (R n ) C Proof The assumption shows that f Q χ Q L p( ) (R n ) C f χ Q L p( ) (R n ) for all cube Q and all f L p( ) (R n ) Using (38) we obtain 1 Q χ Q L p( ) (R ) χ n Q L p ( ) (R n ) 1 Q χ Q L p( ) (R n ) sup f(x)χ Q (x) dx : f L p( ) (R n ) 1 R n = sup f Q χ Q L p( ) (R n ) : f L p( ) (R n ) 1 C sup f χ Q L p( ) (R n ) : f L p( ) (R n ) 1 C For each ball B we can take a cube Q B such that n 1/2 Q B B Q B Thus we get 1 B χ B L p( ) (R n ) χ B L p ( ) (R n ) C 1 Q B χ Q B L p( ) (R n ) χ QB L p ( ) (R n ) C 4 Vector-valued inequalities for sublinear operators In this section we prove the vector-valued inequalities for some sublinear operators on Herz spaces with variable exponent under proper assumptions Theorem 41 Let 1 < r <, p( ) B(R n ), 0 < q < and nδ 1 < α < nδ 2, where δ 1, δ 2 > 0 are the constants appearing in (39) and (311) Suppose that T is a sublinear operator satisfying the vector-valued inequality on L p( ) (R n ) ( ) 1/r ( ) 1/r (41) Tg h r C g h r h=1 L p( ) (R n ) h=1 L p( ) (R n ) for all sequences of functions g h h=1 satisfying g h h l r L p( ) (R n ) <, and size condition (42) Tf(x) C x y n f(y) dy R n

9 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 483 for all f L 1 (R n ) with compact support and ae x / suppf Then we have the vector-valued inequality on K (43) ( ) 1/r Tf h r h=1 K ( ) 1/r C f h r h=1 K for all sequences of functions f h h=1 satisfying f h h l r K < p( ) Moreover T also satisfies the same vector-valued inequality as (43) on K Remark 42 1 There do exist some operators satisfying vectorvalued inequality (41) provided p( ) B(R n ), for example, the Hardy Littlewood maximal operator and singular integral operators (see [2]) 2 Theorem 41 is known when p( ) equals to a constant p (1, ) Hernández, Li, Lu and Yang ([7,17,19]) have proved (43) on the usual Herz spaces provided n/p < α < n/p In order to prove Theorem 41, we additionally introduce the next lemma which is well-known as the generalized Minkowski inequality Lemma 43 If 1 < r <, then there exists a constant C > 0 such that for all sequences of functions f h h=1 satisfying f h h l r L 1 (R n ) <, (44) ( ) r 1/r f h (y) dy C R n h=1 1/r f h (y) r dy R n h=1 Proof of Theorem 41 Our method is based on [18, Proof of Theorem 1] and [25] We give the proof for the homogeneous case while the non-homogeneous case is similar Because T is sublinear, we have that Tf h h l r K = 2 αqj χ j Tf h h l r q L p( ) (R n ) j= j= 2 αqj χ j l= T(f h χ l ) h l r 1/q q L p( ) (R n ) 1/q,

10 484 M IZUKI for every f h h=1 with f h h l r K < Thus we can decompose as follows Tf h h l r K ( j 2 ) q C 2 αqj χ j T(f h χ l ) h l r L p( ) (R n ) +C +C j= j= j= 2 αqj 2 αqj =: E 1 + E 2 + E 3 l= j+1 l=j 1 l=j+2 1/q q χ j T(f h χ l ) h l r L p( ) (R n ) q χ j T(f h χ l ) h l r L p( ) (R n ) For convenience below we denote F := f h h l r Since T satisfies (41), we can easily obtain E 2 C F K We consider the estimates of E 1 and E 3 For each j Z and l j 2 and ae x R j, size condition (42), generalized Minkowski s inequality (44) and generalized Hölder s inequality (37) imply (45) T(f h χ l )(x) h l r C x y n f h (y) dy R l h C 2 R jn f h (y) dy l C 2 jn Fχ l L p( ) (R n ) χ l L p ( ) (R n ) On the other hand, Proposition 32 and Lemma 37 lead h l r l r 1/q 1/q 2 jn χ j L p( ) (R n ) χ l L p ( ) (R n ) 2 jn χbjl p( ) (R n ) χ B l L p ( ) (R n ) C 2 jn B j χ Bj 1 L p ( χ ) (R n ) Bl L p ( ) (R n ) (46) = C χ B l L p ( ) (R n ) χ L C 2 nδ2(l j) Bj p ( ) (R n )

11 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 485 Applying (45) and (46) to E 1, we get E 1 C j= ( j 2 2 αqj l= χ j L p( ) (R n ) ) q 1/q (47) C j= ( j 2 l= 2 jn F χ l L p( ) (R n ) χ l L p ( ) (R n ) ) q 2 αl F χ l L p( ) (R )2 b(l j) n 1/q, where b := nδ 2 α > 0 Similarly we obtain (48) E 3 C q 2 αl F χ l L p( ) (R )2 d(j l) n j= l=j+2 1/q, where d := nδ 1 + α > 0 To continue calculations for (47) and (48), we consider the two cases 1 < q < and 0 < q 1 If 1 < q <, then we use Hölder s inequality and obtain E 1 C = C = C = C j= ( j 2 l= j 2 j= l= l= l= = C F K 2 αql F χ l q L p( ) (R n ) 2bq(l j)/2 ) ( j 2 l= 2 bq (l j)/2 2 αql F χ l q L p( ) (R n ) 2bq(l j)/2 2 αql F χ l q L p( ) (R n ) 2 αql F χ l q L p( ) (R n ) j=l+2 1/q 2 bq(l j)/2 ) q/q 1/q 1/q 1/q

12 486 M IZUKI If 0 < q 1, then we get E 1 C = C = C j 2 j= l= l= l= = C F K 2 αql F χ l q L p( ) (R n ) 2bq(l j) 2 αql F χ l q L p( ) (R n ) 2 αql F χ l q L p( ) (R n ) j=l+2 1/q Applying the same calculations to (48), we obtain We have finished the proof E 3 C F K 2 bq(l j) Theorem 41 leads further results The Hardy Littlewood maximal operator M satisfies the vector-valued inequality on L p( ) (R n ) provided p( ) B(R n ) The next proposition is [2, Corollary 21] Proposition 44 If p( ) B(R n ) and 1 < r <, then we have ( ) 1/r ( ) 1/r Mg h r C g h r h=1 L p( ) (R n ) h=1 1/q 1/q L p( ) (R n ) for all sequences of functions g h h=1 satisfying g h h l r L p( ) (R n ) < Because the Hardy Littlewood maximal operator M is sublinear and satisfies the size condition Mf(x) C x y n f(y) dy, R n we immediately the following Corollary 45 Suppose that α, q and p( ) satisfy the same assumptions as Theorem 41 Then we have ( ) 1/r ( ) 1/r (49) Mf h r C f h r h=1 K h=1 K for all sequences of functions f h h=1 satisfying f h h l r K < In addition, M also satisfies the same vector-valued inequality as (49) on K

13 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 487 In particular M becomes a bounded operator on Herz spaces with variable exponent Thus the same argument as [22] leads the density of the set of all differentiable functions with compact support C c (Rn ) Theorem 46 Let s N 0 and suppose that α, q and p( ) satisfy the same assumptions as Theorem 41 Then Cc (Rn ) is dense in K p( ) W s (R n ) and in K p( ) W s (R n ) 5 Characterizations of Herz Sobolev spaces with variable exponent In this section we will give some equivalent norms on Herz Sobolev spaces with variable exponent 51 Characterizations of Herz spaces We first define a class of systems Φ(R n ) and Herz-type Triebel Lizorkin spaces with variable exponent Using them, we will characterize Herz spaces with variable exponent Definition 51 The set Φ(R n ) consists of all systems ϕ j j=0 S(Rn ) satisfying the following conditions (i) suppϕ 0 x R n : x 2 (ii) suppϕ j x R n : 2 j 1 x 2 j+1 if j N 0 (iii) For every γ N 0 n there exists a constant c γ > 0 such that D γ ϕ j (x) c γ 2 j γ for all j N 0 and all x R n (iv) j=0 ϕ j(x) 1 on R n Definition 52 Suppose s N 0, 0 < q <, p( ) P(R n ) and α R Take ϕ = ϕ j j=0 Φ(Rn ) Then define K 2 (R n ) := f S (R n ) : f K 2 (R n ) <, K 2 (R n ) := f S (R n ) : f K 2 (R n ) <, where f K 2 (R n ) := f K 2 (R n ) := 1/2 2 sj F 1 2 [ϕ j ˆf], j=0 K 1/2 2 sj F 1 2 [ϕ j ˆf] j=0 K Now we give characterizations of Herz spaces with variable exponent

14 488 M IZUKI Lemma 53 Let 1 < q <, p( ) B(R n ) and α R with α < n minδ 1, δ 2, where δ 1, δ 2 > 0 are the constants appearing in (39) and (311) Then we have K p( ) F 0,ϕ 2 (R n ) = K, K p( ) F 0,ϕ 2 (R n ) = K with equivalent norms In particular K p( ) F 0,ϕ 2 (R n ) and K p( ) F 0,ϕ 2 (R n ) are independent of the choice of ϕ = ϕ j j=0 Φ(Rn ) In order to prove Lemma 53, we introduce weighted Lebesgue spaces, Muckenhoupt s A p1 class and the extrapolation theorem A non-negative and locally integrable function is said to be a weight Definition 54 Let w be a weight and p 1 (1, ) a constant The weighted Lebesgue space L p1 w (R n ) is defined by L p1 w (R n ) := f is measurable : f p L 1 w (R n ) <, where f L p 1 w (R n ) := (R n f(x) p1 w(x)dx) 1/p1 Now we define Muckenhoupt s A p1 weights Definition 55 The class of weights A p1 consists of all weights w satisfying w 1/(p1 1) L 1 loc (Rn ) and ( ) p A p1 (w) := sup w(x) dx w 1/(p1 1) (x)dx < Q:cube Q Q Q Q The next lemma is the extrapolation theorem due to Cruz-Uribe, Fiorenza, Martell and Pérez ([2, Corollary 111]) They have proved the boundedness of many important operators on variable Lebesgue spaces by applying the result, provided that M is bounded Lemma 56 Let p( ) B(R n ) and A be a family of ordered pairs of nonnegative and measurable functions (f, g) Suppose that there exists a constant p 1 (1, p ) such that for all w A p1 and all (f, g) A with f L p1 w (R n ), f p L 1 w (R n ) C g L p 1 w (R n ), where C > 0 is a constant depending only on n, p 1 and A p1 (w) Then it follows that for all (f, g) A such that f L p( ) (R n ), f L p( ) (R n ) C g L p( ) (R n ), where C > 0 is a constant independent of (f, g) We also describe the duality of Herz spaces with variable exponent The duality for usual Herz spaces is proved in [7] By virtue of Lemma 35, the same argument as [7] leads the next lemma

15 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 489 Lemma 57 Suppose α R, p( ) P(R n ) and 1 < q < Then K coincides with the dual space of K p ( ) (Rn ) Moreover the quasinorm f K is equivalent to the value sup f(x)g(x)dx R n : g K The same duality is also true for K Proof of Lemma 53 Define Tf := F 1 2 [ϕ j ˆf] j=0 1 p ( ) (Rn ) Take a constant exponent p 1 (1, ) and w A p1 arbitrarily By virtue of [16], we have 1/2 C 1 f L p 1 w (R n ) Tf L p 1 w (R n ) C f L p 1 w (R n ) for all f C c (Rn ) Using Lemma 56, we get (51) C 1 f L p( ) (R n ) Tf L p( ) (R n ) C f L p( ) (R n ) Because C c (Rn ) is dense in L p( ) (R n ), (51) also holds for all f L p( ) (R n ) On the other hand, the operator T satisfies size condition (42) (cf [27, Proof of Theorem 21]) Thus Theorem 41 leads the estimate (52) Tf K C f K for all f K Next we show f K C Tf K for all f to prove (53) K By virtue of Theorem 46 and Lemma 57, it suffices f(x)g(x)dx C Tf K R n

16 490 M IZUKI for all f, g Cc (Rn ) with g K 1 Using the Plancherel formula p ( ) (Rn ) and the properties of ϕ j j=0, we obtain f(x)g(x)dx = ˆf(ξ)ĝ(ξ)dξ R R n n = ϕ j (ξ) ˆf(ξ) ϕ l (ξ)ĝ(ξ)dξ R n j=0 l=0 = ϕ 0 (ξ) ˆf(ξ) ϕ 0 (ξ) + ϕ 1 (ξ) ĝ(ξ)dξ R n + ϕ j (ξ) ˆf(ξ) ϕ j 1 (ξ) + ϕ j (ξ) + ϕ j+1 (ξ) ĝ(ξ)dξ (54) R n j=1 ϕ j (ξ) ˆf(ξ)ϕ j (ξ)ĝ(ξ)dξ + ϕ j (ξ) ˆf(ξ)ϕ j 1 (ξ)ĝ(ξ)dξ j=0 R n j=1 R n + ϕ j (ξ) ˆf(ξ)ϕ j+1 (ξ)ĝ(ξ)dξ R n j=0 By the Cauchy Schwarz inequality and generalized Hölder inequality we have ϕ j (ξ) ˆf(ξ)ϕ j (ξ)ĝ(ξ)dξ = F 1 [ϕ j ˆf](x)F 1 [ϕ j ĝ](x)dx R n R n j=0 (55) Applying (52), we get j=0 C Tf K Tg K p ( ) (Rn ) (56) Tg K p ( ) (Rn ) C g K p ( ) (Rn ) C Combining (54), (55) and (56), we have (53) The non-homogeneous case follows by the same argument 52 Characterizations of Herz Sobolev spaces We need preparations in order to get equivalent norms of Herz Sobolev spaces with variable exponent In this subsection we refer to [26,28] Theorem 58 Let d j > 0, Ω j j=0 be a sequence of compact sets of R n defined by Ω j := x R n : x d j for each j N 0, 0 < q <, p( ) P(R n ), 0 < r < p and nr 2 δ 1 < α < nr 1 δ(r), where δ 1, δ(r) > 0 are the constants appearing in (39) and (312) Suppose the following (I) or (II) (I) r = p 0, p( ) B(R n ) and p < 2, where p 0 (1, p ) is the constant appearing in Proposition 32 (II) p( ) satisfies (31) and (32) in Proposition 31

17 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 491 Then there exists a constant C > 0 such that f j ( ξ) sup (57) ξ R n 1 + (d j ξ ) n/r j= l2 (Z) K p( ) (Rn ) C f j j l 2 K p( ) (Rn ) for all sequences of functions f j j=0 satisfying fj j l 2 K < and suppf j Ω j for each j N 0 The same inequality as (57) also holds for the non-homogeneous case Proof Take f j j=0 satisfying f j j < and suppf l 2 K j Ω j for each j N 0 arbitrarily Denote g j (x) := f j (d 1 j x) Because ĝ j (ξ) = n d j ˆfj (d j ξ) and suppĝ j ξ 1, we get (58) g j (x ξ) 1 + ξ n/r C M( g j r )(x) 1/r for ae x, ξ R n (cf [26, p 22]) By M( f j r )(x) = M( g j (d j ) r )(x) = M( g j ) r )(d j x), (58) leads f j (x ξ) 1 + (d j ξ ) n/r C M( f j r )(x) 1/r Denote u r ( ) := r 1 p( ) Note that 1 < 2/r < and nδ 1 r 1 < rα < nδ(r) Applying Corollary 45 we obtain f j ( ξ) sup ξ R n 1 + (d j ξ ) n/r C M( f j r ) 1/r j = C M( f j r l ) j 2/r C f r l j j 2/r l = C f j j 2 j l 2 K l 2 K 1/r K r/r ur( ) (Rn ) 1/r K r/r ur( ) (Rn ) K Namely we have proved (57) The non-homogeneous case is obtained by the same argument as above

18 492 M IZUKI Definition 59 Let L N 0 A L (R n ) consists of all systems ϕ j j=0 S(R n ) of functions with compact supports such that C(ϕ j j ) := sup x L D γ ϕ 0 (x) x R n < γ L + sup x 0, j N 0 ( x L + x L ) γ L D γ (ϕ j (2 j ))(x) Note that Φ(R n ) A L (R n ) for all L N 0 The next lemma is due to [26, p 53] Let r > 0, L N, ϕ j j=0 A L(R n ) and f S (R n ) We define the maximal function for each j N 0, (59) (ϕ jf)(x) := sup y R n F 1 [ϕ j ˆf](x y) j y n/r Lemma 510 Let r > 0, s R and L N such that L > s +3n/r+n+2 Then there exists a constant C > 0 such that 1/2 ( ) 2 2 js sup (ϕ τ j f)(x) 0<τ<1 j=0 C sup C(ϕ τ ( j j ) 2 js (ϕ 0<τ<1 jf)(x) ) 2 for all ϕ j j=0 Φ(Rn ), all ϕ τ j j=0 A L(R n ) with 0 < τ < 1, all f S (R n ) and all x R n ] Applying Lemma 510 and Theorem 58 with f j = F [ϕ 1 j ˆf and d j = j=0 2 j+2, we immediately obtain the following lemma Lemma 511 Let 0 < q <, p( ) P(R n ), 0 < r < p, nr 2 δ 1 < α < nr 1 δ(r), where δ 1, δ(r) > 0 are the constants appearing in (39) and (312), s N 0 and L N with L > s + 3n/r + n + 2 Suppose (I) or (II) in Theorem 58 Then there exists a constant C > 0 such that (510) j=0 ( 2 js sup (ϕ τ j f) 0<τ<1 ) 1/2 2 K C sup C(ϕ τ j j ) f K 0<τ<1 2 (R n ) for all ϕ = ϕ j j=0 Φ(Rn ), all ϕ τ j j=0 A L(R n ) with 0 < τ < 1 and all f K 2 (R n ) The same inequality as (510) also holds for the non-homogeneous case 1/2

19 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 493 The next theorem is the Fourier multiplier theorem for Herz Sobolev spaces with variable exponent Theorem 512 Let ϕ = ϕ j j=0 Φ(Rn ), 0 < q <, p( ) P(R n ), 0 < r < p, nr 2 δ 1 < α < nr 1 δ(r), where δ 1, δ(r) > 0 are the constants appearing in (39) and (312), s N 0 and L N with L > s + 3n/r + n + 2 Suppose (I) or (II) in Theorem 58 Then there exists a constant C > 0 such that [ (511) F 1 m ˆf ] C m K L f K 2 (R n ) 2 (R n ) for all ϕ j j=0 Φ(Rn ), all m C (R n ) with m L < and all f K 2 (R n ), where m L := sup γ L sup (1 + x 2 ) γ /2 D γ m(x) x R n The same inequality as (511) also holds for the non-homogeneous case A function m is said to be a Fourier multiplier for holds for all f K 2 (R n ) Proof For all m C (R n ) with m L < and all f we have [ F 1 m ˆf ] K 2 (R n ) = = [ F 1 ϕ j F j=0 [ F 1 ϕ j m ˆf j=0 1/2 ((ϕ j m) f) 2 j=0 [ F 1 [ m ˆf 1/2 ] 2 1/2 ]]] 2 K K Because ϕ j m j=0 A L(R n ), Lemma 511 leads [ F 1 m ˆf ] K 2 (R n ) K 2 (R n ) if (511) K C C(ϕ j m j ) f K 2 (R n ) C m L f K 2 (R n ) The non-homogeneous case follows by the same argument K 2 (R n ),

20 494 M IZUKI Now we have some equivalent norms of Herz Sobolev spaces with variable exponent Theorem 513 Let ϕ = ϕ j j=0 Φ(Rn ), 1 < q <, p( ) P(R n ) and α R Suppose the following (I) or (II) (I) p( ) B(R n ), p < 2, (p ) < 2 and n minp 2 0 δ 1, δ 2 < α < n minδ(p 0 )p 1 0, δ 1, δ 2, where δ 1, δ 2 > 0 and p 0 (1, p ) are the constants appearing in (39), (311) and Proposition 32, respectively, and δ(p 0 ) > 0 appears in (312) with r = p 0 (II) p( ) satisfies (31) and (32) in Proposition 31, and α < n minδ 1, δ 2 Then the following four values f K p( ) W s (R n ), γ =0,s D γ f K, [ F f K 2 (R n ), 1 (1 + x 2 ) ˆf] s/2 K are equivalent on K p( ) W s (R n ) The same equivalence also holds for the non-homogeneous case In particular f K 2 (R n ) and f K 2 (R n ) are independent of the choice of ϕ = ϕ j j=0 Φ(Rn ) Remark 514 Theorem 513 is proved by Xu and Yang ([27, 28]) for constant p in the setting of Herz-type Triebel Lizorkin spaces provided n/p < α < n/p ] Proof Below we denote I s f := F [(1 1 + x 2 ) s/2 ˆf For every f K p( ) W s (R n ) it obviously follows that γ =0,s D γ f K f K p( ) W s (R n ) We also note that Theorem 46 shows the density of Cc (Rn ) in Thus we have only to prove (512) f K (513) (514) f K (515) I s f K C f K 2 (R n ), 2 (R n ) C I s f K, p( ) W s (R n ) C I s f K, I s f K C γ =0,s D γ f K K p( ) W s (R n ) for all f C c (R n ) We write ϕ s j (x) := 2js (1 + x 2 ) s/2 ϕ j (x) Then we see that ϕ s := ϕ s j j=0 A L(R n ) for every L N

21 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 495 Proof of (512): By Lemma 53 we have I s f K C I sf K p( ) F0,ϕs 2 (R n ) = C F 1 [ ϕ s jf[i s f] ] j l 2 K = C [ ] F 1 2 js (1 + x 2 ) s/2 ϕ j (1 + x 2 ) s/2 ˆf = C f K 2 (R n ) j l 2 K Proof of (513): We define ( ϕ s j) f by (59) with r := p0 if we suppose (I), r := 1 if we suppose (II) Using Lemmas 511 and 53 we obtain f K 2 (R n ) = I s (I s f) K 2 (R n ) = [ ] 2 sj F 1 (1 + x 2 ) s/2 ϕ j F[I s f] = F 1 [ ϕ s j F[I sf] ] j (ϕ ) s j (Is f) j l 2 l 2 K K j l 2 C I s f K p( ) F0,ϕs 2 (R n ) C I sf K K Proof of (514): Note that x γ (1 + x 2 ) s/2 is a Fourier multiplier for K p( ) F 0,ϕ 2 (R n ) for every γ s By virtue of Lemma 53 and Theorem 512, we get f K p( ) W s (R n ) = D γ f K C [ ] F 1 x γ ˆf K = C γ s C γ s γ s γ s F 1 [ x γ (1 + x 2 ) s/2 F[I s f] ] K I s f K p( ) F0,ϕ 2 (R n ) C I sf K p( ) F0,ϕ 2 (R n ) p( ) F0,ϕ 2 (R n ) Proof of (515): We can take Fourier multipliers ρ 1 (x),, ρ n (x) for K p( ) F 0,ϕ 2 (R n ) such that n G(x) := 1 + ρ h (x)x s h C (1 + x 2 ) s/2 h=1 (see [26, p 60]) Define H(x) := (1 + x 2 ) s/2 G(x) 1, then H(x) is also a Fourier multiplier for K p( ) F 0,ϕ 2 (R n ) Therefore using Lemma

22 496 M IZUKI 53 and Theorem 512 again, we obtain I s f K = C C I s f K [ F 1 H F C p( ) F0,ϕ 2 (R n ) [ F 1 G ˆf ] K [ F 1 [ G ˆf p( ) F0,ϕ 2 (R n ) ]]] K p( ) F0,ϕ 2 (R n ) n C f K p( ) F0,ϕ 2 (R n ) + [ F 1 s ρ h (x)x h ˆf h=1 n C f K p( ) F0,ϕ 2 (R n ) + [ρ F 1 h (x)f h=1 n C f K p( ) F0,ϕ 2 (R n ) + s x s f h h=1 C D γ f K γ =0,s ] K p( ) F0,ϕ 2 (R n ) [ s x h s f]] K p( ) F0,ϕ 2 (R n ) K p( ) F0,ϕ 2 (R n ) Consequently we have proved the theorem for K p( ) W s (R n ) The case of K p( ) W s (R n ) is proved by the same argument 6 Wavelet characterization of Herz Sobolev spaces Based on the fundamental wavelet theory (cf [21]), we can construct functions ϕ, ψ 1, ψ 2,, ψ 2n 1 satisfying the following 1 ϕ, ψ l S(R n ) for every l = 1, 2,,2 n 1 2 The sequence ϕ0,k, ψ l j,k : l = 1, 2,,2n 1, j N 0, k Z n forms an orthonormal basis in L 2 (R n ) 3 R n x γ ψ l (x)dx = 0 for all γ N 0 n and every l = 1, 2,,2 n 1 4 ϕ is real-valued and band-limited with supp ˆϕ [ 4 3 π, 4 3 π] n 5 Each ψ l is real-valued and band-limited with supp ˆψ l ξ R n : a 1 ξ a for some constant a > 8π n 3 The function ϕ is called the Meyer scaling function and each ψ l is called the Meyer wavelet in terms of multiresolution analysis

23 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 497 Using the Meyer scaling function ϕ and the Meyer wavelets ψ l : l = 1, 2,,2 n 1 we define ( ) 1/2 I 0 f := f, ϕ 0,k χ 0,k 2 k Z n 2 n 1/2 1 I 1 f := 2 js f, ψj,k l χ 2 j,k where l=1 Uf := I 0 f + I 1 f, χ j,k := 2 jn/2 χ Qj,k, Q j,k := j=0 k Z n n [ 2 j k h, 2 j (k h + 1) ) (k = (k 1,, k n ) Z n ), h=1 and f, g denotes the L 2 -inner product, namely f, g := R n f(x)g(x) dx In terms of the square function Uf, we have the following wavelet characterization of Herz Sobolev spaces with variable exponent Theorem 61 Let s N 0, 1 < q <, p( ) P(R n ) and α R Suppose the following (I) or (II) (I) p( ) B(R n ), p < 2, (p ) < 2 and α < n minδ(p 0 )p 1 0, p 2 0 δ 1, δ 2, where δ 1, δ 2 > 0 and p 0 (1, p ) are the constants appearing in (39), (311) and Proposition 32, respectively, and δ(p 0 ) > 0 appears in (312) with r = p 0 (II) p( ) satisfies (31) and (32) in Proposition 31, and α <n minδ 1, δ 2 Then we have that for all f K p( ) W s (R n ), (61) C 1 f K p( ) W s (R n ) Uf K C f K p( ) W s (R n ) Additionally if α satisfies α < n(p ) q (p ) +, then the same wavelet characterization as (61) also holds for K p( ) W s (R n ) Remark 62 Before the proof, we have to check that the wavelet coefficients f, ϕ 0,k, f, ψj,k l l,j,k are well-defined in Theorem 61 We first consider the homogeneous case If S(R n ) K p ( ) (Rn ) is true, then f, g C f K g K p ( ) (Rn ) < K holds for all f p( ) W s (R n ) and all g S(R n ) Thus we have only to show that S(R n ) K p ( ) (Rn ) Take g S(R n ) arbitrarily and denote m 0 := (p ) + (p ) Because g(x) C (1 + x ) K for any positive number K >

24 498 M IZUKI αq + nm 0, we get g χ Rl L p ( ) (R n ) C (1 + 2l 1 ) K χ Rl L p ( ) (R n ) Hence we obtain the estimate g q K p ( ) (Rn ) C C C l= 2 αlq (1 + 2 l 1 ) K χ Rl L p ( ) (R n ) 2 αlq (1 + 2 l 1 ) K 2 lnm0 + C l=1 2 l( αq +nm 0 K) + C l=1 0 l= 0 l= 2 l( αq +nm 1 0 ) 2 αlq (1 + 2 l 1 ) K 2 lnm 1 0 By virtue of K > αq + nm 0 and α < q m 0, both of the two series are finite Therefore we have proved S(R n ) K p ( ) (Rn ) In the non-homogeneous case, the similar calculations imply S(R n ) K p ( ) (Rn ) without the assumption α < n n q m 0 We will prove Theorem 61 for the non-homogeneous case while the homogeneous case is similar In order to prove the theorem, we need the following two lemmas Lemma 63 Let γ N 0 n with γ s Then we have that for all f K p( ) W s (R n ), I 0 (D γ f)(x) C Mf(x) Lemma 64 Let N N 0 and γ N n 0 such that γ N Then we have that for all f K p( ) W N γ (R n ), 1/2 2 n 1 2 j(n γ ) f, (D γ ψ l 2 ) j,k χ j,k l=1 j=0 k Z n K C f K p( ) W N γ (R ) n Proof of Lemma 63 We see that ( ) 1/2 I 0 (D γ f)(x) = f, (D γ ϕ) 0,k χ 0,k (x) 2 k Z n χ 0,k (x)(d γ ϕ) 0,k (y) f(y) dy R n k Z n

25 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 499 By virtue of [13, Lemma 28], we can take a bounded and radial decreasing function H L 1 (R n ) so that k Z n χ 0,k (x)(d γ ϕ) 0,k (y) H(x y) In addition, we get R n H(x y) f(y) dy C Mf(x) (cf [24, p 63]) Therefore we have proved the lemma Proof of Lemma 64 Denote φ l j (y) := 2jn (D γ ψ l )( 2 j y) for j Z and l = 1, 2,,2 n 1 Then we have f, (D γ ψ l ) j,k χ j,k (x) 2 k Z n = 2 f(z)2 jn/2 φ l j (2 j k z)dz χ j,k (x) 2 k Z n R n = φ l j f(2 j k) 2 χ Qj,k (x) k Z n sup φ l j f(y) 2 χ Qj,k (x) y Q j,k sup φ l j f(x z) 2 k Z n z 2 j n sup z 2 j n C (F[φ l j ] f)(x) 2, φ l j f(x z) (1 + 2 j z ) n/r 2 sup (1 + 2 j z n/r ) 2 z 2 j n where F[φ l j ] f is defined by (59) with r := p 0 if we suppose (I), r := 1 if we suppose (II) Hence by virtue of Lemma 511 and Theorem 512, we get 1/2 2 n 1 2 j(n γ ) f, (D γ ψ l 2 ) j,k χ j,k l=1 j=0 k Z n K 1/2 2 n 1 C 2 j(n γ ) (F[φ l j ] f) 2 l=1 j=0 K C f K p( ) W N γ (R ) n

26 500 M IZUKI Proof of Theorem 61 By virtue of Lemma 63, we obtain I 0 f(x) C Mf(x) for all f K p( ) W s (R n ) Because p( ) B(R n ), we get I 0 f K C f K C f K p( ) W s (R n ) Additionally if we apply Lemma 64 with N = s and γ = (0,,0), then we immediately obtain Therefore we have I 1 f K C f K p( ) W s (R n ) Uf K I 0f K + I 1f K C f K p( ) W s (R n ) Next we prove (62) f K p( ) W s (R n ) C Uf K We follow a duality argument (cf [5, Chapter 6]) applying Lemma 57 Note that Cc (Rn ) is dense in K p( ) W s (R n ) and in K p ( ) (Rn ) by virtue of Theorem 46 Hence it suffices to show that for all f, g Cc (R n ) with g K 1 and all γ N p ( ) (Rn ) 0 n with γ s, D γ f(x)g(x)dx C Uf K R n Because f, D γ g L 2 (R n ), we obtain the wavelet expansions f = D γ g = k Z n f, ϕ 0,k ϕ 0,k + 2 n 1 l=1 k Z n D γ g, ϕ 0,k ϕ 0,k + j=0 2 n 1 l=1 f, ψj,k l ψl j,k, k Z n D γ g, ψj,k l ψl j,k k Z n Thus we have D γ f(x)g(x)dx = f(x)d R R γ g(x)dx n n 2 f, ϕ 0,k D γ g, ϕ 0,k + n 1 (63) f, ψj,k l k Z Dγ g, ψj,k l n l=1 j=0 k Z n j=0

27 HERZ SOBOLEV SPACES WITH VARIABLE EXPONENT 501 Now we estimate the first sum Using the Cauchy Schwarz inequality and the generalized Hölder inequality, we get f, ϕ 0,k D γ g, ϕ 0,k f, ϕ 0,k D γ g, ϕ 0,k χ 0,k (x)dx k Z n k Z n R n I 0 f(x)i 0 (D γ g)(x)dx R n Applying Lemma 63, we see that C I 0 f K I 0(D γ g) K p ( ) (Rn ) I 0 (D γ g) K p ( ) (Rn ) C Mg K p ( ) (Rn ) C g K p ( ) (Rn ) C Therefore we have the estimate (64) f, ϕ 0,k D γ g, ϕ 0,k C I 0f K k Z n Next we estimate the second sum of (63) Using the Cauchy Schwarz inequality and the generalized Hölder inequality again, we obtain 2 n 1 f, ψj,k D l γ g, ψj,k l l=1 j=0 k Z n 2 n 1 f, ψ l j,k D γ g, ψj,k l χ j,k (x) 2 dx R n = l=1 R n R n j=0 k Z n 2 n 1 l=1 2 n 1 l=1 f, ψj,k χ l j,k (x) g, 2 j γ (D γ ψ l ) j,k χ j,k (x) dx j=0 k Z n j=0 k Z n I 1 f(x) R n C I 1 f K l=1 2 n 1 l=1 2 n 1 2 js f, ψ l j,k χ j,k(x) g, (D γ ψ l ) j,k χ j,k (x) dx j=0 k Z n j=0 k Z n g, (D γ ψ l ) j,k χ j,k (x) 1/2 2 dx 1/2 g, (D γ ψ l ) j,k χ j,k (x) 2 K p ( ) (Rn )

28 502 M IZUKI Thus by virtue of Lemma 64 we get 2 n 1 f, ψj,k l Dγ g, ψj,k l l=1 j=0 k Z n (65) Combing (64) and (65), we obtain (62) Consequently we have proved the theorem C I 1 f K g K p ( ) (Rn ) C I 1 f K Remark 65 In [11] a wavelet characterization of different type of K p( ) W s (R n ) without the restriction on α in Theorem 61 is given Let s N, 1 < q <, p( ) B(R n ) and α R Using a C s+1 -smooth and compactly supported scaling function ϕ of a multiresolution analysis and an associated wavelets ψ l : l = 1, 2,,2 n 1 it was proved that the quasinorm f K p( ) W s (R n ) is equivalent to ( ) 1/2 f, ϕ J,k χ J,k + 1/2 2 n js f, ψ j,k χ l j,k 2, k Z n l=1 j=j k Z n K where J is a sufficiently large integer depending only on s In order to prove above he does not use boundedness of sublinear operators but local properties of wavelets But this argument is not applicable to the homogeneous case Acknowledgements The author is thankful to Professor Naohito Tomita for his advices He is also grateful to the referees whose valuable comments improved the first manuscript References [1] A Baernstein II and E T Sawyer, Embedding and multiplier theorems for H p (R n ), Mem Amer Math Soc 53 (1985), no 318 [2] D Cruz-Uribe, SFO, A Fiorenza, J M Martell and C Pérez, The boundedness of classical operators on variable L p spaces, Ann Acad Sci Fenn Math 31 (2006), [3] D Cruz-Uribe, A Fiorenza and C J Neugebauer, The maximal function on variable L p spaces, Ann Acad Sci Fenn Math 28 (2003), , and 29 (2004), [4] L Diening, Maximal functions on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull Sci Math 129 (2005), [5] E Hernández and G Weiss, A first course on wavelets, CRC Press, Boca Raton, 1996 [6] E Hernández, G Weiss and D Yang, The ϕ-transform and wavelet characterizations of Herz-type spaces, Collect Math 47 (1996), [7] E Hernández and D Yang, Interpolation of Herz spaces and applications, Math Nachr 205 (1999), [8] C S Herz, Lipschitz spaces and Bernstein s theorem on absolutely convergent Fourier transforms, J Math Mech 18 (1968/1969),

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