ON LIPSCHITZ-LORENTZ SPACES AND THEIR ZYGMUND CLASSES

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1 Hacettepe Journal of Mathematics and Statistics Volume 39(2) (2010), ON LIPSCHITZ-LORENTZ SPACES AND THEIR ZYGMUND CLASSES İlker Eryılmaz and Cenap Duyar Received 24:10 :2008 : Accepted 04 :01 :2010 Abstract Let G be a metrizable locally compact abelian group. We prove that (L 1(G),lip (α, pq)), lip (α, pq), (L1(G),Lip (α, pq)) and Lip(α, pq) are isometrically isomorphic, where Lip(α, pq) and lip (α, pq) denote the Lipschitz-Lorentz spaces defined on G, (L 1(G), A) is the space of multipliers from L 1(G) to A and lip (α, pq) denotes the relative completion of lip (α, pq). Also, we characterize the space of multipliers from Lorentz spaces to the Lipschitz-Lorentz-Zygmund classes LΛ (α, pq; G) and Lλ (α, pq; G). Keywords: Lorentz spaces, Lipschitz spaces, Zygmund classes, Relative completion, Multipliers, Translation operator AMS Classification: Primary: 46 E 30, 26 A 16. Secondary: 43 A 22, 54 D Introduction and preliminaries Let G be a metrizable locally compact abelian group with Haar measure µ. In [12], Quek and Yap defined the relative completion à for a linear subspace A of the usual Lebesgue spaces L p(g) for 1 p <. They proved that (L 1(G), A) L 1(G) if p 1, and (L 1(G), A) is isometrically isomorphic to à if p > 1. In the same paper, they defined the subspaces Lip (α, p) and lip (α, p) of the Lebesgue spaces, which are named as Lipschitz spaces. They found the relative completion of lip (α, p), i.e. lip (α, p), and stated the relationship between lip (α, p) and Lip (α, p). In addition to [12], we can find more interesting results concerning multipliers from L 1(G) to Lipschitz spaces in [5-6]. Quek and Yap used a fixed translation operator while constructing the Lipschitz spaces. However, in [13], they gave the definition of a generalized translation operator, and defined the Lipschitz-Zygmund classes. Also, some theorems about multipliers Ondokuz Mayıs University, Faculty of Science and Arts, Department of Mathematics, Kurupelit, Samsun, Turkey. (İ. Eryılmaz) rylmz@omu.edu.tr (C. Duyar) cenapd@omu.edu.tr Corresponding Author.

2 160 I. Eryılmaz, C. Duyar from Lebesgue spaces to these classes were proved in the same paper. We conclude these introductory remarks by noting that there is considerably more literature available than that cited above, e.g. in [1,11]. Certain well-known terms such as multiplier, module homomorphism, (semi) homogeneous Banach space, rearrangement invariant Banach function space etc. are used frequently in the paper. We will not give their definitions and properties explicitly. The reader is referred to [2,3,8,16] for more information. For the convenience of the reader, we will now review briefly what we need from the theory of Lorentz spaces. Let (G, Σ, µ) be a positive measure space and f a complex-valued, measurable function on G. Then the rearrangement function of f on (0, ) is defined by f (t) inf{y > 0 : λ f (y) µ{x G : f(x) > y} t}, t > 0, where inf φ. Also the average (maximal) function of f is defined by f (t) 1 t t If the norms are defined as (1.1) f pq f pq,µ and 0 f (s) ds, t > 0. ( q p (1.2) f p, sup t p 1 f (t) for 0 < p,q, t>0 0 [f (t)] q t q p 1 dt) 1q for p, q (0, ) then the Lorentz spaces denoted by L(p, q)(g) is defined to be the vector space of all (equivalence classes of) measurable functions f on G such that f pq <. We know that, for 1 p, f pp f p and L p(g) L(p, p)(g). It is also known that using f instead of f in (1.1) and (1.2) leads to a norm pq on L(p, q)(g) for 1 < p < and 1 q satisfying (1.3) f pq f pq p p 1 f pq for each f L(p, q)(g). The space (L(p,q)(G), pq) is a reflexive rearrangement-invariant Banach function space with associate space (L(p, q )(G), p q ), where p, q (1, ) and 1 p + 1 p 1 1 q + 1 q. Again, a well-known feature of L(p, q)(g), is that there is an approximate identity {e α} α I in L 1(G) such that e α 1 1 for all α I and f e α f pq 0 for every f L(p,q)(G) with 1 < p <, 1 q <. Therefore, it can be derived easily that L(p,q)(G) is an essential Banach L 1(G)-module under the usual convolution product. For further properties of Lorentz spaces, we refer the reader to [2,3,4,9,10,15] Definition. Let A be a linear subspace of L(p, q)(g) for 1 < p, q < with the following properties: (i) There is a norm A on A such that pq A and (A, A) is a Banach L 1(G)-module with respect to convolution. (ii) There is an approximate identity {e α} in L 1(G) such that e α 1 1 for all α, and f e α f A 0 for all f A.

3 On Lipschitz-Lorentz spaces 161 Then the relative completion of A is the space (1.4) à {f L(p, q)(g) : f e α A for all α and sup f e α A < }, α with f à sup f e α A. α 1.2. Remark. It is easy to check that the relative completion does not depend on the choice of the sequence or net {e α} α I. The following lemma and theorem can be proved easily by using the methods, mutatis mutandis, used in [12] Lemma. Let A be as in Definition 1.1. (i) If f à and g L1(G), then f g à and f g à f à g 1. (ii) f A f à for f A. (iii) A is a closed subspace of à Theorem. Let A be a linear subspace of L(p, q)(g) for 1 < p,q <, as in Definition 1.1. Then (L 1(G), A) is isometrically isomorphic to à Lemma. [16] Every semi homogeneous Banach space contains a maximal homogeneous Banach space. More precisely, if B c(g) is the set of all f in B(G) such that the map x τ xf of G into (B(G), B) is continuous at 0, then B c(g) is the maximal homogeneous Banach space in (B(G), B). 2. Basic properties and multipliers for Lipschitz-Lorentz spaces In this section, p and q will be in (1, ), G will be a metrizable locally compact abelian group with Haar measure µ and d a translation invariant metric with y d(0, y) for all y G. We will assume that there is a decreasing countable (open) basis {V n} n N of the identity 0 in G such that (2.1) µ((y + V n) V n)/ 0 as y 0 for all α (0,1) and y G, where denotes the symmetric difference. Denoting by χ the characteristic function of a set, it is easy to see that {e n} n N defined by e n µ(v n) 1 χ Vn is an approximate identity in L 1(G). For any f L(p, q)(g) and δ > 0, let us define the modulus of continuity as (2.2) ω pq(f; δ) sup{ τ yf f pq : y δ}, where τ yf(x) f(x y) Definition. Define two subspaces of L(p,q)(G) by (2.3) (2.4) Lip (α, pq) {f L(p, q)(g) : ω pq(f; δ) O(δ α )}, lip (α, pq) {f Lip (α, pq) : ω pq(f; δ) o(δ α )}, for all δ > 0 and 0 < α < 1. These spaces are named Lipschitz-Lorentz spaces, and the function (α,pq) defined by τ yf f pq (2.5) f (α,pq) f pq + sup y 0 is a norm in both Lipschitz-Lorentz spaces. The next lemma and theorem illustrate some basic properties of the little Lipschitz spaces lip (α, pq) which are easy to prove by the methods used in [12] Lemma. If f lip (α, pq), then

4 162 I. Eryılmaz, C. Duyar (i) τ yf f (α,pq) 0 as y 0. (ii) f e n f (α,pq) 0 as n. (iii) ( ) lip (α, pq), (α,pq) is a Banach L1(G)-module under convolution. (iv) lip (α, pq), the relative completion of lip (α, pq), is Lip (α, pq), i.e. lip (α, pq) Lip (α, pq) Theorem. Lip (α, pq) is a semi homogeneous Banach space Theorem. lip (α, pq) is a maximal homogeneous Banach space in Lip (α, pq). Proof. Let Lip c (α, pq) be the set of all functions f such that the mapping x τ xf of G into Lip (α, pq) is continuous at 0 G. Let us take f lip (α, pq) and ε > 0. Then there exists a number δ > 0 such that τ yf f pq (2.6) < ε 6, where y 0 and y δ by (2.4). Also, we can write that (2.7) τ y(τ xf f) (τ xf f) pq 2 τ yf f pq, and similarly (2.8) τ y(τ xf f) (τ xf f) pq 2 τ xf f pq. Using the inequalities (2.6), (2.7) and (2.8), we get (2.9) τ y(τ xf f) (τ xf f) pq τ xf f (α,pq) τ xf f pq + sup y 0 τ y(τ xf f) (τ xf f) pq τ xf f pq + sup y δ τ y(τ xf f) (τ xf f) pq + sup y >δ τ xf f pq + ε 2 τxf f pq +. 3 δ α On the other hand, since the translation mapping on L(p, q)(g) is continuous, there exists a neighbourhood U of 0 G such that (2.10) τ xf f pq < 2δα ε 3(δ α + 2) for all x U and ε > 0. If one combines (2.10) with (2.9), we obtain τ xf f (α,pq) < ε for all x U. This shows that f Lip c (α, pq). Conversely, assume that f Lip c (α, pq). By Lemma 1.5, Lip c (α, pq) is a homogeneous Banach space. Since Lip (α, pq) is a Banach L 1(G)-module, f e n Lip (α, pq) for all n N. Separately, since the mapping y τ yf of G into Lip (α, pq) is continuous at 0 G, there exists a positive integer N and a neighbourhood V N of 0 G such that (2.11) τ yf f (α,pq) < ε 2 for all ε > 0 and y V N. Now let y G and y 0. Then τ yf f pq τyf τy(f en) (f f en) pq f f e n (α,pq) + τ yf f (α,pq) + + τy(f en) f en pq τy(f en) f en pq, τy(f en) f en pq

5 On Lipschitz-Lorentz spaces 163 and by (2.11), we get (2.12) τ yf f pq ε 2 + τy(f en) f en pq for all y V N. Since L(p,q)(G) is a Banach L 1(G)-module, it follows that τ y(f e n) f e n pq τy(en) en 1 f pq. Since τ y(e n) e n 1 µ(v n) 1 µ(y +V n V n), we get τy(en) en 1 0 as y 0 by the assumption at the beginning of this section. Thus, there exists at least one δ > 0 such that {y : y δ} V N. Therefore τ y(e n) e n 1 f pq < ε 2 for all y G, where the condition y δ is satisfied. Hence τ y(f e n) f e n pq (2.13) sup < ε y δ 2. Combining (2.12) with (2.13), we get τ yf f pq sup < ε. y δ Thus, this gives that ω pq(f; δ) o(δ α ), i.e. f lip (α, pq). Consequently, lip (α, pq) Lip c (α, pq) Lemma. The space of multipliers (L 1(G), Lip (α, pq)) is contained in Lip (α, pq). Proof. Assume that T (L 1(G), Lip(α, pq)). Hence there exists a unique f L(p, q)(g) such that T(g) f g for all g L 1(G) by [14, Corollary 8.10]. Also, T (L 1(G), Lip (α, pq)) implies that T(g) (α,pq) T g 1. Then we find sup n f e n (α,pq) sup T(e n) (α,pq) sup T e n 1 T n Definition 1.1 and Lemma 2.2 (iv) imply that f lip (α, pq) Lip(α, pq) Theorem. The spaces (L 1(G), lip (α, pq)), lip (α, pq), Lip (α, pq) and (L1(G), Lip (α, pq)) are isometrically isomorphic. Proof. By using Theorem 1.4, Lemma 2.2 (iv) and Lemma 2.5, one can easily see that Lip(α, pq) lip (α, pq) (L 1(G), lip (α, pq)) (L 1(G), Lip (α, pq)) Lip (α, pq). The result follows Theorem. L 1(G) lip (α, pq) lip (α, pq) L 1(G) Lip(α, pq). Proof. Let us take any f lip (α, pq) and ε > 0. If we use the approximate identity {e n} n N of L 1(G), then we can say that L 1(G) lip (α, pq) is dense in lip (α, pq) by Lemma 2.2 (ii). Therefore, the Module Factorization Theorem [8, 32.22] says that L 1(G) lip (α, pq) lip (α, pq). Again, since (L 1(G), lip (α, pq)) Lip (α, pq) due to Theorem 2.6, we get L 1(G) Lip(α, pq) lip (α, pq), and hence lip (α, pq) L 1(G) lip (α, pq) L 1(G) Lip (α, pq) lip (α, pq). This shows that L 1(G) Lip (α, pq) lip (α, pq). n

6 164 I. Eryılmaz, C. Duyar 3. Convolution theorems and multipliers from L(p, q)(g) to the Lipschitz-Lorentz-Zygmund classes In this section, we will recall the definition of generalized translation operators and define the Lipschitz-Lorentz-Zygmund classes LΛ (α, pq;g) and Lλ (α, pq;g) by using the techniques mentioned in [13] Definition. [10,13] Let G be a locally compact abelian group with Haar measure µ, and L(G) {L(p, q)(g) : 1 p, q }. A function τ : G L(G) L(G) is called a translation operator if it has the following properties: (i) For each a G, τ(a, ) is a linear mapping from L(p, q)(g) into L(p, q)(g) for 1 p, q. (ii) If 1 p,q, 1 p, q, 1 p + 1 p 1, 1 q + 1 q 1 and f L(p,q)(G), g L(p, q )(G), then τ(a,f g) τ(a,f) g for all a G Example. For any a G and f L(G), let us define τ i(a, f), i 1,2,... as follows: τ 1(a,f)(x) f(x a) f(x) τ 2(a,f)(x) f(x + a) + f(x a) 2f(x). It is easy to see that τ 1 and τ 2 are translation operators. Also, (τ 1 τ 2)(a, f) τ 1(a, τ 2(a, f)) is a translation operator. For the remainder of this section, G will denote a metrizable locally compact abelian group with translation-invariant metric d such that x d(0, x) for all x G. For 1 p, q, 0 < δ < and f L(p, q)(g), we define the modulus of continuity as ω pq(f; δ) sup{ τ(y,f) pq : y δ}, where the symbol τ denotes an arbitrary translation operator satisfying the conditions in Definition Definition. When 1 p, q, 0 < α < and τ is a translation operator on G L(G), we define the Lipschitz-Lorentz-Zygmund classes LΛ (α, pq;g) and Lλ (α, pq;g) by (3.1) (3.2) LΛ (α, pq; G) {f L(p, q)(g) : ω pq(f; δ) O(δ α )} Lλ (α, pq; G) {f LΛ (α, pq;g) : ω pq(f; δ) o(δ α )}. for all 0 < α, δ <. It is easy to see that the function (α,pq) defined by { } τ(y, f) pq (3.3) f (α,pq) max f pq, sup y 0 is a norm in both classes Claim. If τ τ 1, then LΛ (α, pq;g) and Lλ (α, pq; G) are the Lipschitz-Lorentz spaces Lip (α, pq) and lip (α, pq), respectively Theorem. If f LΛ (α, pq; G), 1 < p,q < and g L 1(G), then f g LΛ (α, ps; G), where s q. Also, f g (α,ps) f (α,ps) g 1. Proof. Let f LΛ (α, pq;g) and g L 1(G). Then by [2, Theorem 2.9], we have f g L(p,s)(G) where s q and τ(y, f g) ps τ(y,f) g ps C τ(y,f) ps g 1 C f (α,ps) g 1,

7 On Lipschitz-Lorentz spaces 165 where C is a constant depending on p and s. Thus, f g LΛ (α, pq;g) by (3.3). Using [2, Theorems 2.10 and 2.12], the following two theorems can be proved easily Theorem. If f LΛ (α, pq;g), 1 < p, q <, and g LΛ (α, p q ; G), where 1 p + 1 p 1, 1 q + 1 q 1, then f g LΛ (α, ;G) and f g (α, ) C f (α,pq) g (α,p q ), where C is a constant depending on q and q Theorem. Let 1 < p, p <, 0 < q, q and > 1. If f LΛ p p (α, pq;g) and g LΛ (α, p q ; G), then f g LΛ (α, rs; G), where and s > 0 is a r p p number such that Furthermore, q q s f g (α,rs) C f (α,pq) g (α,p q ), where C is a constant depending on q, q, r and s Theorem. Let G be a metrizable locally compact abelian group, 0 < α <, 1 < 1 p,q <, and Then the spaces of L p p q q 1(G)-module homomorphisms, Hom L1 (G)(L(p,q)(G), LΛ (α, ; G)) and the classes LΛ (α, p q ; G), are algebraically isomorphic and topologically homeomorphic. Proof. Let f LΛ (α, p q ) and g L(p, q)(g). Then clearly f g L (G) by [2, Theorem 2.10]. Next, we note that (3.4) f g f p q g pq f (α,p q ) g pq, and for any 0 y G, (3.5) τ(y,f g) τ(y, f) g τ(y, f) (α,p q ) g pq f (α,p q ) g pq by (3.3). Thus f g LΛ (α, ; G) by (3.4) (3.5), and (3.6) f g (α, ) f (α,p q ) g pq. Hence, for any f LΛ (α, p q ; G) we can define a mapping T f : L(p, q)(g) LΛ (α, ; G) by T f (g) f g. It follows from (3.6) that T f f (α,p q ) and T f Hom L1 (G)(L(p,q)(G), LΛ (α, ; G)). Conversely, let T Hom L1 (G)(L(p,q)(G), LΛ (α, ; G)). Then by [14, Theorem 4.5], there exists an f L(p, q ) such that T(g) f g for all g L(p, q)(g). Therefore, for any g L(p, q)(g) we get (3.7) and τ(y,f g) sup y 0 τ(y, T(g)) sup y 0 T g pq (3.8) f g T(g) T g pq. Since L(p, q )(G) is a Banach function space, we get the following inequality { } (3.9) f p q p f p q pc sup f(x)ϕ(x) dµ(x) : ϕ pq 1, G where C is a constant. Since G is abelian and µ a Haar measure, µ(e) µ( E) for all E G. Therefore, if we make the substitution ϕ(x) g( x) for any ϕ L(p,q)(G), then we get g L(p,q)(G) and ϕ pq g pq. Since ϕ(x) g( x) implies that

8 166 I. Eryılmaz, C. Duyar λ g(y) µ(e) µ( E) λ ϕ(y), the rearrangement functions coincide for ϕ( x) and g(x). Thus, we get { } f p q p C sup f(x)g( x) dµ(x) : g pq 1 G p C sup{ (f g)(0) : g pq 1} p C sup{ f g : g pq 1} p C sup{ T g pq : g pq 1} pp C T by using (3.8) and (3.9). Similarly by (3.7), we get τ(y, f) p q pp C T. Then, f LΛ (α, p q ; G) and f (α,p q ) pp C T. Hence the spaces LΛ (α, p q ; G) and Hom L1 (G)(L(p,q)(G), LΛ (α, ; G)) are algebraically isomorphic and topologically homeomorphic Theorem. Let 0 < α <. Then the spaces (L 1(G), LΛ (α, pq; G)) and LΛ (α, pq; G) are isometrically isomorphic if 1 < p, q <. Also, if p q 1, then (L 1(G), LΛ (α, pq;g)) L 1(G). Proof. Let f LΛ (α, pq;g). Then by Theorem 3.5, we write f g LΛ (α, pq;g) for all g L 1(G). Hence we can define a mapping T f : L 1(G) LΛ (α, pq; G) by T f (g) f g for all g L 1(G). Since L(p, q)(g) is a Banach L 1(G)-module, we get (3.10) T f (g) pq f g pq f pq g 1 f (α,pq) g 1 and for any y G with y 0, (3.11) τ(y,t f (g)) pq τ(y,f g) pq f (α,pq) g 1. τ(y, f) g pq τ(y,f) pq g 1 Therefore, T f (L 1(G), LΛ (α, pq;g)) and T f f (α,pq) by (3.10) and (3.11). Conversely, let T (L 1(G), LΛ (α, pq;g)). Then, there exists f L(p,q)(G) such that T(g) f g for all g L 1(G) by [4, Theorem 3.4]. So, by the definition of f g (α,pq), we get (3.12) f g pq T(g) pq T g 1, and similarly τ(y,f g) pq (3.13) sup y 0 τ(y,t(g)) pq sup T g y 0 1 for all g L 1(G). Now, we will show that f LΛ (α, pq;g) and f (α,pq) T. Let {a α} α I L 1(G) be an approximate identity for L(p, q)(g) with a α 1 1 for all α I. Then, we have (3.14) τ(y, f a α) pq τ(y, f) aα pq τ(y, f) pq T by (3.13), for y G with y 0. Similarly, we can get f pq T by (3.12). Hence, f LΛ (α, pq;g) and f (α,pq) T by (3.3) and (3.14). Thus the mapping f T f is an isometric isomorphism from LΛ (α, pq;g) onto (L 1(G), LΛ (α, pq;g)). The case (L 1(G), LΛ (α, pq; G)) L 1(G) was already proved in [13].

9 On Lipschitz-Lorentz spaces 167 We know that there is a decreasing countable family {V n} n N of open sets at the identity element 0 G such that e n µ(v n) 1 χ Vn for all n N. Now, let us assume that the sequence {e n} n N is contained in Lλ (α, 1; G). We will refer to this property of G as (E α1). By using the same methods as in [13, Lemma 4.1], one can easily prove the completeness of Lλ (α, pq; G) with respect to (α,pq) Lemma. Let G be a metrizable locally compact abelian group with property (E α1), 0 < α <, 1 < p, q < and 1 p + 1 p 1, 1 q + 1 q 1. Then we have L(p, q)(g) LΛ (α, p q ; G) Lλ (α, ; G). Proof. Let f L(p, q)(g), g LΛ (α, p q ; G). Since, f g L (G), we have (3.15) τ(y, f g e n) τ(y,en) (f g) τ(y,en) 1 f g for all y G, y 0. Since e n Lλ (α, 1; G) for each n N, we get f g e n Lλ (α, ; G) for all n N. If we write (3.16) τ(y,f g e n f g) τ(y,g) (f en f) C τ(y,g) p q f en f pq C g (α,p q ) f e n f pq. for y G, y 0. Since {e n} n N is an approximate identity for L(p,q)(G), we get f g e n f g in Lλ (α, ; G) as n by (3.16). Since Lλ (α, ; G) is complete, we get the result f g Lλ (α, ; G) Theorem. Let G be a metrizable locally compact abelian group with property (E α1), 0 < α < and 1 < p,q <. Then Hom L1 (G)(L(p, q)(g), Lλ (α, ;G)) and LΛ (α, p q ; G) are algebraically isomorphic and topologically homeomorphic spaces, where 1 p + 1 p 1 and 1 q + 1 q 1. Proof. Let f LΛ (α, p q ; G). Then we can define a mapping T f : L(p, q)(g) Lλ (α, ; G) such that T f (g) f g for all g L(p, q)(g) by Lemma For g L(p,q)(G), we have τ(y, f g) τ(y, f) g τ(y,f) p q g pq f (α,p q ) g pq. It follows that T f (g) (α, ) f g (α, ) f (α,p q ) g pq for all g L(p, q)(g). Therefore, T f Hom L1 (G)(L(p,q)(G), Lλ (α, ; G)) and T f f (α,p q ). Conversely, let T Hom L1 (G)(L(p,q)(G), Lλ (α, ; G)). Then, by using Theorem 3.8, there exists f LΛ (α, p q ; G) L(p, q )(G) such that T(g) f g for all g L(p, q)(g) and f (α,p q ) k T, where k is a constant. The rest of the proof of this assertion is the same as for Theorem 3.8. Thus, LΛ (α, p q ; G) and Hom L1 (G)(L(p,q)(G), Lλ (α, ; G)) are algebraically isomorphic and topologically homeomorphic spaces.

10 168 I. Eryılmaz, C. Duyar Lemma. Let G be as in the preceding theorem. Then we have L 1(G) LΛ (α, pq; G) Lλ (α, pq;g) for 1 < p <, 1 q <. Proof. Let f L 1(G) and g LΛ (α, pq;g). Then (3.17) τ(y, f g e n) pq τ(y, en) (f g) pq τ(y, en) 1 f g pq. Since e n Lλ (α, 1; G), (3.17) implies that f g e n Lλ (α, pq; G). Now note that if y G and y 0, then the inequality (3.18) τ(y,f g e n f g) pq τ(y, g) (f en f) pq τ(y, g) pq f e n f 1 is obtained. Since {e n} n N is an approximate identity for L 1(G), and f g e n f g in L(p,q)(G), it follows from (3.18) that τ(y,f g e n f g) pq/ converges to zero. Hence f g e n f g in Lλ (α, pq; G), and f g Lλ (α, pq; G) by the completeness of Lλ (α, pq; G) Theorem. Let G be a metrizable locally compact abelian group with property (E α1), 0 < α < and 1 < p, q <. Then (L 1(G), Lλ (α, pq;g)) is isometrically isometric to LΛ (α, pq; G) if 1 < p, q <. Also, if p q 1, then (L 1(G), Lλ (α,1; G)) L 1(G). Proof. Let f LΛ (α, pq; G). By the preceding lemma, we can define a mapping T f : L 1(G) Lλ (α, pq; G) by T f (g) f g. It is easy to see that T f (L 1(G), Lλ (α, pq;g)) with T f f (α,pq). Conversely, let T (L 1(G), Lλ (α, pq;g)). Then, there exists f L(p, q)(g) such that T(g) f g for all g L 1(G) by [4, Theorem 3.4]. For y G with y 0, we can write that τ(y, f) pq τ(y,f) τ(y,f) en pq + τ(y,t(en) pq τ(y,f) τ(y,f) en pq + T e n 1 for each positive integer n. Since τ(y,f) τ(y,f) e n pq 0 as n, we see that τ(y,f) pq sup y 0 T. This shows that f LΛ (α, pq;g). Using the same methods in Theorem 3.9, f pq T and so f (α,pq) T. Thus the mapping f T f is an isometric isomorphism from LΛ (α, pq;g) onto (L 1(G), Lλ (α, pq;g)). The case (L 1(G), Lλ (α,1; G)) L 1(G) was shown in [13]. References [1] Bloom, W. R. Multipliers of Lipschitz spaces on zero dimensional groups, Math. Z. 176, , [2] Blozinski, A. P. On a convolution theorem for L(p, q) spaces, Trans. Amer. Math. Soc. 164, , [3] Blozinski, A. P. Convolution of L(p, q) functions, Proc. Amer. Math. Soc. 32(1), , 1972.

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