NOTES ON THE HERZ TYPE HARDY SPACES OF VARIABLE SMOOTHNESS AND INTEGRABILITY

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1 M athematical I nequalities & A pplications Volume 19, Number , doi: /mia NOTES ON THE HERZ TYPE HARDY SPACES OF VARIABLE SMOOTHNESS AND INTEGRABILITY DOUADI DRIHEM AND FAKHREDDINE SEGHIRI Communicated by I. Perić Abstract. The aim of this paper is twofold. First we give a new norm equivalents of the variable Herz spaces K α Rn and K α Rn. Secondly we use these results to prove the atomic decomposition for Herz-type Hardy spaces of variable smoothness and integrability. Also, we prove the boundedness of a wide class of sublinear operators on these spaces, which includes maximal, potential and Calderón-Zygmund operators. 1. Introduction It is well-nown that function spaces have been a central topic in modern analysis, and are now of increasing applications in areas such as harmonic analysis and partial differential equations. Some examples of these spaces can be mentioned such as: Herz spaces. It is well nown that these spaces play an important role in Harmonic Analysis. After they have been introduced in [11], the theory of these spaces had a remarable development in part due to its usefulness in applications. For instance, they appearin the characterizationof multipliers on Hardy spaces [3], in the summability of Fourier transforms [10] and in regularity theory for elliptic and parabolic equations in divergence form [20], [21]. In recent years, there has been growing interest in generalizing classical spaces such as Lebesgue, Sobolev spaces, Besov spaces, Triebel-Lizorin spaces to the case with either variable integrability or variable smoothness. The motivation for the increasing interest in such spaces comes not only from theoretical purposes, but also from applications to fluid dynamics [22], image restoration [4] and PDE with non-standard growth conditions. Herz spaces K α,q Rn and K α,q Rn with variable exponent p but fixed α R and q 0,] were recently studied by Izui [12, 13]. These spaces with variable exponents α and were studied in [2], where they gave the boundedness results for a wide class of classical operators on these function spaces. The spaces K α Rn and K α Rn,werefirst introduced by Izui and Noi in [14]. Many authors are interested in Herz spaces with variable exponents, for example, [8], [9], [23], [24] and Mathematics subject classification 2010: 42B20, 42B35, 46E30. Keywords and phrases: Herz-type Hardy space, atom, variable exponent, sublinear operator. c D l,zagreb Paper MIA

2 146 D. DRIHEM AND F. SEGHIRI [26]. See also [5] and[27] for further results on Herz type Besov and Triebel-Lizorin spaces with fixed exponents. The main purpose of this paper is to consider Herz-type Hardy spaces HK α Rn and H K α Rn, where the exponents α and q are variables. The paper is organized as follows. First we give some preliminaries where we fix some notations and recall some basic facts on function spaces with variable integrability. In the preliminary section we also give some ey technical lemmas needed in the proofs of the main statements. We then define the Herz spaces K α Rn and K α Rn in Section 3 and give several basic properties. We show that for some special parameters, the spaces K α Rn are just the Herz spaces K α,q R n and f K α {2 α0 f χ } q0 + l < L {2α f χ } l q > L. Also we define the Herz-type Hardy spaces HK α Rn and H K α Rn and we present the relation between these function spaces and Herz spaces by using the boundedness results for a wide class of classical operators, where for maing the presentation clearer, we give the proofs of the boundedness of these class of operators later in Section 5. The main statements are formulated in Section 4, where we give the atomic decomposition of these function spaces. 2. Preliminaries As usual, we denote by R n the n-dimensional real Euclidean space, N the collection of all natural numbers and N 0 = N {0}. The letter Z stands for the set of all integer numbers. For a multi-index α =α 1,...,α n N n 0, we write α = α α n. The Euclidean scalar product of x =x 1,...,x n and y =y 1,...,y n is given by x y = x 1 y x n y n. The expression f g means that f cg for some independent constant c and non-negative functions f and g, and f g means f g f.asusual for any x R, [x] stands for the largest integer smaller than or equal to x. For x R n and r > 0 we denote by Bx,r the open ball in R n with center x and radius r. By supp f we denote the support of the function f, i.e., the closure of its non-zero set. If E R n is a measurable set, then E stands for the Lebesgue measure of E and χ E denotes its characteristic function. The symbol S R n is used in place of the set of all Schwartz functions on R n and we denote by S R n the dual space of all tempered distributions on R n. The variable exponents that we consider are always measurable functions on R n with range in [c,[ for some c > 0. We denote the set of such functions by P 0.The subset of variable exponents with range [1, is denoted by P.Forp P 0 R n,we use the notation p + = esssup px, p = essinf px. x R n x R n

3 NOTES ON THE HERZ-TYPE HARDY SPACES 147 Everywhere below we shall consider bounded exponents. Let p P 0 R n.thevariable exponent Lebesgue space L R n is the class of all measurable functions f on R n such that the modular ρ f := f x px dx R n is finite. This is a qausi-banach function space equipped with the norm 1 } f := inf {μ > 0:ρ μ f 1. If px p is constant, then L R n =L p R n is the classical Lebesgue space. A useful property is that ρ f 1 if and only if f 1unit ball property. This property is clear for constant exponents due to the obvious relation between the norm and the modular in that case. We say that a function g : R n R is locally log-hölder continuous, if there exists a constant c log > 0 such that gx gy c log lne + 1/ x y for all x,y R n.if c log gx g0 lne + 1/ x for all x R n, then we say that g is log-hölder continuous at the origin or has a log decay at the origin. If, for some g R and c log > 0, there holds gx g c log lne + x for all x R n, then we say that g is log-hölder continuous at infinity or has a log decay at infinity. By P0 lnrn and P lnrn we denote the class of all exponents p PR n which have a log decay at the origin and at infinity, respectively. The notation P ln R n is used for all those exponents p PR n which are locally log-hölder continuous and have a log decay at infinity, with p := lim x px. Obviously we have P ln R n P0 lnrn P lnrn. Note that p P ln R n if and only if p P ln R n, and since p =p we write only p for any of these quantities. Let p,q P 0. The mixed Lebesgue-sequence space l q L is defined on sequences of L -functions by the modular { ρ l q L f fv } v v =inf λ v > 0:ρ 1. v λ 1/q v The quasi-norm is defined from this as usual: 1 f v v l q L {μ = inf > 0:ρ l q L μ f v v }

4 148 D. DRIHEM AND F. SEGHIRI Since q + <, then we can replace 2.1 by the simpler expression ρ l q L f v v = fv q. Furthermore, if p and q are constants, then l q L =l q L p. It v q is nown, cf. [1] and[15], that l q L is a norm if q 1 is constant almost everywhere a.e. on R n 1 and 1, or if px + 1 qx 1a.e. onrn,orif1 qx px < a.e. on R n. Very often we have to deal with the norm of characteristic functions on balls or cubes when studying the behavior of various operators in Harmonic Analysis. In classical L p spaces the norm of such functions is easily calculated, but this is not the case when we consider variable exponents. Nevertheless, it is nown that for p P log we have χ B χ B p B. 2.2 Also, χ B B 1 px, x B 2.3 for small balls B R n B 2 n, and χ B B 1 p 2.4 for large balls B 1, with constants only depending on the log-hölder constant of p see, for example, [7, Section 4.5]. Here p denotes the conjugate exponent of p given by 1/+1/p =1. We refer the reader to the recent monograph [7, Section 4.5] for further details, historical remars and more references on variable exponent spaces. The following lemma is from [6, Lemma 2.11], see also [17, Lemma 2.6]. LEMMA 1. Let p P log. For any cubes balls P and Q, such that P Q, we have Q 1/p + C χ Q Q 1/p c P χ P P with c,c > 0 are independent of Q and P. The next lemma is a Hardy-type inequality which is easy to prove. LEMMA 2. Let 0 < a < 1 and 0 < q. Let{ε } Z be a sequence of positive real numbers, such that {ε } Z l q = I <. Then the sequences { δ : δ = j a j } ε j Z and { η : η = j a j } ε j Z belong to l q, and {δ } Z l q + {η } l Z q ci, with c > 0 only depending on a and q. The proof of the following results are given in [2], where the second lemma is a generalization of 2.2, 2.3 and2.4 to the case of dyadic annuli.

5 NOTES ON THE HERZ-TYPE HARDY SPACES 149 LEMMA 3. Let α L R n and r 1 > 0. Ifα is log-hölder continuous both at the origin and at infinity, then α + r1 r2 if 0 < r 2 r 1 2 r αx 1 r αy 2 1 if r 1 2 < r 2 2r 1 α r1 r2 if r 2 > 2r 1 for any x B0,r 1 \ B0, r 1 2 and y B0,r 2 \ B0, r 2 2, with the implicit constant not depending on x,y,r 1 and r 2. LEMMA 4. Let p P lnrn and let R = B0,r \ B0, r 2.If R 2 n,then χ R R 1 px R p 1 with the implicit constants independent of r and x R. The left-hand side equivalence remains true for every R > 0 if we assume, additionally, p P0 lnrn P lnrn. For convenience, we set 3. Variable Herz-type Hardy spaces B := B0,2, R := B \ B and χ = χ R, Z. DEFINITION 1. Let p,q P 0 R n and α : R n R with α L R n.theinhomogeneous Herz space K α Rn consists of all f L loc Rn such that f α := f χ K B0 + 2 α f χ 1 <. 3.1 l q L Similarly, the homogeneous Herz space K α Rn is defined as the set of all f L loc Rn \{0} such that 2 α f χ f K α := If α and p,q are constant, then K α K p,q α Rn are the classical Herz spaces. Let us denote Z <. 3.2 l q L Rn =Kp,q α R n and K α Rn = {g } l q > L := =0 g q 1/q and {g } l q < L := 1/q g q = for sequences {g } Z of measurable functions with the usual modification if q =. Now we present the main result of this section.

6 150 D. DRIHEM AND F. SEGHIRI PROPOSITION 1. Let α L R n,p,q P 0 R n.ifα and q are log-hölder continuous at infinity, then K α Rn =K α,q R n. Additionally, if α and q have a log decay at the origin, then f K α {2 α0 f χ } q0 + l < L {2α f χ } l q > L. 3.3 Rn, which is equiva- for any f K α,q R n. By the scaling argument, we Proof. Step 1. We will prove that K α,q R n K α lent to f α f K K α,q see that it suffices to consider the case f K α = 1 and show that the modular of f,q on the left-hand side is bounded. In particular, we will show that =1 c 2 α q f χ q for some constant c > 0. Since α has logarithmic decay at infinity, then for 1and x R we have αx α 1. lne + x Therefore, 2 αx 2 α with constants independent of and x, and hence c 2 α q f χ c 2 α q f χ. =1 q =1 q Our estimate 3.4, clearly follows from the inequality c 2 α q f χ 2 α q f χ + 2 = δ. 3.5 q This claim can be reformulated as showing that δ c 2 α q f χ 1, q which is equivalent to c δ 1 q 2 α f χ 1. We have for any x R δ 1 qx =2 δ 1 q 1 qx 2 1 qx 1 q δ 1 q.

7 NOTES ON THE HERZ-TYPE HARDY SPACES 151 Since q has logarithmic decay at infinity, then for 1andx R we have qx q q qx qx q q q 1. lne + x Therefore, 2 q 1 1 qx 1 with constants independent of and x. Also, since 1 < 2 δ < 2 +1, 2 δ q 1 1 qx 2 +1 q 1 1 qx 1. Hence, with an appropriate choice of c > 0 c δ 1 q 2 α f χ δ q 1 2 α f χ 1, because of 2 α f χ δ q 1. Step 2. We will prove that K α Rn K α,q R n, which is equivalent to f K α,q f K α that it suffices to consider the case f K α for any f K α Rn. By the scaling argument, we see =1 c 2 α q f χ = 1andshowthat for some constant c > 0. As before, we have for 1 2 α q 2 f χ α q f χ. Now, our estimate 3.6, clearly follows from the inequality c 2 α q f χ 2 α q f χ + 2 = δ. 3.7 q This claim can be reformulated as showing that c δ q 1 2 α f χ 1. From above, δ q 1 δ 1 qx, then with an appropriate choice of c > 0 c δ q 1 2 α f χ δ 1 q 2 α f χ. The left-hand side is less than or equal 1 if and only if δ 1 q q 2 α f χ 1. q

8 152 D. DRIHEM AND F. SEGHIRI We see that the right-hand side can be rewritten us δ 2 α q f χ 1 q which follows immediately from the definition of δ. Step 3. Let us prove that {2 α0 f χ } q0 + l < L {2α f χ } l q > L f K α. We suppose that f K α 1. If, in addition, α has a log decay at the origin, then we also have 2 αx 2 α0 for < 0andx R. Thus {2 α0 f χ } {2α q0 f χ l < L } q0. l < L As in Step 2 we can prove that c 2 α f χ q0 2 α f χ q q for any < 0 and for some constant c > 0. Then {2 α0 f χ } q0 1. Using l < L the estimate 3.7 we obtain {2 α f χ } l q > L 1. Therefore, + 2 {2 α0 f χ } q0 + l < L {2α f χ } l q > L 1. The desired estimate can be obtained by the scaling argument. Now let {2 α0 f χ } q0 l < L 1and {2α f χ } l q > L 1. As in Step1 we have for any < 0 and for some constant c > 0 c 2 α q f χ 2 α0 q0 f χ + 2 q and using 3.5 we obtain = 2 α f χ q q 1. Therefore, f K α 1 and hence the result follows by the scaling argument. Let G N f be the grand maximal function of f defined by G N f x= sup ϕ A N ϕ N f x, where A N = {ϕ S R n :sup α N, β N x α β ϕx 1} and ϕn f x=sup t>0 ϕ t f x, with ϕ t = t n ϕ t.

9 NOTES ON THE HERZ-TYPE HARDY SPACES 153 DEFINITION 2. Let p,q P 0 R n and α : R n R with α L R n and N > n + 1. The inhomogeneous Herz-type Hardy space HK α Rn consists of all f S R n such that G N f K α Rn and we define f α = G HK N f α. K Similarly, the homogeneous Herz-type Hardy space H K α Rn is defined as the set of all f S R n such that G N f K α Rn and we define f H K α = G N f K α. We consider sublinear operators satisfying the size condition T fx R n f y dy, x / supp f 3.8 x y n for integrable and compactly supported functions f. Condition 3.8 is satisfied by several classical operators in Harmonic Analysis, such as Calderón-Zygmund operators, the Carleson maximal operator and the Hardy-Littlewood maximal operator see [18] and[25]. Using the same of arguments as in [2], we obtain the following results. THEOREM 1. Let q P 0 R n,p P lnrn with 1 < p p + <, and let α and q are log-hölder continuous at infinity, with α L R n and n < α < n p p. Suppose that T is a sublinear operator satisfying estimate 3.8. If T is bounded on L R n, then T is bounded on K α Rn. For homogeneous spaces we have the following statement: THEOREM 2. Let q P 0 R n,p P0 lnrn P lnrn with 1 < p p + <, and let α and q are log-hölder continuous, both at the origin and at infinity, such that α L R n and n p + < α α + < n 1 1 p. Then every sublinear operator T satisfying 3.8 which is bounded on L R n is also bounded on K α Rn. The proof of Theorems 1 and 2 is postponed to the Appendix. We have G N satisfies the size condition 3.8. Let α,q, p as in Theorem 1, with p P ln R n.then HK α Rn L loc Rn =K α Rn. Let α,q, p as in Theorem 2, with p P ln R n.then H K α Rn L loc Rn \{0}= K α Rn.

10 154 D. DRIHEM AND F. SEGHIRI 4. Atomic decomposition In recent years, it turned out that atomic decomposition of some function spaces are extremely useful in many aspects. This concerns, for instance, the investigation of compact embeddings between function spaces. But this applies equally to questions of mapping properties of some operators, such as Calderón-Zygmund operators, the commutator of Calderón-Zygmund operator with a BMO function and to trace problems, where arguments can be equivalently transferred to the sequence space, which is often more convenient to handle. The main goal of this section is to prove an atomic decomposition result for HK α Rn and H K α Rn. First we introduce the basic notation. DEFINITION 3. Let α L R n, p PR n, q P 0 R n and s N 0. A function a is said to be a central α, -atom, if i suppa B0,r={x R n : x r}, r > 0. ii a B0,r α0/n,0< r < 1. iii a B0,r α/n, r 1. iv R n xβ axdx = 0, β s. A function a on R n is said to be a central α, -atom of restricted type, if it satisfies the conditions iii, vi above and suppa B0,r, r 1. Now we come to the atomic decomposition theorems. THEOREM 3. Let α andqarelog-hölder continuous at infinity and p P ln R n with 1 < p p + <. For any f HK α Rn, we have f = =0 λ a, 4.1 where the series converges in the sense of distributions, λ 0, eacha is a central α, -atom of restricted type with suppa B and 1/q λ q c f HK α. =0 Conversely, if α n1 1 p and s [α + n 1 p 1], and if 4.1 holds, then f HK α Rn, and f HK α inf λ q =0 1/q, where the infimum is taen over all the decompositions of f as above.

11 NOTES ON THE HERZ-TYPE HARDY SPACES 155 THEOREM 4. Let α and q are be log-hölder continuous, both at the origin and at infinity and p P ln R n with 1 < p p + <. For any f H K α Rn,we have f = λ a, 4.2 = where the series converges in the sense of distributions, λ 0, eacha is a central α, -atom with suppa B and 1/q0 1/q λ q0 + λ q c f H K α. = =0 Conversely,if α n1 1 p and s [α + + n 1 p 1], and if 4.2 holds, then f H K α Rn, and f H K α inf 1/q0 λ q0 + = λ q =0 where the infimum is taen over all the decompositions of f as above. 1/q, REMARK 1. Corresponding statements to Theorems 3 and 4 were proved by Liu and Wang [19], with α and q constants, under the assumption that the maximal operator M is bounded on L R n both in the homogeneous and the inhomogeneous situation. Here we are requiring the log-hölder continuity at two points only zero and infinity. Proof. By similarity, we only consider H K α Rn. The proof follows the ideas in [18], see also [19]. Step 1. We prove the necessity part of the theorem. Let f S R n, ϕ D R n such that ϕ 0, ϕxdx = 1 and suppϕ B 0.Setϕ j = 2 jn ϕ2 j and f j = f ϕ j, j N 0. It is well now that f j C R n and lim j f j = f.letψ be a radial smooth function such that suppψ {x : 1 2 ε x 1 + ε}, 0 < ε < 1 4,andψx =1, if 1 2 x 1. Let R,ε = {x :2 2 ε x ε} and ψ = ψ2. It is easy to see that suppψ { R,ε, ψ x =1ifx R and 1 = ψ x 2, ψ x x 0. Set Φ x= j= ψ jx, x 0 0, x = 0.Then = Φ x=1, x 0. We denote by P m the class of all the real polynomials with the degree less than m. LetP j x= P R f j Φ,ε xχ R x P m be the unique polynomial satisfying,ε R,ε f j xφ x P j x x β dx = 0, [ ] 1 β α + + n p 1 = m.

12 156 D. DRIHEM AND F. SEGHIRI Observe that f j x = f j xφ x P j x + = = I j + II j. = P j x For I j,letg j x= f jxφ x P j x and a j λ = b +1 l= = g j λ,where B +1 α /n G N f χ l and b is a constant which will be chosen later. It is easy to see that suppg j B +1 and I j = = λ a j. R,ε Let us prove that a j is a central α, -atom. Let {ϕd : d m} be the 1 orthogonal polynomials restricted to R,ε with respect to the weight, which are R,ε obtained from {x β : β m} by Gram-Schmidt s method, that is ϕd,ϕ v = 1 ϕd R,ε R xϕ v xdx = δ dv.,ε Therefore, P j x = d m f j Φ,ϕd ϕ d x for x R,ε. Observe that ϕv x = ϕv 1 2 x a.e for x R,ε,bytheHölder inequality B +1 αx/n P j x c B +1 αx/n f j y Φ y dy R,ε R,ε c B +1 αx/n χ p f j Φ R,ε. R,ε Using Lemma 3 to obtain B +1 αx/n 2 αx 2 αy for any x,y R,ε. Hence B +1 α /n g j c B +1 α /n f j Φ + c B +1 α /n χ p f j Φ R,ε R,ε χ R,ε c B +1 α /n f j Φ C B +1 α /n G N f χ l, +1 l= p where we used the fact that χ R,ε χ R R,ε,ε, see Lemma 4. Choose b = C, we obtain B+1 α /n a j c.

13 NOTES ON THE HERZ-TYPE HARDY SPACES 157 This relation is equivalent to the inequalities ii and iii in Definition 3 and hence each a j is a central α, -atom with support contained in B +1. Furthermore, since B +1 αx/n 2 αx 2 lαx 2 lα0 for any x R l with and l +1, = λ q0 C C = = +1 l= +1 l= q0 B +1 α /n G N f χ l q0 2 lα0 G N f χ l c G N f q0, K α which is the desired estimate. Similarly, there exists a constant c > 0 such that =0 λ q c G N f q. K α Here we use the fact that B +1 αx/n 2 αx 2 lαx 2 lα for any x R l with 0and l + 1. It remains to estimate II j.let{ψd : d m} be the dual basis of {xβ : β m} 1 with respect to the weight R,on R,ε,ε,thatis Let h j Therefore,,d x= II j = l= = d m = ψd,xβ = 1 ψd R,ε R xxβ dx = δ β d.,ε ψ d xχ x R,ε R,ε d m = l= d m = = ψ +1 d f j Φ,x d ϕ d xχ R,ε x f j Φ l,x d ψ d xχ x R,ε R,ε h j,d x= d m = xχ R x +1,ε R +1,ε f jxφ l xx d dx. R n σ,d a j,d x, ψ +1 d xχ R x +1,ε R +1,ε where and σ,d = b a j +2 l=,d x=h j,d x σ,d B +2 α /n G N f χ l,

14 158 D. DRIHEM AND F. SEGHIRI where b > 0 is a constant which will be chosen later. Observe that f jyφ l yy d dy R n c 2n+ d G N f x, x B +2 and then l= ψd xχ x R,ε R,ε,d ψ +1 d B +2 α /n h j C xχ R x +1,ε c 2 n+ d R +1,ε +2 l= +1 l= χ l x, B +2 α /n G N f χ l. Thus if we tae b = C, thena j,d is a central α, -atom with support contained in R,ε R +1,ε B +2. Furthermore, since B +1 αx/n 2 αx 2 lαx 2 lα0 for any x R l with and l + 2, and = σ,d q0 C = +1 =0 l= by Proposition 1. Hence f j can be written as q0 2 lα0 G N f χ l σ,d q c G N f q, K α f j = = λ a j c G N f q0 K α where a j is a central α, -atom with support contained in R,ε R +1,ε B +2, λ is independent of j and 1/q0 1/q λ q0 + λ q c G N f α, K = =0 where c is independent of j and f. Using the Banach-Alaoglu theorem and the usual diagonal method, we can fined a subsequence { j v } of N such that for each Z, lim v a j v = a in S R n,which is a central α, -atom supported on B +2. Now it remain to prove that f = = λ a 4.3 in S R n. For each ϕ S R n, note that suppa j v +2 i= R i = R and f,ϕ = lim λ a j v xϕxdx. v =

15 NOTES ON THE HERZ-TYPE HARDY SPACES 159 It will now that, see [19] This term is bounded by a j v xϕxdx a c 2m+1 j v 1 χ R. c 2 m+1 α0 2 α a j v 1 χ R c 2 m+1 α0 2 α a j v χ R χ R p c 2 m+1 α0+n1 1 p 2 α a j v χ R, 0 c 2 m+1 α0+n1 1 p where we have used successively 2 α 2 α0, < 0,Hölder s inequality and Lemma 4. If > 0, let 0 N such that 0 + α n + n p > 0, then by the fact that 2 α 2 α, > 0, Hölder inequality and Lemma 4 we have a j v xϕxdx c a j v x x 0 dx R c 2 0+α 2 α a j v χ p χ R R c 2 0+α n+ n p, where c > 0 is independent of. Ifweset then { b = λ 2 m+1 α0+n1 1 p if 0 λ 2 n 0 α p n if > 0, 1/q0 1/q b c λ q0 + λ q c G N f α <. K = = =0 Therefore, f,ϕ = λ a xϕxdx. = This means that 4.3 holds in the sense of distribution. Step 2. We prove the sufficiency part of the theorem. In view of Proposition 1, it suffices to estimate {2 α0 G N f χ } l q0 < L and {2 α G N f χ } l q > L.

16 160 D. DRIHEM AND F. SEGHIRI For any < 0, 2 α0 G N f χ = λ l 2 α0 G N a l χ l= 2 l= + l= + l= By the L boundedness of the grand maximal operator G N, the third term is bounded by c 2 α0 λ l a l l=0 l=0 λ l 2 α0 lα 2 lα a l 2 α0 λ l 2 lα l=0 2 α0 l=0 λ l q 1/q. The l q0 < of this expression is bounded by l=0 λ l q 1/q. Now the second term is bounded c l= λ l 2 lα0 2 lα a l l= λ l 2 lα0. Here we use that fact that 2 lα0 2 lαy, y B l and l 0, since l αx α0 cl loge + 1 x c, x B l,l 0. Again, the l q0 1/q0 < of this expression is bounded by = λ q0 by Lemma 2. Using the similar arguments used in [19] and, again, the fact that 2 lα0 2 lαy y B l and l 0, we obtain 2 α0 G N a l x c 2 lm+1+α0 x n+m+1 a l 1, x R c 2 lα+ 2 lm+1 n+m+1 2 lα a l 1, l + 2. Applying Hölder s inequality and the fact that a l is a central α, -atom, we obtain 2 α0 G N a l χ c 2 lα+ 2 lm+1 n+m+1 χ Bl p χ B c 2 lα+ 2 lm+1 n+m+1 χ Bl p χ B c 2 l m+1+n α+ n p,

17 NOTES ON THE HERZ-TYPE HARDY SPACES 161 where in the last estimate we have used Lemma 1. Hence the first sum in 4.4 is bounded by c 2 l= λ l 2 l m+1+n α+ n p. Therefore, the l q0 < of this expression is bounded by Again, for any 0, we have As before, we have 2 α G N f χ Hence the first sum is bounded by = = λ q0 1/q0. λ l 2 α G N a l χ l= l= 2 + l=0 + l=. 2 α G N a l x 2 l m+1+n α+ n p, 0 > l. c 2 m+1+n α+ n p l= λ l 2 lm+1+n α+ n p 1/q0 2 m+1+n α+ n p λ q0. = Since again, m n α + n p > 0, the l q > -norm of this expression is bounded by 1/q0 c λ q0. = The remaining terms they are essentially similar to the estimate of the first term and the second term in 4.4. Hencethe theoremis proved. REMARK 2. In the necessity part of the theorem, the atoms in the decompositions 4.1 and4.2 can be taen to be supported in dyadic annuli. Also the assumption p P ln R n can be replaced by p P0 lnrn P lnrn, respectively p P lnrn, in the H K α Rn spaces, respectively HK α Rn spaces. 5. Appendix Here we present the more technical proofs of the Theorems 1 and 2. Weomitthe proof of Theorem 1 since is essentially similar to the proof of Theorem 4.2 in [2], then we need only to prove the Theorem 2. Our proofs use partially some decomposition

18 162 D. DRIHEM AND F. SEGHIRI techniques already used in [16] and[2]. In view of Proposition 1 we use the property 3.3. We split the operator into T fx T f χb 2 x + T f χ R x + T f χr n \B +2 x, where R := { x R n :2 2 x < 2 +2} with Z and x R. Estimation of T f χ B 2.Wehave 2 α0 T f χb 2 x 2 α0 B 2 x y n f y dy= 2 α0 2 x y n f y dy j= R j for any x R, < 0. To estimate the last integral we note that x y x y > 2 4 and 2 αx 2 α0 if x R. Hence by Lemma 3 we arrive at the inequality 2 α0 2 T f χb 2 x 2 jα+ n 2 jαy f y dy j= R j for any < 0. After applying Hölder s inequality to the last integral, we get 2 α0 T 2 f χ B 2 χ 2 jα+ n 2 jα f χ j χ j p χ. j= Since p P ln 0 Rn P ln R n implies p P ln 0 Rn P ln R n, then Lemma 4 gives 1 χ j p p R j x j px, x j R j, and χ R, x R. Hence the sum above can be rewritten as 2 2 jα+ n R j 1 px j 1 R px 2 jα f χ j. j= Now we can distinguish three cases as follows here we present the all cases: 0 j 2: by Lemma 4 we get R j 1 px j 1 px R R j p 1 R p 1 2 j p n j n 2 j < 0 2: in this case we obtain j 2 < 0: here we have R j 1 px j 1 px R R j 1 p R 1 p 2 R j 1 px j 1 px R R R j 1 px j R 1 px 1 1 j n p. px j 2 p. j n p.

19 NOTES ON THE HERZ-TYPE HARDY SPACES 163 Indeed, since x < 2, x j < 2 j < 2 we mae use of local log-hölder continuity of p at the origin and get, for 0, 1 px 1 log 1 log 1 2 px j R log e + 1 c 2 with c > 0 independent of, j,x,x j. Therefore, in all cases we have essentially the same bound and hence, combining the estimates above, we arrive at the inequality 2 α0 T 2 f χ B 2 χ 2 jα+ n+ n p 2 jα f χ j. 5.1 j= Since α + n + n p < 0, we apply Lemma 2 and get 2 α0 T 1/q0 1/q0 f χ B 2 χ q0 2 α0 f χ R q0 f K α. = = To estimate 2 α T f χ B 2 in l q > -norm, we have the same estimate 5.1, with 2 α in place of 2 α0. We write 2 2 jα+ n+ n p 2 jα f χ j = j= j= j= for any 0 we put 2 j=1... = 0if = 0,1,2. Since α+ n + n p < 0, thefirst term is bounded by 2 α+ n+ n p 0 2 jα+ n+ n j= p sup 2 jα0 f χ j c 2 α+ n+ n j 0 p f K α The l q > -norm of this expression is bounded by c f K α. Again by Lemma 2, we can estimate the second term in 5.2. Estimation of T f χ R. Using the boundedness of T. {T 2 α0 f χ R } l q0 < L + {T 2 α f χ R } l q > L {2 α0 f χ R } q0 + l < L {2α f χ R } l q > L f K α.,q Estimation of T f χ R n \B +2. Using a combination of the arguments used in the proof of Theorem 1 in [2], we arrive at the inequality 2 α0 T f χ R n \B +2 χ j= 2 jα + n p + 2 jα f χ j 5.3

20 164 D. DRIHEM AND F. SEGHIRI for any < 0. This sum can be rewritten us j=... + j=0 Observing that α + n p + > 0, an application of Lemma 2 yields that the l q0 < -norm of the first sum is bounded by c... 1/q0 2 α0 f χ q0 f K α. = Taing into account that 2 jαy 2 jα for any y R j, j 0, the second sum is bounded by c 2 α + n p + 2 jα + n p + sup2 jα f χ j 2 α + n p + f K α. j=0 j 0 The l q0 < -norm of this expression is bounded by c f K α. To estimate the term 2 α T f χ R n \B +2 in l q > -norm, we have the same estimate 5.3, with 2 α in place of 2 α0, an application of Lemma 2 yields the desired inequality and hence the proof of Theorem 2 is complete. Acnowledgement. The authors than Alexandre Almeida for invaluable discussions and suggestions, where he told that there is a strong possibility that K α Rn is isomorphic to some space K α,q R n, with q constant. Also, we than the anonymous referee for pointing the references [7, 8, 23, 24, 26, 27] out to us and for the valuable comments and suggestions. REFERENCES [1] A. ALMEIDA AND P. HÄSTÖ, Besov spaces with variable smoothness and integrability, J. Funct. Anal , [2] A. ALMEIDA AND D. DRIHEM, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl , [3] A. BAERNSTEIN II AND E. T. SAWYER, Embedding and multiplier theorems for H p R n,mem. Amer. Math. Soc. 53, no. 318, [4] Y. CHEN, S. LEVINE AND R. RAO, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math , no. 4, [5] D. DRIHEM, Embeddings properties on Herz-type Besov and Triebel-Lizorin spaces, Math. Ineq and Appl. 16, , [6] D. DRIHEM, Some properties of variable Besov-type spaces, Funct. Approx. Comment. Math 52, , [7] L. DIENING, P. HARJULEHTO, P. HÄSTÖ AND M. RŮŽIČKA, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, [8] B. DONG AND J. XU, New Herz type Besov and Triebel-Lizorin spaces with variable exponents, J. Funct. Spaces Appl. Volume , Article ID , 27 pages.

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