Atomic Decompositions of Hardy Spaces with ...

Atomic Decompositions of Hardy Spaces with Variable Exponents and its Application to Bounded Linear Operators Yoshihiro Sawano

Integral Equations and Operator Theory ISSN 0378-620X Volume 77 Number 1 Integr. Equ. Oper. Theory (2013) 77:123-148 DOI 10.1007/s00020-013-2073-1

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Author's personal copy Integr. Equ. Oper. Theory 77 (2013), 123–148 DOI 10.1007/s00020-013-2073-1 Published online July 13, 2013 c Springer Basel 2013

Integral Equations and Operator Theory

Atomic Decompositions of Hardy Spaces with Variable Exponents and its Application to Bounded Linear Operators Yoshihiro Sawano Abstract. As applications of atomic decomposition results of Hardy spaces with variable exponents, we shall prove the boundedness of commutators and the fractional integral operators as well as the Hardy operators. There are many ways to prove such boundedness. For example, the boundedness of commutators can be proved by the sharp maximal inequalities. But here, we propose a different method based upon our atomic decomposition. Mathematics Subject Classification (2010). Primary 41A17; Secondary 42B25, 42B35, 26A33. Keywords. Hardy spaces, fractional integral operators, atomic decomposition.

1. Introduction The aim of this paper is to sharpen and apply the atomic decomposition results of Hardy spaces with variable exponents, which was partly obtained in our earlier paper [31], to the boundedness of linear operators and to compare it with the atomic decomposition results of classical Hardy spaces. Before we describe Hardy spaces with variable exponents, let us recall classical Hardy spaces. Let 0 < p < ∞. The Hardy space H p (Rn ) is given n by the set of all distributions f ∈ S (R ) for which the quasi-norm f H p ≡ supt>0 |etΔ f | p is finite, where {etΔ }t>0 denotes the heat semigroup. L In the present paper we replace Lp (Rn ) with Lp(·) (Rn ). The space Lp(·) (Rn ) is called variable Lebesgue spaces and initiated by Nakano [32,33]. As a counterpart for Hardy spaces, we are led to considering Hardy spaces with variable exponents where we work mainly on. Now let us describe Hardy spaces with variable exponents and their decomposition results. Let The author was supported by Grant-in-Aid for Young Scientists (B), No. 24740085, Japan Society for the Promotion of Science.

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p(·) : Rn → (0, ∞) and f : Rn → C be measurable functions. Then define the variable Lebesgue quasi-norm f Lp(·) of f ; ⎧ ⎫ p(x) ⎨ ⎬ |f (x)| dx ≤ 1 , (1.1) f Lp(·) ≡ inf λ > 0 : ⎩ ⎭ λ Rn

where inf ∅ ≡ ∞. The space L (Rn ) is the set of all measurable functions f on Rn for which the quasi-norm f Lp(·) is finite. Here and below, we shall postulate the following conditions on p(·): p(·)

(log-Hölder continuity)

(decay condition)

|p(x) − p(y)| ≤

|p(x) − p(y)| ≤

C log(1/|x − y|)

C log(e + |x|)

1 , 2 (1.2)

for |x − y| ≤

for |y| ≥ |x|.

(1.3)

The Hardy space H p(·) (Rn ) with variable exponent p(·) is given by the set of all distributions f ∈ S (Rn ) for which the quasi-norm tΔ f H p(·) ≡ sup |e f | (1.4) t>0

Lp(·)

is finite. If we assume 1 < p− ≡ inf x∈Rn p(x) ≤ p+ ≡ supx∈Rn p(x) < ∞, (1.2) and (1.3), then, from Proposition 2.2 below, the Hardy–Littlewood maximal operator M is known to be bounded on Lp(·) (Rn ) and from the reflexivity of Lp(·) (Rn ), we can prove Lp(·) (Rn ) = H p(·) (Rn ) with norm equivalence. We use the following notations about function spaces in the present paper: First, denote by Lqcomp (Rn ) the set of all Lq (Rn )-functions with compact support. For L = 0, 1, 2, · · · , PL (Rn ) denotes the set of all polynomials with degree less than or equal to L and P−1 (Rn ) ≡ {0}. The space PL (Rn )⊥ is the set of all integrable functions f satisfying Rn (1 + |x|)L |A(x)| dx < ∞ and Rn xα A(x) dx = 0 for all multiindices α such that |α| ≤ L. By convention, P−1 (Rn )⊥ is the set of all measurable functions. For L = −1, 0, 1, · · · we n q n n ⊥ define Lq,L comp (R ) ≡ Lcomp (R ) ∩ PL (R ) . If C depends on some parameters such that s, then we write A ≤ Cs B. The following decomposition result extends the one of our earlier paper, which is our starting point of the present paper: We define p = min(p− , 1),

dp(·) ≡ max{[n/p− − n], −1}

for p ∈ (0, ∞). Theorem 1.1. Let p(·) satisfy 0 < p− ≤ p+ < ∞ as well as (1.2) and (1.3). Let L ∈ N ∪ {0} and s ∈ (0, ∞). 1. Let q > p+ when p+ ≥ 1 and q ≥ 1 when p+ < 1. Suppose that we are given countable collections of cubes {Qj }∞ j=1 , of non-negative numbers q,d

p n ∞ {λj }∞ j=1 and of Lcomp (R )-functions {aj }j=1 such that

supp(aj ) ⊂ Qj ,

aj Lq ≤ |Qj |1/q

(1.5)

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and that

⎛ ⎞ p1 ∞ ⎝ p⎠ (λj χQj ) < ∞. (1.6) p(·) j=1 L ∞ Then the series f = j=1 λj aj converges in H p(·) (Rn ) and satisfies ⎛ ⎞ p1 ∞ ⎝ f H p(·) ≤ C (λj χQj )p ⎠ . j=1 p(·) L

2. Let f ∈ H

p(·)

n

(R ). Then there exists a decomposition ∞ f= λj aj j=1

in S (R ) by means of countable collections of cubes {Qj }∞ j=1 , of non∞,L n ∞ and of L (R )-functions {a } negative numbers {λj }∞ j comp j=1 j=1 such that n

|aj | ≤ χQj , and that

⎛ ⎞1/s ∞ ⎝ s⎠ (λj χQj ) j=1

≤ Cs f H p(·) .

Lp(·)

Here and below, we use the following convention about cubes: By a “cube” we mean a closed cube whose edges are parallel to the coordinate axes. Its side length is denoted by (Q) and its center by cQ . For c > 0, c Q denotes a cube concentric to Q with sidelength c (Q). In [31, Theorems 4.5 and 4.6], the possibility when dp(·) = −1 was excluded but actually it is possible by Theorem 1.1. Next, we present a decomposition result for compactly supported functions. Theorem 1.2. Let κ > 1, s > 0, max(1, p+ ) < q < ∞ and L ≥ dp(·) . Supn pose f ∈ Lq,L comp (R ) is supported on a cube Q. Then there exists a decomN position f = j=1 λj aj by means of finite collections of cubes {Qj }N j=1 , of q,L n N non-negative numbers {λj }N and of L (R )-functions {a } such that j j=1 comp j=1 aj Lq ≤ |Qj |1/q , and that

supp(aj ) ⊂ Qj ⊂ κQ

⎛ ⎞ p1 N ⎝ p⎠ (λj χQj ) j=1

(j = 1, 2, · · · , N )

≤ Cκ f H p(·) .

Lp(·)

For comparison, we dare repeat to state Theorems 1.1 and 1.2 for Lp (Rn ) spaces as Theorems 1.3 and 1.4. Theorem 1.3. Let p ∈ (1, ∞), L ∈ N ∪ {0, −1} and s ∈ (0, ∞). Suppose p < q ≤ ∞.


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1. Suppose that we are given countable collections of cubes {Qj }∞ j=1 , of nonQ n ∞ and of L (R )-functions {a } negative numbers {λj }∞ j comp j=1 j=1 such that supp(aj ) ⊂ Qj , and that

aj Lq ≤ |Qj |1/q

∞ λ χ j Qj j=1

Then the series f =

∞

< ∞.

Lp

λj aj converges in Lp (Rn ) and satisfies ∞ ≤C λj χQj < ∞. j=1 p

j=1

f Lp

L

2. Let s > 0 and L ≥ dp . Let f ∈ Lp (Rn ). Then there exist a decomposition ∞ p n f = j=1 λj aj in L (R ) by means of countable collections of cubes ∞ ∞,L n {Qj }∞ j=1 , of non-negative numbers {λj }j=1 and of Lcomp (R )-functions ∞ {aj }j=1 such that |aj | ≤ χQj , and that

⎛ ⎞1/s ∞ ⎝ s⎠ (λj χQj ) j=1

≤ Cs f Lp .

Lp

Theorem 1.4. Let p, q, s ∈ (0, ∞) and L ≥ dp satisfy 1 < p < q < ∞. Let N n f ∈ Lq,L comp (R ). Then there exists a decomposition f = j=1 λj aj by means of finite collections of cubes {Qj }N , of non-negative numbers {λj }N j=1 j=1 and q,L n N of Lcomp (R )-functions {aj }j=1 such that |aj | ≤ χQj , and that

⎛ ⎞1/s N ⎝ s⎠ (λj χQj ) j=1

≤ Cs f Lp .

Lp

Remark that Theorems 1.3 and 1.4 are already included in [48] and that Theorems 1.3(i) and 1.4 with s = p are included in [12, Theorems 2.1 and 2.2]. Let us look back on the history of spaces with variable exponents. It seems that the theory dates back to the paper of Orlicz [36]. Later, Nakano and Luxemberg independently considered spaces of variable exponents [25,32,33] in 1950’s. Especially, the definition of the variable exponent Lebesgue spaces can be found in [32]. It had been left intact until Kov´ acˇik and J. R´ akosn´ık investigated Sobolev spaces based on Lebesgue spaces with variable exponents. About the fractional integral operators, much was studied from earlier. From the point of harmonic analysis, Diening paved the

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theory of the boundedness of the Hardy–Littlewood maximal operator in [6]. Based upon the pioneering paper [6], many authors investigated the boundedness of the Hardy–Littlewood maximal operator in [3,5,9,22,24,34]. With the boundedness of the Hardy–Littlewood maximal operator, the boundedness of other related operators (see [22,26–30,37] for example) and the theory of function spaces (see [2,8,14–16,18,19,38,44,45,49,50] for example) are developed rapidly. See also surveys [39,40]. In [38] variable exponent Campanato spaces are defined in the setting of quasimetric measure spaces. As for Hardy spaces with variable exponents, we refer to [20] as well as [31]. Among others, in addition to the recent development about the spaces with variable exponents, the localization principle proved by H¨ asto is important [11], which seems to have a connection with the proof of the Hardy–Littlewood maximal operator. For their precise statements of the key facts, which we use in the present paper, we refer to Proposition 2.1 here. From the long history of this space, we learn that spaces with variable exponents are difficult to analyze. The main reason was the difficulty of the proof of the boundedness of the Hardy–Littlewood maximal operator; Diening works paved the way. See the textbooks [4,7] for more details. Apart from the development of spaces with variable exponents, the classical Hardy space H p (Rn ) has three different aspects as was described in the book by Stein [47]. When 0 < p < 1, H p (Rn ) contains distributions which are not L1loc (Rn ) functions. When p = 1, H p (Rn ) is strictly embedded into L1 (Rn ). When 1 < p < ∞, by virtue of reflexivity of Lp (Rn ) and the boundedness of the Hardy–Littlewood maximal operator, H p (Rn ) and Lp (Rn ) coincide as a subset of S (Rn ). To have a unified understanding of this strange but important phenomenon, we can use Lebesgue spaces with variable exponents. Notice that we did not require that p+ ≤ 1 nor that p− > 1 in Theorems 1.1 and 1.2. So, once we propose a framework of Hardy spaces with variable exponents, we can treat them in a unified manner. Our plan in the present paper is as follows: first, we recall some elementary facts for variable Lebesgue spaces. Then we prove Theorems 1.1 and 1.2. In Sect. 2, we shall review some fundamental facts for variable exponent Lebesgue spaces. Section 2.1 is intended as a quick review of key inequalities in variable Lebesgue spaces and Sect. 2.2 collects a maximal inequality. We recall and supplement some basic facts about Hardy spaces with variable exponents in Sect. 3. Section 4 is the heart of the present paper. Theorem 1.1(i), Theorem 1.1(ii) and Theorem 1.2 are proved in Sects. 4.1, 4.2 and 4.3, respectively. In Sect. 5, we consider applications of Theorems 1.1 and 1.2. Section 5.1 deals with fractional integral operators. Section 5.2 is devoted to the review of the definition and the boundedness of the singular integral operators. Section 5.3 intends as the definition and the boundedness of commutators. The Fefferman–Phong inequality is considered in Sect. 5.4, where we are convinced that we essentially improve the result of [31]. By the Fefferman–Phong inequality, or the trace inequality, we mean g · Iα f X ≤ cgY · f Z

(1.7)


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for some Banach spaces X, Y and Z. When X and Z are Morrey spaces, namely, if their norms are given by ⎛ ⎞1/q1 1 1 − f X = f Mpq11 = sup |Q| p1 q1 ⎝ |f (y)|q1 dy ⎠ x∈Rn

Q

and hZ = hMpq22 = sup |Q| x∈Rn

1 p2

− q1

2

⎛ ⎞1/q2 ⎝ |h(y)|q2 dy ⎠ , Q

then (1.7) is referred to as the Olsen inequality. In [35], for m = 1 Olsen considered (1.7) to investigate the Schr¨ odinger equation with Y being Morrey spaces. Later, many authors considered and sharpened (1.7) with m = 1. We refer to [13,17,41,42,46] for related results. We take up the Hardy operator in Sect. 5.5. Finally in Sect. 5.6 we disprove that the Fourier transform is not bounded from Lp(·) (Rn ) to Lp (·) (Rn ) even when the exponent p(·) satisfies 1 < p− ≤ p+ < ∞ as well as (1.2) and (1.3).

2. Variable Lebesgue Spaces We consider (1.1) under the conditions (1.2) and (1.3). 2.1. Some Inequalities About Exponents for Variable Lebesgue Spaces Note that p∞ = lim p(x) exists in view of (1.3). From p+ < ∞ and (1.2) it x→∞ follows that 1 for all x, y ∈ Rn . |p(x) − p(y)| ≤ C (2.1) log(e + 1/|x − y|) Observe that (1.3) is equivalent to the following estimate; 1 for all x ∈ Rn . |p(x) − p∞ | ≤ C log(e + |x|)

(2.2)

Note that (2.2) is equivalent to | log(e + |x|)(p(x)−p∞ ) | ≤ C, that is, (e + |x|)p(x) ∼ 1 for all x ∈ Rn . (e + |x|)p∞

(2.3)

Among other related inequalities, we recall the following localization principle due to H¨ asto [11]: Proposition 2.1. Under the conditions (2.1) and (2.2), the equivalence 1/p∞ p∞ f Lp(·) ∼ (χm+[0,1]n f Lp(·) ) (f ∈ Lp(·) (Rn )) m∈Zn

holds. 2.2. Fefferman–Stein Type Maximal Inequality In this paper, we still need the following Fefferman–Stein type inequality for the Hardy–Littlewood maximal operator M , which is given by


Atomic Decompositions of H p(·) (Rn ) 1 M f (x) ≡ sup |Q| Q∈Q(x)

129

|f (y)| dy.

(2.4)

Q

Here Q(x) denotes the set of all cubes containing x. We invoke the following estimate: Proposition 2.2. Let p(·) satisfy 1 < p− ≤ p+ < ∞ as well as (1.2) and (1.3). For every q ∈ (1, ∞], ⎛ ⎛ ⎞1/q ⎞1/q ∞ ∞ ⎝ ⎝ q⎠ q⎠ M fj ≤ Cp(·),q |fj | j=1 p(·) j=1

.

Lp(·)

L

Proposition 2.2 seems to have been a hint of defining the sequence norm in Theorems 1.1 and 1.2. An important fact illustrated in [8, p. 1746] was that we can not replace q with variable exponents.

3. Known Results for Variable Hardy Spaces and Some Improvements The following results for variable Hardy spaces are known and in the present paper we take them for granted: first, we recall some of equivalent expression about Hardy spaces with variable exponents. We topologize S(Rn ) by the collection of semi-norms {pN }N ∈N given by sup (1 + |x|)N |∂ α ϕ(x)| pN (ϕ) ≡ |α|≤N

x∈Rn

for each N ∈ N. Define FN ≡ {ϕ ∈ S(Rn ) : pN (ϕ) ≤ 1}.

(3.1)

Let f ∈ S (R ). Denote by Mf the grand maximal operator given by n

Mf (x) ≡ sup{|t−n ψ(t−1 ·) ∗ f (x)| : t > 0,

ψ ∈ FN },

where we choose and fix a large integer N . Below we write B(r) ≡ {x ∈ Rn : |x| ≤ r}. The Fourier transform and its inverse are defined respectively by 1 f (x) exp(−ix · ξ) dx, Ff (ξ) ≡ n (2π) 2 Rn 1 −1 F f (x) ≡ f (ξ) exp(ix · ξ) dξ. n (2π) 2 Rn

Theorem 3.1 ([31, Chapters 3 and 5]). Let p(·) satisfy 0 < p− ≤ p+ < ∞ n as well as (1.2) and (1.3). S(Rn ) satisfy the Let f ∈ S (R ) and let ϕ ∈ n non-degenerate condition Rn ϕ(x) dx = 0. Let ψ ∈ S (R ) be chosen so that χB(1) ≤ ψ ≤ χB(2) . Define Δj f (x) ≡ F −1 [(ψ(2−j ·) − ψ(2−j+1 ·))Ff ](x). Then the following are equivalent:


Y. Sawano

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f ∈ H p(·) (Rn ). Mf Lp(·) is finite. supt>0 |t−n ϕ(t−1 ·) ∗ f | p(·) is finite. L 1 ∞ ∞ n 2 2 4. f = j=−∞ Δj f holds in S (R ) and j=−∞ |Δj f |

1. 2. 3.

< ∞.

Lp(·)

If one of these conditions is satisfied, then f H p(·) ∼ Mf Lp(·)

⎛ ⎞ 12 ∞ −1 ϕ(t ·) ∗ f 2⎠ ⎝ sup ∼ ∼ |Δ f | j p(·) t>0 tn L j=−∞

Lp(·)

holds. Proof. The conditions 1.–3. are equivalent as we can see [31, Chapter 3]. Assume f ∈ H p(·) (Rn ). Then by [31, Chapter 5.3], we have ⎛ ⎞ 12 ∞ ⎝ |Δj f |2 ⎠ < ∞. f H p(·) ∼ j=−∞ p(·) L

N Let us show that f = j=−∞ Δj f holds in S (Rn ). Set fN ≡ j=−N Δj f for each N ∈ N. Then {fN }N ∈N is a Cauchy sequence in H p(·) (Rn ). Denote by g its limit in H p(·) (Rn ). Since it is established in [31, Remark 3.5] that ∞ H p(·) (Rn ) → S (Rn ), it follows that g = j=−∞ Δj f holds in S (Rn ). Since f − g has frequency support in {0}, f − g agree with a polynomial P in S (Rn ). Since P = f − g belongs to H p(·) (Rn ), we must have P = 0. Thus, it follows that ∞

f =g=

∞

Δj f

j=−∞

holds in S (Rn ). If we assume 4., then H

p(·)

N

j=−N Δj f

n

∞ N =1

is a Cauchy sequence in

(R ) because we know that ⎛ ⎞ 12 ∞ ⎝ 2⎠ |Δj g| gH p(·) ∼ j=−∞

Lp(·)

for all g ∈ H p(·) (Rn ). (See [31, Section 5.3].) Hence

N

j=−N Δj f

∞ N =1

is

convergent to an element h ∈ H (R ). The convergence takes place in the topology of S (Rn ) as well. Thus, it follows that p(·)

f = lim

N →∞

N j=−N

n

Δj f = h ∈ H p(·) (Rn ).



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To state some fundamental embeddings, we define ∞ n n n ⊥ S∞ (R ) = S(R ) ∩ . PL (R ) L=1

Theorem 3.2. Let p(·) satisfy 0 < p− ≤ p+ < ∞ as well as (1.2) and (1.3). 1. S∞ (Rn ) → H p(·) (Rn ) → S (Rn ) in the sense of continuous embedding. p+ +1,d 2. Lcomp p(·) (Rn ) is dense in H p(·) (Rn ). Proof. The inclusion H p(·) (Rn ) → S (Rn ) is proved in [31, Remark 3.1]. We p+ +1,d also know that Lcomp p(·) (Rn ) is dense in H p(·) (Rn ). (See [31, Section 4].) The inclusion S∞ (Rn ) → H p(·) (Rn ) holds since S∞ (Rn ) → H p− (Rn ) ∩ H p+ (Rn ) → H p(·) (Rn ).

4. Atomic Decomposition 4.1. Proof of Theorem 1.1 1. We modify the proof of our earlier paper [31]. Actually, the following key lemma is improved: Lemma 4.1. Let q > p+ ≥ 1 Suppose negative

or

(4.1)

that we are given countable collections of cubes {Qj }∞ j=1 , q,dp ∞ n ∞ numbers {λj }j=1 and of Lcomp (R )-functions {aj }j=1 such supp(aj ) ⊂ Qj ,

Then

q = 1 > p+ .

⎛ ⎞ p1 ∞ ⎝ p⎠ |λj aj | j=1

Lp(·)

1/q

aj Lq ≤ |Qj |

.

of nonthat (4.2)

⎛ ⎞ p1 ∞ ⎝ p⎠ ≤C |λj χQj | j=1

< ∞.

Lp(·)

Remark that the condition (4.1) was q 1 in our earlier paper [31, Theorem 4.6]. Proof. As before, we can assume that the sums are essentially finite. Choose a positive function g ∈ L(p(·)/p) (Rn ) so that ⎛ ⎞ p1 ⎛ ⎞ p1 ∞ ∞ ⎝ |λj aj |p ⎠ =⎝ |λj aj (x)|p g(x) dx⎠ . j=1 p(·) Rn j=1 L

Then by the H¨ older inequality, we obtain ∞ ∞ |λj aj (x)|p g(x) dx = |λj |p |aj (x)|p g(x) dx Rn j=1

j=1

≤

∞ j=1

Rn

|λj |p (aj Lq )p gL(q/p) (Qj ) .


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If we invoke (4.2), then we have

∞

Rn j=1

|λj aj (x)|p g(x) dx ≤

∞

|λj |p (|Qj |1/q )p gL(q/p) (Qj )

j=1

≤

∞

|λj |p χQj (x)M [g (q/p) ](x)1/(q/p) dx.

j=1Rn

An arithmetic shows p(x) q < p p

−

⇐⇒

q p+ > ⇐⇒ q > p+ . p p

Thus, we can use Hölder inequality and obtain the desired result.

Now we prove Theorem 1.1 1. of exception. Assume for the time being that λj = 0 with finite number Fix ϕ ∈ S(Rn ) satisfying the non-degenerate condition ϕ(x) dx = 0. As we showed in [31, (5.2)], we have Mϕ aj (x) ≤ Cχ3Qj M aj (x) + M χQj (x)

n+dp +1 − n

(x ∈ Rn ).

This pointwise estimate yields ⎛ ⎞ ∞ Mϕ ⎝ ⎠ λj aj p(·) j=1 L ∞ ∞ n+dp +1 − n ≤C λj χ3Qj M aj + λj (M χQj ) j=1 p(·) j=1 L ∞ ∞ n+d p− +1 n ≤C λj χ3Qj M aj + λj (M χQj ) j=1 p(·) j=1 L

.

Lp(·)

Therefore, by Proposition 2.2 and Lemma 4.1, we obtain ⎛ ⎞ ∞ Mϕ ⎝ ⎠ λ a j j p(·) j=1 L ⎧ ⎫ p1 ∞ ∞ ⎨ ⎬ n+dp +1 − p n λj χ3Qj M aj ≤C + λ (M χ ) j Q j ⎩ ⎭ p(·) j=1 j=1 p(·) L L ⎧ ⎫ p1 ∞ ⎨ p ⎬ λj χQj ≤C . ⎩ ⎭ j=1 p(·) L



In summary, we obtained ⎛ ⎞ ∞ Mϕ ⎝ ⎠ λ a j j j=1

Lp(·)

133

⎧ ⎫ p1 ∞ ⎨ p ⎬ λj χQj ≤C ⎩ ⎭ j=1

.

(4.3)

Lp(·)

Therefore, the result is proved if λ has only a finite number of non-zero entries. Suppose that we are given countable collections of cubes {Qj }∞ j=1 , of q,d

p n ∞ non-negative numbers {λj }∞ j=1 and of Lcomp (R )-functions {aj }j=1 satisfying (1.5) and (1.6). Then from (4.3), we learn that ⎧ ⎫ p1 ⎛ ⎞ N N 2 2 ⎨ p ⎬ Mϕ ⎝ λj χQj λj aj ⎠ ≤C (4.4) ⎩ ⎭ j=N1 j=N 1 Lp(·) p(·)

for 1 ≤ N1 ≤ N2 < ∞. Therefore,

J

j=1

λj aj

∞

L

is a Cauchy sequence

J=1 p(·) n

in H (R ) and converges to an element f ∈ H (R ). Since H p(·) (Rn ) is known to beembedded continuously into S (Rn ), it follows that the sequence ∞ J converges to f in S (Rn ). Note that j=1 λj aj p(·)

n

J=1

⎧ ⎫ p1 J ⎨ p ⎬ lim λj χQj J→∞ ⎩ ⎭ j=1

Lp(·)

⎧ ⎫ p1 ∞ ⎨ p ⎬ λj χQj = ⎩ ⎭ j=1

Lp(·)

by the monotone convergence theorem. Consequently, from (4.4) and the Fatou lemma, we deduce ⎧ ⎫ p1 ⎛ ⎞ ∞ J ⎨ p ⎬ Mϕ ⎝ λj χQj λj aj ⎠ ≤C f H p(·) ≤ C lim J→∞ ⎩ ⎭ p(·) j=1 j=1 L

Lp(·)

and Theorem 1.1 1. was proved. Remark 4.2. q,d

p(·) 1. Let q > max(1, p+ ). The above observation shows that Lcomp (Rn ) is p(·) n included in H (R ). 2. If p+ < 1, a similar argument shows q = 1 will do. The same can be said for Theorem 1.2.

4.2. Proof of Theorem 1.1 2. We invoke the following decomposition result from [47]: Lemma 4.3. Let d ∈ {−1, 0, 1, 2, · · · } and j ∈ Z. Suppose that f ∈ Lq (Rn ) with q ≥ 1. Then there exist collections of cubes {Q∗j,k }k∈Kj and functions ∞ {ηj,k }k∈Kj ⊂ Ccomp (Rn ), and a decomposition f = gj + bj , b = k∈Kj bj,k , such that


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(i) The {Q∗j,k }k∈Kj have the bounded intersection property, and Q∗j,k = {Mf > 2j } = Oj . k∈Kj

(ii) Each function ηj,k is supported in Q∗j,k and ηj,k = χ{Mf >2j } , 0 ≤ ηj,k ≤ 1. k∈Kj

(iii) The distribution gj satisfies the inequality: Mgj (x) ≤ C Mf (x)χ{Mf >2j } (x) + 2j k

j,k n+d+1 (j,k + |x − xj,k |)n+d+1

for x ∈ R . (iv) Each distribution bj,k is given by bj,k ≡ ηj,k (f − cj,k ) with a polynomial cj,k ∈ Pd (Rn ) satisfying Rn bj,k (x)q(x) dx = 0 for all q ∈ Pd (Rn ), and 2j j,k n+d+1 χRn \Q∗j,k (x) (x ∈ Rn ). Mbj,k (x) ≤ C Mf (x)χQ∗j,k (x)+ |x−xj,k |n+d+1 n

In the above, xj,k and j,k denote the center and the side-length of Q∗j,k , respectively, and the implicit constants are dependent only on n. By virtue of Remark 4.2, the routine argument described in [47] and the density result obtained in [31], we can assume that f ∈ Lq,dp(·) (Rn ) with q > max(1, p+ ). For each j ∈ Z, consider the level set Oj ≡ {x ∈ Rn : Mf (x) > 2j }.

(4.5)

Then it follows immediately from the definition that Oj+1 ⊂ Oj .

(4.6)

If we invoke Lemma 4.3, then f can be decomposed; f = gj + bj , bj = bj,k , bj,k = ηj,k (f − cj,k ) k

where each bj,k is supported in a cube Q∗j,k as is described in Lemma 4.3. We have shown in [31, p. 3691] f=

∞

(gj+1 − gj ),

(4.7)

j=−∞

with the sum converging in the sense of distributions. Here, going through the same argument as the one in [47, p108–109], we have an expression; f= Aj,k , gj+1 − gj = Aj,k (j ∈ Z) (4.8) j,k

k

in the sense of distributions, where each Aj,k , supported in Q∗j,k , satisfies the pointwise estimate |Aj,k (x)| ≤ C0 2j for some universal constant C0 and the


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moment condition Rn Aj,k (x)q(x) dx = 0 for every q ∈ Pd (Rn ). With these observations in mind, let us set aj,k ≡

Aj,k , C0 2j

κj,k ≡ C0 2j .

Then we automatically obtain that each aj,k satisfies |aj,k | ≤ χQ∗j,k , and that f=

aj,k ⊥ PL (Rn )

κj,k aj,k

j,k

in the topology of Lq (Rn ) ≈ H q (Rn ), since f ∈ Lq (Rn ). It remains to prove the estimate of coefficients; once this can be achieved, we have only to rearrange {Aj,k }j,k and {κj,k }j,k . From the definition we need to estimate ⎛ ⎞1/s ∞ ⎝ s⎠ |λj χQj | p(·) j=1 L ⎧ ⎫ ⎛ s ⎞ p(x) s ⎪ ⎪ ⎨ ⎬ κj,k χQ∗j,k (x) ⎠ = inf λ > 0 : ⎝ dx ≤ 1 . ⎪ ⎪ λχQ∗j,k Lp(·) ⎩ ⎭ j,k n R

Since {Q∗j,k }k forms a Whitney covering of Oj (see Lemma 4.3 (i)), we have ⎛ ⎞1/s ∞ ⎝ s⎠ |λj χQj | p(·) j=1 L ⎧ ⎫ ⎛ ⎞ p(x) s ⎪ ⎪ s ∞ j ⎨ ⎬ 2 χOj (x) ⎠ ∼ inf λ > 0 : ⎝ dx ≤ 1 . ⎪ ⎪ λ ⎩ ⎭ j=−∞ n R

Recall that Oj ⊃ Oj+1 for each j ∈ Z (see (4.6) above). Consequently we have ⎞s ⎛ ⎞s ⎛ s ∞ j ∞ ∞ 2j χOj \Oj+1 (x) 2 χOj (x) 2j χOj (x) ⎠ ∼⎝ ⎠ . ∼⎝ λ λ λ j=−∞ j=−∞ j=−∞ Thus, we obtain ⎛ ⎞1/s ∞ ⎝ s⎠ |λj χQj | j=1

⎧ ⎪ ⎨

≤ C inf λ > 0 : ⎪ ⎩

Lp(·)

Rn

⎫ ⎞p(x) ⎪ ∞ ⎬ 2j χOj \Oj+1 (x) ⎝ ⎠ dx ≤ 1 . ⎪ λ ⎭ ⎛

j=−∞


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We deduce from (4.5), the definition of Oj that ⎞p(x) ⎛ ∞ ∞ 2j χOj \Oj+1 (x) ⎠ ⎝ dx = λ j=−∞ j=−∞ Rn

∼ Rn

Therefore, we obtain ⎛ ⎞1/s ∞ ⎝ s⎠ |λj χQj | j=1

Oj \Oj+1

Mf (x) λ

2j λ

p(x) dx

p(x) dx.

⎧ ⎫ p(x) ⎨ ⎬ Mf (x) ≤ C inf λ > 0 : dx ≤ 1 ⎩ ⎭ λ

Lp(·)

Rn

= Cf H p(·) ,

(4.9)

q,d

p(·) (Rn ). if f ∈ Lcomp

4.3. Proof of Theorem 1.2 n Under a special setting that f ∈ Lq,L comp (R ), we shall reexamine the proof of Theorem 1.1 2.. Roughly speaking, we need to consider the truncation with respect to j. This consists of two steps. As the first step, we consider a truncation with respect to j. We disregard j ≥ j0 for some j0 . According to the proof of Theorem 4.5 in [31], we know the structure of Mf (x). More precisely, −n−dp −1 (Q) 1 f L1 (Q) + χκQ (x)M f (x) . Mf (x) ≤ Cκ |Q| (Q) + |x − cQ | Recall that Oj is given by (4.5) and that we have (4.6). Therefore, there exists j0 ∈ Z such that Oj ⊂ κQ for all j ≥ j0 . Then

⎛

supp(gj0 ) = supp ⎝f −

j≥j0

and |gj0 (x)| ≤ C0 2 χκQ (x) ≤ C0

⎞ Aj,k ⎠ ⊂ Oj0 ⊂ κQ

k

j0

(4.10)

inf Mf (y) χκQ (x).

y∈Oj

(4.11)

Here C0 is a constant that needs to be specified. Let λ1 = 2j0 C0 . Then λ1 χκQ Lp(·) ≤ CMf Lp(·) ≤ Cf H p(·) .

(4.12)

The next step is, roughly speaking, to truncate of j and k such that j ≤ j0 . Fix x ∈ Rn and write j1 = j1 (x) = [1 + log2 Mf (x)]. Recall that Aj,k


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137

/ Oj . Thus, in view of the is supported in Q∗j,k ⊂ Oj . If j ≥ j1 , then x ∈ expression (4.8) and the bounded overlapping property of Q∗j,k , we have

j1 (x)

|Aj,k (x)| =

2j ≤ C2j1 (x) ≤ CMf (x).

j≤j1 (x)

j=j0 k∈Z

j≥j0 k∈Z

|Aj,k (x)| ≤ C

This means that ⎞ ⎛ |Aj,k |⎠ ⊂ Oj0 and that |Aj,k | ∈ Lq (Rn ). supp ⎝ j≥j0 k∈Z

(4.13)

j≥j0 k∈Z

Choose a finite set F ⊂ ([j0 , ∞) ∩ Z) × Z so that χκQ Lp(·) A j,k f H p(·) (j,k)∈([j0 ,∞)∩Z)×Z\F

1

≤ |κQ| q .

(4.14)

Lq

Set h≡

χκQ Lp(·) f H p(·)

Aj,k .

(j,k)∈([j0 ,∞)∩Z)×Z\F

Then, from (4.9), (4.11), (4.12), (4.13) and (4.14), we conclude gj f H p(·) κj,k aj,k + ·h f = λ1 0 + λ1 χκQ Lp(·) (j,k)∈F

is the desired finite decomposition.

5. Applications of Atomic Decompositions 5.1. Boundedness of Fractional Integral Operators Now we investigate the boundedness of fractional integral operator Iα of order α, which is given by f (y) dy (x ∈ Rn ). Iα f (x) ≡ |x − y|n−α Rn

Theorem 5.1. Let p(·) satisfy 0 < p− ≤ p+
1 j q(·)

L

⎞ ⎠

Lq(·)

Now we prove Theorem 5.4. n Proof. We may assume f ∈ L∞,L comp (R ) with L 1 in view of the density of n p(·) L∞,L (Rn ). Then we have comp (R ) in H


Y. Sawano

f=

N

IEOT

λj aj

j=1

as we described in Theorem 1.2. By virtue of the moment condition |Iα f (x)| ≤ C

N

λj (Qj )L+n ((Qj ) + |x − c(Qj )|)L+n−α

j=1

≤C

N

λj (Qj )α M χQj (x)

L+n−α n

.

j=1

Since we can take L large enough, we can assume L+n−α > 1. n Thus, by Proposition 2.2, it follows that N L+n−α α n Iα f Lq(·) ≤ C λ (Q ) (M χ ) j j Qj j=1 q(·) L N ≤C λj (Qj )α χQj . j=1 q(·) L

If we invoke Lemma 5.2, then we have N Iα f Lq(·) ≤ C λ χ j Qj j=1

≤ Cf Lp(·) .

Lp(·)

This is the desired inequality.

5.2. Boundedness of Singular Integral Operators By a “singular integral operator”, we mean an L2 (Rn )-bounded linear operator T equipped with the kernel K satisfying the following properties: (0) K is a C-valued measurable function on Rn × Rn \ diag, where diag is a diagonal set given by diag ≡ {(x, x) ∈ Rn × Rn : x ∈ Rn }. (1) On Rn × Rn \ diag, the size estimate |K(x, y)| ≤ C|x − y|−n holds. (2) If (x, y), (x, z) ∈ Rn × Rn \ diag satisfy 2|y − z| ≤ |x − z|, then the H¨ olmander estimate |y − z| |K(x, y) − K(x, z)| + |K(y, x) − K(z, x)| ≤ C |x − y|n+1 holds. (3) If f ∈ L2comp (Rn ), then T f (x) =

K(x, y)f (y) dy Rn

for almost all x ∈ R \ supp(f ). n



141

A well-known fact in harmonic analysis is that T can be extended to a bounded linear operator on Lq (Rn ) for # all 1 < q < ∞. Thus, with this fact, we tacitly assume that T is defined on 1 0 so that (n + 1 − ε)p− > n. With Assuming p− > this choice, the above series is summable and ⎧ ⎧ ⎫ p1 ⎫ p1 N N ∞ ⎨ ⎨ ⎬ ⎬ 1+ε (k2−k(n+1) λj χ2k Qj )p ≤C (λj (M χQj ) p )p . ⎩ ⎩ ⎭ ⎭ j=1 p p j=1 k=1 n n+1 ,

L (·)

L (·)

Since ε > 0, if we use Proposition 2.2 and the John–Nirenberg inequality, then we obtain (5.5). If we modify the proof, we have a similar assertion when p− = 1; the range space will be replaced by the weak space w − Lp(.) (Rn ). We omit the detail. 5.4. Olsen’s Inequality Applying the improved atomic decomposition, we can prove the following theorem: Theorem 5.6. Let 0 < α < n and 1 ≤ q ≤ 1 ≤ p− ≤ p+ < ∞,

n α.

Let p(·) satisfy q > p+

n/α

as well as (1.2) and (1.3). Then for g ∈ Mq have

(Rn ) and f ∈ H p(·) (Rn ), we

g · Iα f Lp(·) ≤ CgMn/α f H p(·) . q


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IEOT

n Proof. By density, we can assume that f ∈ L∞,d comp (R ). Then we have

f=

N

λj aj

j=1

as we described in Theorem 1.2. Here we take d ∈ N so that d > α + With this decomposition, ⎛ ⎞ ∞ N |g · Iα f (x)| ≤ C 2−kL ⎝ |g(x)|(Qj )α λj χ2k Qj (x)⎠ .

n p− .

j=1

k=1

Here L = d − α. Observe that N N α −kα k α |g|(Q ) λ χ = 2 |g|(2 Q ) λ χ k k j j 2 Qj j j 2 Qj p(·) j=1 p(·) j=1 L L N ≤ CgMn/α λ χ kQ j 2 j q j=1 p(·) L N kn ≤ C2 p− gMn/α λ χ j Q j q j=1 p(·) in view of Theorem 1.1 and Proposition 2.2. Since d > α + estimate is summable over k.

L n p− ,

the above

5.5. Boundedness of the Hardy Operator We place ourselves in R+ and consider the Hardy operator. 1 Hf (t) = t

t f (s) ds

(t ∈ R+ ).

0

Although H is defined for functions defined on (0, ∞), we shall use the zero extension to define Hf (t) for functions on R. That is, for f ∈ L∞ comp (R), we define t 1 f (s) ds. Hf (t) = χ(0,∞) (t) × t 0

Here and below we assume that p(·) satisfies 1 ≤ p− ≤ p + < ∞ as well as (1.2) and (1.3). If a ∈ L∞,1 comp (0, ∞) is supported on a cube Q contained in (0, ∞), then a simple calculation shows |Ha(t)| ≤ aL∞ χQ (t). Since L∞,1 comp (0, ∞) is dense p(·) ∞,1 in H (R), and any function f ∈ Lcomp (0, ∞) admits a finite decomposition N f = j=1 λj aj in the way described in Theorem 1.2, we can recapture [10, (1.1)] with α = 0.


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Theorem 5.7. Assume that p : R → [1, ∞] satisfies 1 ≤ p− ≤ p+ < ∞ as well as (1.2) and (1.3). Then the Hardy operator H is bounded from H p(·) (R) to Lp(·) (R). 5.6. Unboundedness of the Fourier Transform Here we disprove that a natural extension of the Hausdorff–Young inequality is available for variable Lebesgue spaces. Proposition 5.8. Let

$

p(x) ≡ min

3 , max {|x| − 0.1, 1} 2

% (x ∈ Rn ).

Then there does not exist a constant C > 0 such that Ff Lp (·) ≤ Cf Lp(·)

(5.7)

for all f ∈ Lp(·) (Rn ). ∞ (B((2n)−1 )) satisfy Proof. Assume that inequality (5.7) holds. Let a ∈ Ccomp 2 n Fa(0) = 1 and κ = {κm }m∈Zn be an (Z )-sequence such that κm = 0 if |m| ≥ 3. Then Fκ (x) ≡ κm a(x − m) (x ∈ Rn ) m∈Zn 2

n

belongs to L (R ) and Fκ Lp(·) ≤ CaL2 · κL3/2 (Zn ) .

(5.8)

On the other hand,

|FFκ (0)| = κm ≤ FFκ Lp (·) . n

(5.9)

m∈Z

If we combine (5.7), (5.8) and (5.9), then κm ≤ CκL3/2 (Zn ) . n

(5.10)

m∈Z

Since (5.10) is valid for all 2 (Zn )-sequences κ = {κm }m∈Zn such that κm = 0 if |m| ≥ 3, (5.10) is a contradiction. Acknowledgments I am indebted to Professor Yoshihiro Mizuta for his kind suggestion that lead me to considering Subsection 5.6. I am grateful to Dr. Yohei Tsutsui for his knowledge about Hardy spaces. I am thankful to Professor Kenji Nakanishi for his hint to obtain the condition (4.1). This work is supported by Grantin-Aid for Young Scientists (B) No. 21740104 and No. 24740085, which is from Japan Society for the Promotion of Science. I am thankful to Professor Luong Dang Ky for his helpful comment about Theorem 5.4.


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[42] Sawano, Y., Sugano, S., Tanaka, H.: Orlicz-Morrey spaces and fractional operators. Potential Anal. 36, 517–556 (2012) [43] Samko, S.: Convolution and potential type operators in the space Lp(x) . Integral Transform Special Funct. 7(3–4), 261–284 (1998) [44] Samko, N., Vakulov, B.: Spherical fractional and hypersingular integrals in generalized H¨ older spaces with variable characteristic. Math. Nachrichten 284, 355–369 (2011) [45] Samko, N., Samko, S.G., Vakulov, B.: Fractional integrals and hypersingular integrals in variable order H¨ older spaces on homogeneous spaces. Armen. J. Math. 2(2), 38–64 (2009) [46] Sugano, S.: Some inequalities for generalized fractional integral operators on generalized Morrey spaces. Math Inequal. Appl. 14(4), 849–865 (2011) [47] Stein, E.M.: Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993) [48] Str¨ omberg, J.O., Torchinsky, A.: Weighted Hardy spaces. Lecture Notes in Mathematics, vol. 1381. Springer, Berlin (1989) [49] Vakulov, B.G.: Spherical potentials in weighted H¨ older spaces of variable order. Dokl. Akad. Nauk, 400(1) (2005), 7–10. Translated in Doklady Mathematics 71(1), 1–4 (2005) [50] Vakulov, B.G.: Spherical potentials of complex order in the variable order H¨ older spaces. Integral Transforms Spec. Funct. 16(5-6), 489–497 (2005) Yoshihiro Sawano (B) 1-1 Minami-Ohsawa Hachioji, Tokyo 192-0397 Japan e-mail: [email protected] Received: February 22, 2013. Revised: June 10, 2013.