Variable Triebel-Lizorkin-type spaces

Variable Triebel-Lizorkin-type spaces arXiv:1508.05980v2 [math.FA] 13 Jan 2016

Douadi Drihem

∗

January 14, 2016

Abstract In this paper we study Triebel-Lizorkin-type spaces with variable smoothness and integrability. We show that our space is well-defined, i.e., independent of the choice of basis functions and we obtain their atomic characterization. Moreover the Sobolev embeddings for these function spaces are obtained. MSC 2010 : 46E35. Key words and phrases. Atom, Embeddings, Triebel space, maximal function, variable exponent.

1

Introduction

Function spaces with variable exponents have been intensively studied in the recent years by a significant number of authors. The motivation to study such function spaces comes from applications to other fields of applied mathematics, such that fluid dynamics and image processing. Some examples of these spaces can be mentioned such as: variable exponent Lebesgue spaces Lp(·) and variable Sobolev spaces W k,p(·) where the study of these function spaces was initiated in [23]. Almeida and Samko [2] and Gurka, Harjulehto and Nekvinda [19] introduced variable exponent Bessel potential spaces, wich generalize the classical Bessel potential spaces. Leopold [24-27] and Leopold & Schrohe [28] studied pseudo-differential operators, they introduced related Besov spaces α(·) with variable smoothness Bp,p . Along a different line of study, J.-S. Xu [49], [50] has studied Besov spaces with variable p, but fixed q and α. More general function spaces with variable smoothness were explicitly studied by Besov [3], including characterizaα(·) tions by differences. Besov spaces of variable smoothness and integrability, Bp(·),q(·) , initially appeared in the paper of A. Almeida and P. H¨ast¨o [1]. Several basic properties were established, such as the Fourier analytical characterisation and Sobolev embeddings. When p, q, α are constants they coincide with the usual function spaces s Bp,q . Also, A. I. Tyulenev [40], [41] has studied some new function spaces of variable smoothness. Triebel-Lizorkin spaces with variable exponents were introduced by [7]. They proved a discretization by the so called ϕ-transform. Also atomic and molecular decomposition of these function spaces are obtained and used it to derive trace results. The Sobolev M’sila University, Department of Mathematics, Laboratory of Functional Analysis and Geometry of Spaces , P.O. Box 166, M’sila 28000, Algeria, e-mail: [email protected] ∗

1

embedding of these function spaces was proved by J. Vyb´ıral, [42]. Some properties of these function spaces such as local means characterizations and characterizations by ball means of differences can be found in [21] and [22]. When α, p, q are constants they α coincide with the usual function spaces Fp,q . In recent years, there has been increasing interest in a new family of function spaces s,τ which generalize the Triebel-Lizorkin spaces, called Fp,q spaces, were introduced and s studied in [43]. When τ = 0, they coincide with the usual function spaces Fp,q . Various properties of these function spaces including atomic, molecular or wavelet decompositions, characterizations by differences, have already been established in [10, 12-13, 29, 33-35, 43-46, 48]. Moreover, these function spaces, including some of their special cases related to Q spaces. Based on Triebel-Lizorkin-type spaces and Triebel-Lizorkin spaces with variable expoα(·) nents Fp(·),q(·) , we present another Triebel-Lizorkin-type spaces with variable smoothness and integrability, which covers Triebel-Lizorkin-type spaces with fixed exponents. These type of function spaces are introduced in [47], where several properties are obtained such as atomic decomposition and the boundedness of trace operator. The paper is organized as follows. First we give some preliminaries where we fix some notations and recall some basics facts on function spaces with variable integrability and we give some key technical lemmas needed in the proofs of the main statements. For making the presentation clearer, we give their proofs later in Section 6. In Section α(·),τ (·) 3 we define the spaces Fp(·),q(·) where several basic properties such as the ϕ-transform characterization are obtained. In Section 4 we prove elementary embeddings between these functions spaces, as well as Sobolev embeddings. In Section 5, we give the atomic decomposition of these function spaces.

2

Preliminaries

As usual, we denote by Rn the n-dimensional real Euclidean space, N the collection of all natural numbers and N0 = N ∪ {0}. The letter Z stands for the set of all integer numbers. The expression f . g means that f ≤ c g for some independent constant c (and non-negative functions f and g), and f ≈ g means f . g . f . As usual for any x ∈ R, [x] stands for the largest integer smaller than or equal to x.

By supp f we denote the support of the function f , i.e., the closure of its non-zero set. If E ⊂ Rn is a measurable set, then |E| stands for the (Lebesgue) measure of E and χE denotes its characteristic function.

The Hardy-Littlewood maximal operator M is defined on L1loc by Z 1 Mf (x) = sup |f (y)| dy r>0 |B(x, r)| B(x,r) R 1 and MB f (x) = |B| |f (y)| dy, x ∈ B. The symbol S(Rn ) is used in place of the set of B all Schwartz functions ϕ on Rn . We denote by S ′ (Rn ) the dual space of all tempered distributions on Rn . The Fourier transform of a tempered distribution f is denoted by F f while its inverse transform is denoted by F −1 f . 2

For v ∈ Z and m = (m1 , ..., mn ) ∈ Zn , let Qv,m be the dyadic cube in Rn , Qv,m = {(x1 , ..., xn ) : mi ≤ 2v xi < mi + 1, i = 1, 2, ..., n}. For the collection of all such cubes we use Q := {Qv,m : v ∈ Z, m ∈ Zn }. For each cube Q, we denote its center by cQ , its lower left-corner by xQv,m = 2−v m of Q = Qv,m and its side length by l(Q). For r > 0, we denote by rQ the cube concentric with Q having the side length rl(Q). + Furthermore, we put vQ = − log2 l(Q) and vQ = max(vQ , 0). For v ∈ Z, ϕ ∈ S(Rn ) and x ∈ Rn , we set ϕ e (x) := ϕ(−x), ϕv (x) := 2vn ϕ(2v x), and ϕv,m (x) := 2vn/2 ϕ(2v x − m) = |Qv,m |1/2 ϕv (x − xQv,m ) if Q = Qv,m .

The variable exponents that we consider are always measurable functions p on Rn with range in [c, ∞[ for some c > 0. We denote the set of such functions by P0 . The subset of variable exponents with range [1, ∞[ is denoted by P. We use the standard notation p(x), p+ := ess-sup p(x). p− := ess-inf n x∈R

x∈Rn

The variable exponent Lebesgue space Lp(·) is the class of all measurable functions f on Rn such that the modular Z ̺p(·) (f ) := |f (x)|p(x) dx Rn

is finite. This is a quasi-Banach function space equipped with the quasi-norm o 1  n kf kp(·) := inf µ > 0 : ̺p(·) f ≤ 1 . µ If p(x) := p is constant, then Lp(·) = Lp is the classical Lebesgue space. A useful property is that ̺p(·) (f ) 6 1 if and only if kf kp(·) 6 1 (unit ball property). This property is clear for constant exponents due to the obvious relation between the norm and the modular in that case. Let p, q ∈ P0 . The space Lp(·) (ℓq(·) ) is defined to be the set of all sequences (fv )v≥0 of functions such that





(fv (x)) q(x)

(fv ) p(·) q(·) :=

p(·) < ∞.

v≥0 ℓ v≥0 L (ℓ ) L

It is easy to show that Lp(·) (ℓq(·) ) is always a quasi-normed space and it is a normed space, if min(p(x), q(x)) ≥ 1 holds point-wise. log We say that g : Rn → R is locally log-H¨older continuous, abbreviated g ∈ Cloc , if there exists clog (g) > 0 such that |g(x) − g(y)| ≤

clog (g) log(e + 1/ |x − y|)

(1)

for all x, y ∈ Rn . We say that g satisfies the log-H¨older decay condition, if there exists g∞ ∈ R and a constant clog > 0 such that |g(x) − g∞ | ≤

clog log(e + |x|)

for all x ∈ Rn . We say that g is globally-log-H¨older continuous, abbreviated g ∈ C log , if it is locally log-H¨older continuous and satisfies the log-H¨older decay condition. The 3

constants clog (g) and clog are called the locally log-H¨older constant and the log-H¨ older log decay constant, respectively. We note that all functions g ∈ Cloc always belong to L∞ .

We define the following class of variable exponents o n 1 P log := p ∈ P : ∈ C log , p

were introduced in [9, Section 2]. We define 1/p∞ := lim|x|→∞ 1/p(x) and we use the 1 convention ∞ = 0. Note that although 1p is bounded, the variable exponent p itself can be unbounded. It was shown in [8], Theorem 4.3.8 that M : Lp(·) → Lp(·) is bounded if p ∈ P log and p− > 1, see also [9], Theorem 1.2. Let p ∈ P log , ϕ ∈ L1 and Ψ (x) := sup|y|≥|x| |ϕ (y)|. We suppose that Ψ ∈ L1 . Then it was proved in [8, Lemma 4.6.3] that kϕε ∗ f kp(·) ≤ ckΨk1 kf kp(·)
for all f ∈ Lp(·) , where ϕε := ε1n ϕ ε· . We also refer to the papers [5] and [6], where various results on maximal function in variable Lebesgue spaces were obtained. −m Recall that η v,m (x) := 2nv (1 + 2v |x|) , for any x ∈ Rn , v ∈ N0 and m > 0. Note that

η v,m ∈ L1 when m > n and that η v,m 1 = cm is independent of v, where this type of function was introduced in [20] and [8]. By c we denote generic positive constants, which may have different values at different occurrences.

2.1

Some technical lemmas

In this subsection we present some results which are useful for us. The following lemma is from [22, Lemma 19], see also [7, Lemma 6.1]. log Lemma 1 Let α ∈ Cloc and let R ≥ clog (α), where clog (α) is the constant from (1) for α. Then 2vα(x) η v,m+R (x − y) ≤ c 2vα(y) η v,m (x − y)

with c > 0 independent of x, y ∈ Rn and v, m ∈ N0 .

The previous lemma allows us to treat the variable smoothness in many cases as if it were not variable at all, namely we can move the term inside the convolution as follows: 2vα(x) η v,m+R ∗ f (x) ≤ c η v,m ∗ (2vα(·) f )(x). Lemma 2 Let r, R, N > 0, m > n and θ, ω ∈ S (Rn ) with suppF ω ⊂ B(0, 1). Then there exists c = c(r, m, n) > 0 such that for all g ∈ S ′ (Rn ), we have   N m  |θR ∗ ω N ∗ g (x)| ≤ c max 1, (η N,m ∗ |ω N ∗ g|r (x))1/r , R

x ∈ Rn ,

(2)

where θ R (·) = Rn θ(R·), ω N (·) = N n ω(N·) and η N,m := N n (1 + N |·|)−m . The proof of this lemma is given in [15]. We will make use of the following statement, see [9], Lemma 3.3 and [8], Theorem 3.2.4.

4

Theorem 1 Let p ∈ P log . Then for every m > 0 there exists β ∈ (0, 1) only depending on m and clog (p) such that p(x)  β Z |f (y)| dy |Q| Q Z 1 ≤ |f (y)|p(y) dy |Q| Q Z   1 m −m (e + |y|)−m dy , + min (|Q| , 1) (e + |x|) + |Q| Q for every cube (or ball) Q ⊂ Rn , all x ∈ Q ⊂ Rn and all f ∈ Lp(·) + L∞ with kf kp(·) + kf k∞ ≤ 1. Notice that in the proof of this theorem we need only Z |f (y)|p(y) dy ≤ 1 Q

and/or kf k∞ ≤ 1. We set

 f



v k(fv )v kLτ (·) (ℓq(·) ) := sup χP ,

τ (·) + p(·) |P | v≥vP Lp(·) (ℓq(·) ) P ∈Q

where, vP = − log2 l(P ) and vP+ = max(vP , 0). The following result is from [7, Theorem 3.2]. log Theorem 2 Let p, q ∈ Cloc with 1 < p− ≤ p+ < ∞ and 1 < q − ≤ q + < ∞. Then the inequality

(η v,m ∗ fv )v p(·) q(·) ≤ c k(fv )v k p(·) q(·) . L (ℓ ) L (ℓ )

holds for every sequence (fv )v∈N0 of L1loc -functions and constant m > n + clog (1/q).

We will make use of the following statement (we use it, since the maximal operator is in general not bounded on Lp(·) (ℓq(·) ), see [7, Section 5]). log Lemma 3 Let τ ∈ Cloc , τ − > 0, p, q ∈ P log with 0 < enough sucth that

τ+ τ−

< min (p− , q − ). For m large

m > nτ + + 2n + w, w > n + clog (1/q) + clog (τ ) and every sequence (fv )v∈N0 of L1loc -functions, there exists c > 0 such that

(ηv,m ∗ fv )v τ (·) q(·) ≤ c k(fv )v k τ (·) q(·) . L (ℓ ) L (ℓ ) p(·)

p(·)

The proof of this lemma is postponed to the Appendix. g p(·) p(·) Let Lτ (·) be the collection of functions f ∈ Lloc (Rn ) such that

fχ

kf k g := sup

τP(·) < ∞, p ∈ P0 , τ : Rn → R+ , p(·) Lτ (·) |P | p(·)

where the supremum is taken over all dyadic cubes P with |P | ≥ 1. Notice that

f q(·)



≤ 1. kf k g ≤ 1 ⇔ sup

τ (·) χP p(·) Lτ (·) |P | p(·)/q(·) P ∈Q,|P |≥1 Let θv = 2vn θ (2v ·). The following lemma is from [16]. 5

(3)

log Lemma 4 Let v ∈ Z, τ ∈ Cloc , τ − > 0, p ∈ P0log , 0 < r < p− and θ, ω ∈ S(Rn ) with suppF ω ⊂ B(0, 1). For any f ∈ S ′ (Rn ) and any dyadic cube P with |P | ≥ 1, we have

θ ∗ ω ∗ f

v v , χP ≤ c kω v ∗ f k g

p(·) τ (·) Lτ (·) p(·) |P |

such that the right-hand side is finite, where c > 0 is independent of v and l(P ).

The next three lemmas are from [7] where the first tells us that in most circumstances two convolutions are as good as one. Lemma 5 For v0 , v1 ∈ N0 and m > n, we have ηv0 ,m ∗ ηv1 ,m ≈ η min(v0 ,v1 ),m with the constant depending only on m and n. Lemma 6 Let v ∈ N0 and m > n. Then for any Q ∈ Q with l(Q) = 2−v , y ∈ Q and x ∈ Rn , we have   χQ η v,m ∗ (x) ≈ η v,m (x − y) |Q| with the constant depending only on m and n. Lemma 7 Let v, j ∈ N0 , r ∈ (0, 1] and m > nr . Then for any Q ∈ Q with l(Q) = 2−v , we have + (η j,m ∗ ηv,m ∗ χQ )r ≈ 2(v−j) n(1−r) η j,mr ∗ η v,mr ∗ χQ , where the constant depends only on m, n and r.

The next lemma is a Hardy-type inequality which is easy to prove. Lemma 8 Let 0 < a < 1, J ∈ Z P and 0 < q ≤ ∞. Let (εk )k be a sequences of positive k k−j εj , k ≥ J + . Then there exists constant real numbers and denote δ k = j=J + a c > 0 depending only on a and q such that ∞ ∞ 1/q 1/q X X q εqk . δk ≤c k=J +

k=J +

log Lemma 9 Let α, τ ∈ Cloc , τ − ≥ 0 and p, q ∈ P0log with 0 < q − ≤ q + < ∞. Let (fk )k∈N0 be P a sequence of measurable functions on Rn . For all v ∈ N0 and x ∈ Rn , −|k−v|δ let gv (x) = ∞ fk (x). Then there exists a positive constant c, independent of k=0 2 (fk )k∈N0 such that

k(gv )v kLτ (·) (ℓq(·) ) ≤ c k(fv )v kLτ (·) (ℓq(·) ) ,

δ > τ +.

p(·)

p(·)

The proof of Lemma 9 can be obtained by the same arguments used in [15, Lemma 2.10]. The following statement can be found in [4], that plays an essential role later on. Lemma 10 Let real numbers s1 < s0 be given, and σ ∈]0, 1[. For 0 < q ≤ ∞ and J ∈ N0 there is c > 0 such that

(σs +(1−σ)s )j

1−σ

σ 1

(2 0 aj )j≥J ℓq ≤ c (2s0 j aj )j≥J ℓ∞ (2s1 j aj )j≥J ℓ∞

holds for all complex sequences (2s0 j aj )j∈N0 in ℓ∞ . 6

3

α(·),τ (·)

The spaces Fp(·),q(·)

In this section we present the Fourier analytical definition of Triebel-Lizorkin-type spaces of variable smoothness and integrability and we prove the basic properties in analogy to the Triebel-Lizorkin-type spaces with fixed exponents. Select a pair of Schwartz functions Φ and ϕ satisfy suppF Φ ⊂ B(0, 2) and |F Φ(ξ)| ≥ c if |ξ| ≤

5 3

(4)

and

3 5 ≤ |ξ| ≤ (5) 5 3 where c > 0. Let α : Rn → R, p, q, τ ∈ P0 and Φ and ϕ satisfy (4) and (5), respectively and we put ϕv = 2vn ϕ(2v ·). suppF ϕ ⊂ B(0, 2)\B(0, 1/2) and |F ϕ(ξ)| ≥ c if

Definition 1 Let α : Rn → R, τ : Rn → R+ , p, q ∈ P0 and Φ and ϕ satisfy (4) and α(·),τ (·) (5), respectively and we put ϕv = 2vn ϕ(2v ·). The Triebel-Lizorkin-type space Fp(·),q(·) is the collection of all f ∈ S ′ (Rn ) such that kf kFα(·),τ (·) p(·),q(·)

 2vα(·) ϕ ∗ f 

v χP < ∞, := sup

+ τ (·) |P | v≥vP Lp(·) (ℓq(·) ) P ∈Q

(6)

where ϕ0 is replaced by Φ.

Using the system (ϕv )v∈N0 we can define the norm α,τ kf kFp,q

 ∞  1

X vαq

q := sup 2 |ϕv ∗ f | χP |1/q τ p P ∈Q |P | + v=vP

α,τ for constants α and p, q ∈ (0, ∞]. The Triebel-Lizorkin-type space Fp,q consist of all ′ n α,τ < ∞. It is well-known that these spaces distributions f ∈ S (R ) for which kf kFp,q do not depend on the choice of the system (ϕv )v∈N0 (up to equivalence of quasinorms). Further details on the classical theory of these spaces can be found in [10] and [48]; see also [13] for recent developments. α(·),τ (·) α,τ One recognizes immediately that if α, τ , p and q are constants, then Fp(·),q(·) = Fp,q . α(·),τ (·)

When, q := ∞ the Triebel-Lizorkin-type space Fp(·),∞ f ∈ S ′ (Rn ) such that



sup + P ∈Q,v≥vP

consist of all distributions

2vα(·) ϕv ∗ f

χP < ∞. p(·) |P |τ (·)

Let BJ be any ball of Rn with radius 2−J , J ∈ Z. In the definition of the spaces α(·),τ (·) Fp(·),q(·) if we replace the dyadic cubes P by the balls BJ , then we obtain equivalent quasi-norms. From these if we replace dyadic cubes P in Definition 1 by arbitrary cubes P , we then obtain equivalent quasi-norms.

7

α(·)

The Triebel-Lizorkin space of variable smoothness and integrability Fp(·),q(·) is the collection of all f ∈ S ′ (Rn ) such that




kf kF α(·) := 2vα(·) ϕv ∗ f v≥0 p(·) q(·) < ∞, L

p(·),q(·)

(ℓ

)

which introduced and investigated in [7], see [22] and [30] for further results. Obviously, α(·),0 α(·) Fp(·),q(·) = Fp(·),q(·) . We refer the reader to the recent paper [46] for further details, historical remarks and more references on embeddings of Triebel-Lizorkin-type spaces with fixed exponents. More information about Triebel-Lizorkin spaces can be found in [35-38]. Sometimes it is of great service if one can restrict supP ∈Q in the definition to a supremum taken with respect to dyadic cubes with side length ≤ 1. log , τ − ≥ 0 and p, q ∈ P0log with Lemma 11 Let α, τ ∈ Cloc α(·),τ (·)

 − τ − p1 ≥ 0 and 0
0 or τ − p1 ≥ 0 and q := ∞, then τ − p1 α(·),τ (·)

α(·)+n(τ (·)−1/p(·)) Fp(·),q(·) = B∞,∞

with equivalent quasi-norms.  − −  Proof. We consider only τ − 1p > 0. The case τ − p1 ≥ 0 and q := ∞ can  − be proved analogously with the necessary modifications. Since τ − p1 > 0, then we use the equivalent norm given in the previous lemma. First let us prove the following estimate kf kFα(·),τ (·) . kf kB∞,∞ α(·)+n(τ (·)−1/p(·)) p(·),q(·)

α(·)+n(τ (·)−1/p(·))

for any f ∈ B∞,∞ obtain that

. Let P be a dyadic cube with volume 2−nvP , vP ∈ N. We

2vα(x) |ϕv ∗ f (x)| ≤ c 2v(α(x)+n(τ (x)−1/p(x))+n(vP −v)(τ (x)−1/p(x))+nvP /p(x) |ϕv ∗ f (x)| |P |τ (x) ≤ c 2n(vP −v)(τ (x)−1/p(x))+nvP /p(x) kf kB∞,∞ α(·)+n(τ (·)−1/p(·)) for any x ∈ P . Then for any v ≥ vP

q(·)  ∞ vα(·)

 X 1/q(·) 2 ϕv ∗ f

χ

P |P |τ (·) p(·) v=vP

∞

 X n(v −v)(τ (·)−1/p(·))+nv /p(·) q(·) 1/q(·)

P 2 P . kf kB∞,∞ χP α(·)+n(τ (·)−1/p(·))

p(·)

v=vP

nv /p(·)

2 P χP p(·) . 1, . c kf kB∞,∞ α(·)+n(τ (·)−1/p(·)) q(·)

−  α(·),τ (·) > 0. Let f ∈ Fp(·),q(·) , with kf kFα(·),τ (·) = 1. By Lemma 2 we have for since τ − p1 p(·),q(·)

any x ∈ Rn , m > n

2v(α(x)+n(τ (x)−1/p(x)) |ϕv ∗ f (x)|

−

−

≤ c 2v(α(x)+n(τ (x)−1/p(x)) (η v,m ∗ |ϕv ∗ f |p (x))1/p

v(α(x)+nτ (x)) v −m/2p− ≤ c 2 ϕv ∗ f (·)(1 + 2 |x − ·|)

p(·)

by H¨older’s inequality, with

1 p−

=

1 p(·)



vn/t(x) v −m/2p− (1 + 2 |x − ·|)

2

t(·)

1 + t(·) . The second norm on the right-hand side is

−

bounded if m > 2pt− (n + clog (1/t)) (this is possible since m can be taken large enough). To show that, we investigate the corresponding modular: Z − vn/t(x) v −m/2p− ̺t(·) (2 (1 + 2 |x − ·|) ) = 2vnt(y)/t(x) (1 + 2v |x − y|)−mt(y)/2p dy Rn Z − − vn (1 + 2v |x − y|)−(m−clog (1/t))t /2p dy < ∞, ≤ 2 Rn

9

,

where we used Lemma 1. Again by the same lemma the first norm is bounded by

v(α(·)+nτ (·))

2 ϕv ∗ f (·)(1 + 2v |x − ·|)−h p(·) ,

where h = 2pm− − clog (α + nτ ). Let now prove that this expression is bounded. We investigate the corresponding modular: ̺p(·) (2v(α(·)+nτ (·)) ϕv ∗ f (·)(1 + 2v |x − ·|)−h ) Z = 2v(α(y)+nτ (y))p(y) |ϕv ∗ f (y)|p(y)(1 + 2v |x − y|)−hp(y)dy Rn

=

Z

|y−x| 0 independent of v and m such that

−1

|λv,m | ≤ c 2−v(α(x)+n/2) |Qv,m |τ (x) kλkfα(·),τ (·) χv,m p(·) . p(·),q(·)

log Lemma 14 Let α, τ ∈ Cloc , τ − ≥ 0, p, q ∈ P0log with 0 < q + , p+ < ∞ and Ψ, α(·),τ (·) ψ ∈ S(Rn ) satisfy, respectively, (4) and (5). Then for all λ ∈ fp(·),q(·)

Tψ λ :=

X

λ0,m Ψm +

∞ X X

λv,m ψ v,m ,

v=1 m∈Zn

m∈Zn α(·),τ (·)

converges in S ′ (Rn ); moreover, Tψ : fp(·),q(·) → S ′ (Rn ) is continuous. 13

For a sequence λ = {λv,m ∈ C}v∈N0 ,m∈Zn , 0 < r ≤ ∞ and a fixed d > 0, set 1/r X |λv,h |r λ∗v,m,r,d := (1 + 2v |2−v h − 2−v m|)d h∈Zn and λ∗r,d := {λ∗v,m,r,d ∈ C}v∈N0 ,m∈Zn . log Lemma 15 Let α, τ ∈ Cloc , τ − > 0, p, q ∈ P0log , 0 < q + , p+ < ∞, w > n + clog (1/q) + min(p− ,q − ) + clog (τ ) and ττ − < . Let a = r max(clog (α), α+ − α− ). Then r

∗

λ α(·),τ (·) ≈ kλk α(·),τ (·) r,d f f p(·),q(·)

p(·),q(·)

where

d > nrτ + + nτ − + n + a + w. The proof of this lemma is postponed to the Appendix. By this result and by the same arguments given in [15, Theorem 3.14] we obtain the following statement. log Theorem 5 Let α, τ ∈ Cloc , τ − > 0, p, q ∈ P0log and 0 < q + , p+ < ∞. Suppose that n Φ, Ψ ∈ S(R ) satisfying (4) and ϕ, ψ ∈ S(Rn ) satisfy (5) such that (9) holds. The opα(·),τ (·) α(·),τ (·) α(·),τ (·) α(·),τ (·) erators Sϕ : Fp(·),q(·) → fp(·),q(·) and Tψ : fp(·),q(·) → Fp(·),q(·) are bounded. Furthermore, α(·),τ (·)

Tψ ◦ Sϕ is the identity on Fp(·),q(·) .

α(·),τ (·)

From this theorem, we obtain the next important property of spaces Fp(·),q(·) . log Corollary 1 Let α, τ ∈ Cloc , τ − > 0, p, q ∈ P0log and 0 < p+ , q + < ∞, The definition α(·),τ (·) of the spaces Fp(·),q(·) is independent of the choices of Φ and ϕ.

4

Embeddings α(·),τ (·)

For the spaces Fp(·),q(·) introduced above we want to show some embedding theorems. We say a quasi-Banach space A1 is continuously embedded in another quasi-Banach space A2 , A1 ֒→ A2 , if A1 ⊂ A2 and there is a c > 0 such that kf kA2 ≤ c kf kA1 for all f ∈ A1 . We begin with the following elementary embeddings.

log , τ − > 0 and p, q, q0 , q1 ∈ P0log with p+ , q + , q0+ , q1+ < ∞. Theorem 6 Let α, τ ∈ Cloc (i) If q0 ≤ q1 , then α(·),τ (·) α(·),τ (·) Fp(·),q0 (·) ֒→ Fp(·),q1 (·) .

(ii) If (α0 − α1 )− > 0, then

α (·),τ (·)

α (·),τ (·)

0 1 Fp(·),q ֒→ Fp(·),q . 0 (·) 1 (·)

The proof can be obtained by using the properties of Lp(·) (ℓq(·) ) spaces. We next consider embeddings of Sobolev-type. It is well-known that Fpα00,q,τ ֒→ Fpα11,q,τ . if α0 − n/p0 = α1 − n/p1 , where 0 < p0 < p1 < ∞, 0 ≤ τ < ∞ and 0 < q ≤ ∞ (see e.g. [48, Corollary 2.2]). In the following theorem we generalize these embeddings to variable exponent case. 14

log + + Theorem 7 Let α0 , α1 , τ ∈ Cloc , τ − > 0 and p0 , p1 , q ∈ P0log with 0 < p+ 0 , p1 , q < ∞.  + < 1, then If α0 > α1 and α0 (x)− p0n(x) = α1 (x)− p1n(x) with 0 < pp01 α (·),τ (·)

α (·),τ (·)

Fp00(·),q(·) ֒→ Fp11(·),∞ . Proof. We use some idea of [31]. By homogeneity, it suffices to consider the case kf kFα0 (·),τ (·) = 1. We will provet that p0 (·),∞

 2vα1 (·) ϕ ∗ f 

v χ .1

P + τ(·) |P | v≥vP Lp1 (·) (ℓq(·) )

for any dyadic cube P . We will study two cases in particular. Case 1. |P | > 1. Let Qv ⊂ P be a cube, with ℓ (Qv ) = 2−v and x ∈ Qv ⊂ P . By τ − p− Lemma 2 we have for any m > n, 0 < r < τ +0
1/r |ϕv ∗ f (x)| ≤ c η v,m ∗ |ϕv ∗ f |r (x) , where c > 0 independent of v. We have

η v,m ∗ |ϕv ∗ f |r (x) Z |ϕv ∗ f (z)|r vn = 2 m dz v Rn (1 + 2 |x − z|) Z X · · ·dz + = 3Qv

k=(k1 ,...,kn)∈Zn ,maxi=1,...,n |ki |≥2

Z

Qv +kl(Qv )

· · ·dz.

√ Let z ∈ Qv + kl(Qv ) with k ∈ Zn and |k| > 4 n. Then |x − z| ≥ |k| 2−v−1 and the second integral is boundd by Z −m vn |ϕv ∗ f (z)|r dz = |k|−m MQv +kl(Qv ) |ϕv ∗ f |r (x). |k| 2 Qv +kl(Qv )

log We put σ = pp10 ∈]0, 1[ and s0 = α1 − n/p1 ∈ Cloc . Since σα0 + (1 − σ) s0 = α1 , Lemma n 10 gives for any x ∈ R

 2α1 (x)v ϕ ∗ f (x) 

v χ (x)

P τ (x) v≥0 ℓq(x) |P |

 2α0 (x)v ϕ ∗ f (x)  σ(x)  2s0 (x)v ϕ ∗ f (x)  1−σ(x)



v v χ (x) χ (x) . ≤



P P τ (x) τ (x) ∞ v≥0 ℓ v≥0 ℓ∞ |P | |P |

This expression in Lp1 (·) -norm is bounded. Indeed,

 2s0 (x)vr |ϕ ∗ f (x)|r 

v χP (x)

τ (x)r v≥0 ℓ∞ |P |

 2s0 (x)vr g  X

v,k,m,N (x) χ (x) .

+

P τ (x)r v≥0 ℓ∞ √ |P | k∈Zn ,|k|≤4 n

 s0 (x)vr |k|−n g  X

v,k,m,N (x) n−m+nτ + r+σ+ 2 χP (x) |k|

,

τ (x)r v≥0 ℓ∞ √ |P | k∈Zn ,|k|>4 n 15

where ̺(·) = n − gv,k,m,N

ndr p0 (·)τ (·)

and

 

if |k| = 0 √ M3Qv |ϕv ∗ f |r r MQv +kl(Qv ) |ϕv ∗ f | if 0 < |k| ≤ 4 n = √  |k|−nτ r MQv +kl(Qv ) |k|−̺(·) |ϕv ∗ f |r if |k| > 4 n.

We will prove that

 2s0 (x)vr |k|−n g 

v,k,m,N (x) χ (x)

.1 P τ (x)r v≥0 ℓ∞ |P |

(10)

√ for any k ∈ Zn with |k| > 4 n. We take d > 0 such that τ + < d < Qv + kl(Qv ) ⊂ Q (x, |k|21−v ), by Theorem 1, |k|−ndr

MQ(x,|k|21−v ) |k|−̺(·) 2vs0 r |ϕv ∗ f |r (x)

dr

|P |

≤ MQ(x,|k|21−v )

 2v s0τdr |ϕ ∗ f |rd/τ  v |k|ndr |P |dr

τ − p− 0 . r

Observe that


d/τ (x)

(x) + C.

By H¨older’s inequality this expression is bounded by 



2v s0 + pn0 dr/τ

rd/τ + C. |ϕv ∗ f | χQ(·,|k|21−v )

dr p0 (·)τ (·)/dr |k|ndr |P |

Observe that Q (x, |k|21−v ) ⊂ Q (cP , |k|21−vP ) for any x ∈ Qv . The last norm is bounded if and only if Z

2

 v s0 (y)+ p

n 0 (y)

 p0 (y)

|ϕv ∗ f (y) |p0 (y)

(|k|n |P |)p0 (y)τ (y)

Q(cP ,|k|21−vP )

dy . 1,

which is equivalent to

2

  v s0 (·)+ p n(·) p0 (·)

|ϕv ∗ f | χQ(cP ,|k|21−vP )

. 1,

1−v τ (·) |Q (cP , |k|2 P ) | p0 (·)

0

due to the fact that s0 (·) + p0n(·) = α0 (·) and |Q (cP , |k|21−vP )| ≥ 1. Therefore, the P sum k∈Zn ,|k|>4√n · · · is bounded by taking m large enough. Similar arguments with √ necessary modifications can be used to prove (10) for any |k| ≤ 4 n. Therefore, Z X ∞

2vα1 (x)q(x)

|ϕv ∗ f (x)|q(x) p1 (x)/q(x)

|P |τ (x)q(x) v=0 Z  α0 (x)v ϕv ∗ f (x) 

σ(x)p1 (x)

2 dx ≤ c

v≥0 ℓ∞ |P |τ (x) P Z  α0 (x)v ϕv ∗ f (x) 

2

p0 (x) = c dx ≤ c.

v≥0 ℓ∞ |P |τ (x) P P

16

dx

Case 2. |P | ≤ 1. Since τ is log-H¨older continuous, we have |P |−τ (x) ≤ c |P |−τ (y) (1 + 2vP |x − y|)clog (τ ) ≤ c |P |−τ (y) (1 + 2v |x − y|)clog (τ ) for any x, y ∈ Rn and any v ≥ vP . Therefore, 1

η τ (·) v,m

|P |


|ϕv ∗ f |χQ ∗ |ϕv ∗ f |χQ . η v,m−clog (τ ) ∗ |P |τ (·)

for any dyadic cube where Q. The arguments here are quite similar to those used in the case |P | > 1, where we did not need to use Theorem 1, which could be used only to move |P |τ (·) inside the convolution and hence the proof is complete. log Let α, τ ∈ Cloc , τ − > 0, p, q ∈ P0log with 0 < p+ , q + < ∞. From (8), we obtain n α(·)+nτ (·)− p(·)

α(·),τ (·)

Fp(·),q(·) ֒→ Fp(·),∞

֒→ S ′ (Rn ).

Similar arguments of [48, Proposition 2.3] can be used to prove that α(·),τ (·)

S(Rn ) ֒→ Fp(·),q(·) . Therefore, we obtain the following result. log , τ − > 0 and p, q ∈ P0log with 0 < p+ , q + < ∞. Then Theorem 8 Let α, τ ∈ Cloc α(·),τ (·)

S(Rn ) ֒→ Fp(·),q(·) ֒→ S ′ (Rn ). α(·),τ (·)

Now we establish some further embedding of the spaces Fp(·),q(·) . log Theorem 9 Let α, τ ∈ Cloc , τ − > 0 and p, q ∈ P0log with 0 < p+ , q + < ∞. If (p2 − p1 )+ ≤ 0, then α(·)+nτ (·)+ p n(·) − p n(·)

Fp2 (·),q(·)

2

1

α(·),τ (·)

֒→ Fp1 (·),q(·) .

Proof. Using the Sobolev embeddings α(·)+nτ (·)+ p n(·) − p n(·)

Fp2 (·),q(·)

2

1

α(·)+nτ (·)

see [42] it is sufficient to prove that Fp1 (·),q(·)

α(·)+nτ (·)

֒→ Fp1 (·),q(·) , α(·),τ (·)

֒→ Fp1 (·),q(·) . We have

 2vα(·) ϕ ∗ f 




vα(·)

v χ ≤ 2 ϕ ∗ f



p (·) q(·) . P v τ (·) + v≥0 p (·) v≥vP L 1 (ℓq(·) ) L 1 (ℓ ) P ∈Q,|P |>1 |P | sup

α(·)

In view of the definition of Fp1 (·),q(·) spaces the last expression is bounded by kf kF α(·)

p1 (·),q(·)

kf kF α(·)+nτ (·) . Now we have the estimates p1 (·),q(·)

 2vα(·) ϕ ∗ f 

v χ

p (·) q(·)

P τ (·) + v≥v L 1 (ℓ ) P ∈Q,|P |≤1 |P | P





≤ sup 2v(α(·)+nτ (·))+nτ (·)(vP −v) ϕv ∗ f

v≥vP Lp1 (·) (ℓq(·) ) P ∈Q,|P |≤1




v(α(·)+nτ (·))

≤ sup 2 ϕv ∗ f v≥0 p (·) q(·) ≤ kf kF α(·)+nτ (·) , sup

L

P ∈Q,|P |≤1

17

1

(ℓ

)

p1 (·),q(·)

≤

which completes the proof. u(·) Let p, u ∈ P0 be such that 0 < p(·) ≤ u(·) < ∞. The variable Morrey space Mp(·) is defined to be the set of all p(·)-locally Lebesgue-integrable functions f on Rn such that

kf kM u(·) = sup p(·)

P ∈Q

f 1

1

|P | p(·) − u(·)

χP

p(·)

< ∞.

One recognizes immediately that if p and u are constants, then we obtain the usual Morrey space Mpu . Let α : Rn → R and p, q, u ∈ P0 be such that 0 < p(·) ≤ u(·) < ∞. Let Φ and ϕ satisfy (4) and (5), respectively and we put ϕv = 2vn ϕ(2v ·). The Triebel-Lizorkin-Morrey α(·),q(·) space Eu(·),p(·) is the collection of all f ∈ S ′ (Rn ) such that

 

kf kE α(·),q(·) := 2vα(·) ϕv ∗ f



u(·)

v≥0 Mp(·) (ℓq(·) )

u(·),p(·)

where ϕ0 is replaced by Φ.

< ∞,

log Theorem 10 Let α ∈ Cloc and p, q, u ∈ P0log with 0 < p− < p(·) ≤ u(·) < u+ < ∞ and 0 < q − ≤ q + < ∞. Then α(·),

1

p(·) Fp(·),q(·)

1 − u(·)

α(·),q(·)

= Eu(·),p(·) ,

with equivalent quasi-norms. α(·),

1

p(·) Proof. Obviously we need to prove that kf kE α(·),q(·) . 1 for any f ∈ Fp(·),q(·)

1 − u(·)

with

u(·),p(·)

kf k

α(·), 1 − 1 p(·) u(·)

Fp(·),q(·)

≤ 1. We must show the inequality

+ vP

 X 2vα(·) ϕv ∗ f q(·) 1/q(·)

χP

.1 1 1 − u(·) )q(·) ( p(·) p(·) v=0 |P |

for any dyadic cube P . Applying the property (8), we obtain vn

2vα(x) |ϕv ∗ f (x)| . 2 u(x) for any x ∈ Rn , with the implicit constant independent of k and x. Hence the last quasi-norm is bounded by +

vP

nvP+  X 1/q(·) + nq(·)

p(·) 2(v−vP ) u(·) χP

2

Lp(·)

v=0

This finishes the proof.

18

nvP+

. 2 p(·) χP

p(·)

. 1.

5

Atomic decomposition

The idea of atomic decompositions leads back to M. Frazier and B. Jawerth in their series of papers [17], [18], see also [38]. The main goal of this section is to prove an α(·),τ (·) atomic decomposition result for Fp(·),q(·) . We now present a fundamental characterization of spaces under consideration. log Theorem 11 Let τ , α ∈ Cloc , τ − > 0 and p, q ∈ P0log with 0 < p− ≤ p+ < ∞ and + 0 < q − ≤ q + < ∞. Let m be as in Lemma 3, a > ττ− pm− and Φ and ϕ satisfy (4) and (5), respectively. Then

kf kHFα(·),τ (·) p(·),q(·)

 ϕ∗,a 2vα(·) f 

v

:= sup χP

+ τ (·) v≥vP Lp(·) (ℓq(·) ) P ∈Q |P | α(·),τ (·)

is an equivalent quasi-norm in Fp(·),q(·) . Proof. It is easy to see that for any f ∈ S ′ (Rn ) with kf kHFα(·),τ (·) < ∞ and any x ∈ Rn p(·),q(·)

we have

vα(·) f (x). 2vα(x) |ϕv ∗ f (x)| ≤ ϕ∗,a v 2

This shows that kf kFα(·),τ (·) ≤ kf kHFα(·),τ (·) . Let t > 0 be such that a > p(·),q(·)

p(·),q(·)

m t

>

m . p−

By

Lemmas 2 and 1 the estimates


1/t 2vα(y) |ϕv ∗ f (y)| ≤ C1 2vα(y) ηv,w ∗ |ϕv ∗ f |t (y)  1/t ≤ C2 η v,w−clog (α) ∗ (2vα(·) |ϕv ∗ f |)t (y)

(11)

are true for any y ∈ Rn , v ∈ N0 and any w > 0. Now divide both sides of (11) by (1 + 2v |x − y|)a , in the right-hand side we use the inequality (1 + 2v |x − y|)−a ≤ (1 + 2v |x − z|)−a (1 + 2v |y − z|)a ,

x, y, z ∈ Rn , α(·),τ (·)

in the left-hand side take the supremum over y ∈ Rn and get for all f ∈ Fp(·),q(·) , any x ∈ P any v ≥ vP+ and any w > at + clog (α) vα(·) ϕ∗,a f (x) v 2


t

≤ C2 η v,at ∗ (2vα(·) |ϕv ∗ f |)t (x)

where C2 > 0 is independent of x, v and f . Lemma 3 gives that

 η ∗ (2vα(·) |ϕ ∗ f |)t 

1/t

v,at

v kf kHFα(·),τ (·) . C sup χ

p(·) q(·) P τ (·)t + t (ℓ t ) p(·),q(·) v≥v P ∈Q |P | P L

vα(·)
≤ C 2 ϕv ∗ f v Lp(·) (ℓq(·) ) = C kf kFα(·),τ (·) . p(·),q(·)

The proof is complete. Atoms are the building blocks for the atomic decomposition.

19

Definition 3 Let K ∈ N0 , L + 1 ∈ N0 and let γ > 1. A K-times continuous differentiable function a ∈ C K (Rn ) is called [K, L]-atom centered at Qv,m , v ∈ N0 and m ∈ Zn , if supp a ⊆ γQv,m |∂ β a(x)| ≤ 2v(|β|+1/2) , for and if

Z

xβ a(x)dx = 0,

for

Rn

(12)

0 ≤ |β| ≤ K, x ∈ Rn

(13)

0 ≤ |β| ≤ L and v ≥ 1.

(14)

If the atom a located at Qv,m , that means if it fulfills (12), then we will denote it by av,m . For v = 0 or L = −1 there are no moment conditions (14) required.

For proving the decomposition by atoms we need the following lemma, see Frazier & Jawerth [17, Lemma 3.3].

Lemma 16 Let Φ and ϕ satisfy, respectively, (4) and (5) and let ̺v,m be an [K, L]atom. Then
ϕj ∗ ̺v,m (x) ≤ c 2(v−j)K+vn/2 1 + 2v x − xQv,m −M if v ≤ j, and


ϕj ∗ ̺v,m (x) ≤ c 2(j−v)(L+n+1)+vn/2 1 + 2j x − xQv,m −M

if v ≥ j, where M is sufficiently large, ϕj = 2jn ϕ(2j ·) and ϕ0 is replaced by Φ. Now we come to the atomic decomposition theorem. log Theorem 12 Let α, τ ∈ Cloc , τ − > 0 and p, q ∈ P0log with 0 < p− ≤ p+ < ∞ and 0 < q − ≤ q + < ∞ Let K, L + 1 ∈ N0 such that

K ≥ ([α+ + nτ + ] + 1)+ , and L ≥ max(−1, [n(

(15)

τ+ − 1) − α− ]). τ − min(1, p− , q − )

(16)

α(·),τ (·)

Then f ∈ S ′ (Rn ) belongs to Fp(·),q(·) , if and only if it can be represented as f=

∞ X X v=0

converging in S ′ (Rn ),

λv,m ̺v,m ,

m∈Zn

(17) α(·),τ (·)

where ̺v,m are [K, L]-atoms and λ = {λv,m ∈ C : v ∈ N0 , m ∈ Zn } ∈ fp(·),q(·) . Furthermore, inf kλkfα(·),τ (·) , where the infimum is taken over admissible representations (17), p(·),q(·)

α(·),τ (·)

is an equivalent quasi-norm in Fp(·),q(·) .

20

The convergence in S ′ (Rn ) can be obtained as a by-product of the proof using the same method as in [38, Corollary 13.9 ] and [15]. If p, q, τ , and α are constants, then the restriction (15), and their counterparts, in the 1 − atomic decomposition theorem are K ≥ ([α + nτ ] + 1)+ and L ≥ max(−1, [n( min(1,p,q) 1) − α]), which are essentially the restrictions from the works of [13, Theorem 3.12]. Proof. The proof follows the ideas in [17, Theorem 6]. α(·),τ (·)

Step 1. Assume that f ∈ Fp(·),q(·) and let Φ and ϕ satisfy, respectively (4) and (5). There exist functions Ψ ∈ S(Rn ) satisfying (4) and ψ ∈ S(Rn ) satisfying (5) such that for all ξ ∈ Rn ∞ X e f =Ψ∗Φ∗f + ψv ∗ ϕ e v ∗ f, v=1

see Section 3. Using the definition of the cubes Qv,m we obtain ∞ X Z X X Z vn e ϕ e (2v (x − y))ψv ∗ f (y)dy, 2 Φ(x − y)Ψ ∗ f (y)dy + f (x) = m∈Zn

Q0,m

v=1

m∈Zn

Qv,m

with convergence in S ′ (Rn ). We define for every v ∈ N and all m ∈ Zn λv,m = Cθ sup |ψ v ∗ f (y)|

(18)

y∈Qv,m

where Cϕ = max{ sup |D α ϕ(y)| : |α| ≤ K}. |y|≤1

Define also ̺v,m (x) =



1 2vn λv,m

R

Qv,m

ϕ e v (2v (x − y))ψv ∗ f (y)dy if λv,m = 6 0 . 0 if λv,m = 0

(19)

Similarly we define for every m ∈ Zn the numbers λ0,m and the functions ̺0,m taking e respectively. Let us now in (18) and (19) v = 0 and replacing ψ v and ϕ e by Ψ and Φ, check that such ̺v,m are atoms in the sense of Definition 3. Note that the support and moment conditions are clear by (4) and (5), respectively. It thus remains to check (13) in Definition 3. We have −1  v(n+|β|) Z β β D ̺v,m (x) ≤ 2 (D ϕ e )(2v (x − y)) |ψ v ∗ f (y)| dy sup |ψ v ∗ f (y)| Cϕ y∈Qv,m Qv,m v(n+|β|) Z β 2 (D ϕ ≤ e )(2v (x − y)) dy ≤ 2v(n+|β|) |Qv,m | ≤ 2v|β| . Cϕ Qv,m The modifications for the terms with v = 0 are obvious.

Step 2. Next we show that there is a constant c > 0 such that kλkfα(·),τ (·) ≤ c kf kFα(·),τ (·) . p(·),q(·)

p(·),q(·)

For that reason we exploit the equivalent quasi-norms given in Theorem 11 involving Peetre’s maximal function. Let v ∈ N. Taking into account that |x − y| ≤ c 2−v for x, y ∈ Qv,m we obtain 2v(α(x)−α(y)) ≤

clog (α)v clog (α)v ≤ ≤c log(e + 1/ |x − y|) log(e + 2v /c) 21

if v ≥ [log2 c] + 2. If 0 < v < [log2 c] + 2, then 2v(α(x)−α(y)) ≤ 2v(α

+ −α− )

≤ c. Therefore,

2vα(x) |ψ v ∗ f (y)| ≤ c 2vα(y) |ψ v ∗ f (y)| for any x, y ∈ Qv,m and any v ∈ N. Hence, X X 2vα(x) sup |ψ v ∗ f (y)| χv,m (x) λv,m 2vα(x) χv,m (x) = Cθ y∈Qv,m

m∈Zn

m∈Zn

X

2vα(x−z) |ψ v ∗ f (x − z)| sup ≤ c (1 + 2v |z|)a χv,m (x) v a −v (1 + 2 |z|) m∈Zn |z|≤c 2 X vα(·) vα(·) χv,m (x) = c ψ ∗,a f (x), ≤ c ψ ∗,a f (x) v 2 v 2 m∈Zn

P

where we have used

m∈Zn

χv,m (x) = 1. This estimate and its counterpart for v = 0

(which can be obtained by a similar calculation) give


vα(·) f v Lτ (·) ℓq(·) ≤ c kf kFα(·),τ (·) , kλkfα(·),τ (·) ≤ c ψ ∗,a v 2 ) p(·),q(·) p(·),q(·) p(·) ( by Theorem 11.

Step 3. Assume that f can be represented by (17), with K and L satisfying (15) α(·),τ (·) and (16), respectively. We will show that f ∈ Fp(·),q(·) and that for some c > 0, kf kFα(·),τ (·) ≤ c kλkfα(·),τ (·) . The arguments are very similar to those in [15]. For the p(·),q(·)

p(·),q(·)

convenience of the reader, we give some details. We write f=

∞ X X

λv,m ̺v,m =

v=0 m∈Zn

j X v=0

···+

∞ X

v=j+1

· · ·.

From Lemmas 16 and 6, we have for any M sufficiently large and any v ≤ j X 2jα(x) |λv,m | ϕj ∗ ̺v,m (x) m∈Zn

+)

. 2(v−j)(K−α

X

m∈Zn

. 2

(v−j)(K−α+ )

X

m∈Zn

2v(α(x)−n/2) |λv,m | η v,M (x − xQv,m )

2v(α(x)+n/2) |λv,m | η v,M ∗ χv,m (x).

Lemma 1 gives 2vα(·) η v,M ∗ χv,m . ηv,T ∗ 2vα(·) χv,m , with T = M − clog (α) and since K > α+ + τ + we apply Lemma 9 to obtain j

 X i h X

(v−j)(K−α+ ) v(α(·)+n/2) |λv,m | χv,m 2 η v,T ∗ 2

 i h X

|λv,m | χv,m . η v,T ∗ 2v(α(·)+n/2)

τ (·)

v Lp(·) (ℓq(·) )

m∈Zn

22

τ (·)

j Lp(·) (ℓq(·) )

m∈Zn

v=0

.

The right-hand side can be rewritten us ir  h P v(α(·)+n/2) |λ | χ η ∗ 2 v,m v,m

1/r

 v,T 

m∈Zn χP sup

rτ (·) + v≥vP Lp(·)/r (ℓq(·)/r ) P ∈Q |P | i h P r v(α(·)+n/2)r |λ | χ η ∗ 2 v,m v,m

 v,T r

1/r 

m∈Zn . sup , χP

rτ (·) + v≥vP Lp(·)/r (ℓq(·)/r ) P ∈Q |P | −

by Lemma 7, since η v,T ≈ ηv,T ∗ ηv,T and 0 < r < ττ + min(1, p− , q − ). The application of Lemma 3 and the fact that



1/r




(gv )v≥vP+ p(·)/r q(·)/r = |gv |1/r v≥v+ p(·) q(·) L

(ℓ

)

P

L

(ℓ

)

give that the last expression is bounded by kλkfα(·),τ (·) . Now from Lemma 16, we have p(·),q(·)

for any M sufficiently large and v ≥ j X 2jα(x) |λv,m | ϕj ∗ ̺v,m (x) m∈Zn

X

. 2(j−v)(L+1+n/2)

m∈Zn

X

. 2(j−v)(L+1+n/2)

m∈Zn

2j(α(x)−n/2) |λv,m | ηj,M (x − xQv,m ) 2j(α(x)−n/2) |λv,m | ηj,M ∗ η v,M (x − xQv,m ),

where the last inequality follows by Lemma 5, since η j,M = η min(v,j),M . Again by Lemma 6, we have η j,M ∗ ηv,M (x − xQv,m ) . 2vn ηj,M ∗ η v,M ∗ χv,m (x). P jα(x) 2 |λv,m | ϕj ∗ ̺v,m (x) is bounded by Therefore, m∈Zn

c 2(j−v)(L+1−n/2)

X

m∈Zn

. 2

(j−v)(L+1−α− )

2j(α(x)+n/2) |λv,m | η j,M ∗ η v,M ∗ χv,m (x)

i h X |λv,m | χv,m (x), η j,T ∗ η v,T ∗ 2v(α(·)+n/2) m∈Zn

by Lemma 1, with T = M − clog (α). Let 0 < r < such that L > n/r − 1 − α− − n. We have ∞ X

≤

v=j ∞ X

min(1, p− , q − ) be a real number

ir h X − |λv,m | χv,m 2(j−v)(L+1−α ) η j,T ∗ η v,T ∗ 2v(α(·)+n/2)

2(j−v)(L−n/r+1−α

v=j

τ− τ+

m∈Zn

− +n)r

i h X |λv,m |r χv,m , η j,T r ∗ η v,T r ∗ 2v(α(·)+n/2)r m∈Zn

where we have used Lemma 7. The application of Lemma 3 gives that ∞

 X i h X −

|λv,m | χv,m 2(j−v)(L+1−α ) η j,T ∗ ηv,T ∗ 2v(α(·)+n/2)

m∈Zn

v=j

23

τ (·)

j Lp(·) (ℓq(·) )

is bounded by



c sup

P∞

v=j

2

(j−v)Hr

i h P r v(α(·)+n/2)r |λvm | χv,m ηv,T r ∗ 2 m∈Zn

|P |rτ (·)

P ∈Q

χP



1/r

p(·)/r +

j≥jP

L

(ℓq(·)/r )

,

where H := L − n/r + n + 1 − α− . Observing that H > 0, an application of Lemma 9, yields that the last expression is bounded by i h P r v(α(·)+n/2)r |λ | χ η ∗ 2 v,m v,m

 v,T r

1/r 

m∈Zn c sup χ

p(·)/r q(·)/r . kλkfα(·),τ (·) , P rτ (·) + p(·),q(·) (ℓ ) v≥v P ∈Q |P | P L where we used again Lemma 3 and hence the proof is complete.

6

Appendix

Here we present more technical proofs of the Lemmas. Proof of Lemma 3. By the scaling argument, we see that it suffices to consider when k(fv )v kLτ (·) (ℓq(·) ) ≤ 1 and show that for any dyadic cube P p(·)

∞

 X η v,m ∗ |fv | q(·) 1/q(·)

χP . 1.

|P |τ (·) p(·)

(20)

+ v=vP

Case 1. |P | > 1. Let Qv ⊂ P be a cube, with ℓ (Qv ) = 2−v and x ∈ Qv ⊂ P . We have η v,m ∗ |fv | (x) Z |fv (z)| vn = 2 m dz v Rn (1 + 2 |x − z|) Z X · · ·dz + = 3Qv

k=(k1 ,...,kn)∈Zn ,maxi=1,...,n

= Jv1 (fv χ3Qv )(x) +

X

|ki |≥2

k=(k1 ,...,kn )∈Zn ,maxi=1,...,n

Z

Qv +kl(Qv )

· · ·dz

2 Jv,k (fv χQv +kl(Qv ) )(x).

(21)

|ki |≥2

√ Let z ∈ Qv + kl(Qv ) with k ∈ Zn and |k| > 4 n. Then |x − z| ≥ |k| 2−v−1 and 2 Jv,k (fv χQv +kl(Qv ) )(x) . |k|−m MQv +kl(Qv ) (fv ) (x).

Let d > 0 be such that τ + < d ≤ τ − min (p− , q − ). Therefore, the left-hand side of (20) is bounded by ∞

 X X gv,k q(·) 1/q(·)

(22) χP

τ (·) p(·) |P | √ n v=0 k∈Z ,|k|≤4 n ∞

 X X gv,k q(·) 1/q(·)

w−m+n(1+ 1− )τ + t χP , + log |k| |k| τ (·) p(·) |P | √ n v=0 k∈Z ,|k|>4 n

24

where gv,k =

  

M3Qv (fv ) MQv +kl(Qv ) (fv )

  MQ +kl(Q ) v v

if k=0 √  if 0 < |k| ≤√4 n −n(1+1/t(·))τ (·) |k| fv if |k| > 4 n.

1 with d1 = p(·)τ1 (·) + t(·) . By similarity we only estimate the second norm in (22). This term is bounded if and only if ∞



 X 

(bk gv,k )d/τ (·) q(·)τ (·)/d d/τ (·)q(·) . 1, χ

P |P |d p(·)τ (·)/d v=0

(23)

 τ (·)/d where bk = |k|w 1log|k| . Observe that Qv + kl(Qv ) ⊂ Q(x, |k| 21−v ). By H¨older’s inequality,   Q(x, |k| 21−v ) MQ(x,|k|21−v ) |k|−n(1+1/t(·))d |fv |d/τ (·) (x)

d

d

|fv |1/τ (·)



−n(1+1/t(·)) 1−v . |Q(x, |k| 2 )| |k| χ χ 1−v 1−v

Q(x,|k|2 ) Q(x,|k|2 ) 1−v |Q(x, |k| 2 )| t(·) p(·)τ (·)

for any x ∈ Q(x, |k| 21−v ). The second norm is bounded. The first norm is bounded if and only if

fv

. 1, χ 1−v

Q(x,|k|2 ) 1−v τ (·) p(·) |Q(x, |k| 2 )|

which follows since k(fv )v kLτ (·) (ℓq(·) ) ≤ 1. Therefore we can apply Lemma 1, p(·)

   d/τ (x) −n(1+1/t(·))τ (·) MQ(x,|k|21−v ) |k| fv (x)  
−n(1+1/t(·))τ (·) d/τ (·) (x) + min 1, |k|ns 2n(1−v)s ω(x) ≤ MQ(x,|k|21−v ) |k| fv

for any s > 0 large enough, where


ω(x) = (e + |x|)−s + MQ(x,|k|21−v ) (e + |·|)−s (x)
≤ (e + |x|)−s + M (e + |·|)−s (x) = h (x) .

Therefore (gv,k )d/τ (·) can be estimated by   c MQ(·,|k|21−v ) |k|−nd |fv |d/τ (·) + σ v,k h, where σ v,k = Therefore, the quantity



···



1 if 2v ≤ 2 |k| |k| 2−vns if 2v > 2 |k| . ns

d/τ (·)q(·)

of the term (23) is bounded by

∞ X  |k|−nd |f |d/τ (·)  q(·)τ (·)/d d/τ (·)q(·) v |k| χ −v+1 MQ(·,|k|2−v+1 ) Q(·,|k|2 ) d |P | v=0 ∞ d/τ (·)q(·) 1 X (σ v,k |h|)q(·)τ (·)/d + . log |k| v=0 −w

25

(24)

The first term is bounded by ∞ X  |k|−nd |f |d/τ (·)  q(·)τ (·)/d d/τ (·)q(·) v η ∗ . χ v,w Q(·,|k|21−v ) d |P | v=0

where w > n + clog (1/q) + clog (τ ). Applying Theorem 2, the Lp(·)τ (·)/d -norm of this expression is bounded by ∞ −nτ (·)

 X d/τ (·)q(·) fv q(·)

|k| . χ

1−v

Q(·,|k|2 ) τ (·) p(·)τ (·)/d |P | v=0

Observe that Q(·, |k| 2−v+1 ) ⊂ Q(cP , |k| 2−vP +1 ) and the measure of the last cube is greater that 1, the last norm is bounded by 1. The summation (24) can be estimated by  c |h|q(·)τ (·)/d c + . |h|

q(·)τ (·)/d

X

v≥1,2v ≤2|k|

X  2v −nsq(·)τ (·)/d  1 + log |k| |k| v 2 >2|k|

.

This expression, with power d/τ (·)q(·), in the Lp(·)τ (·)/d -norm is bounded by 1. Therefore, the second sum in (22) is bounded by taking m large enough such that m > nτ + + 2n + w, with w > n + clog (1/q) + clog (τ ). Case 2. |P | ≤ 1. As before, η v,m ∗ |fv | (x) . Jv1 (fv χ3P )(x) +

X

2 Jv,k (fv χP +kl(P ))(x).

k=(k1 ,...,kn )∈Zn ,maxi=1,...,n |ki |≥2

We see that Jv1 (fv χ3P )(x) . η v,m ∗ (|fv | χ3P ) (x). Since τ is log-H¨older continuous, |P |−τ (x) ≤ c |P |−τ (y) (1 + 2vP |x − y|)clog (τ ) ≤ c |P |−τ (y) (1 + 2v |x − y|)clog (τ ) for any x ∈ P and any y ∈ 3P . Hence

  |P |−τ (x) Jv1 (fv χ3P )(x) . ηv,m−clog (τ ) ∗ |P |−τ (·) |fv | χ3P (x).

Also, we have −τ (x)

|P |

2 Jv,k (fv χP +kl(P ))(x)

  −τ (·) . η v,m−clog (τ ) ∗ |P | |fv | χP +kl(P ) (x)

√ for x ∈ P and z ∈ P + kl(P ) with k ∈ Zn and |k| > 4 n. Our estimate follows by Lemma 1. The proof is complete.

Proof of Lemma 15. Obviously, kλk α(·),τ (·) ≤ λ∗ α(·),τ (·) . Let us prove the converse fp(·),q(·)

r,d f p(·),q(·)

inequality. By the scaling argument, it suffices to consider the case kλkfα(·),τ (·) ≤ 1 and p(·),q(·)

26

show that the modular of a the sequence on the left-hand side is bounded. It suffices to prove that P v(α(·)+n/2) ∗ 2 λv,m,r,d χv,m  ∞

 X q(·) 1/q(·)

m∈Zn .1 (25) χ

P p(·) |P |τ (·) + v=vP

for any dyadic cube P ∈ Q. For each k ∈ N0 we define Ωk := {h ∈ Zn : 2k−1 < 2v |2−v h − 2−v m| ≤ 2k } and Ω0 := {h ∈ Zn : 2v |2−v h − 2−v m| ≤ 1}. Then for any P 2vrα(x) |λ |r x ∈ Qv,m ∩ P , h∈Zn (1+2v |2−v h−2v,h−v m|)d can be rewritten as ∞ X X

. =

k=0 h∈Ωk ∞ X −dk

2

k=0 ∞ X

2

2vrα(x) |λv,h |r

(1 + 2v |2−v h − 2−v m|)d X

h∈Ωk

2vrα(x) |λv,h |r

(n−d)k+(v−k)n+vrα(x)

k=0

Z

∪z∈Ωk Qv,z

X

h∈Ωk

|λv,h |r χv,h (y)dy.

(26)

Let x ∈ Qv,m ∩ P and y ∈ ∪z∈Ωk Qv,z , then y ∈ Qv,z for some z ∈ Ωk and 2k−1 < 2v |2−v z − 2−v m| ≤ 2k . From this it follows that y is located in some cube Q(x, 2k−v+3 ). In addition, from the fact that |yi − (cP )i | ≤ |yi − xi | + |xi − (cP )i | ≤ 2k−v+2 + 2−vP −1 < 2k−vP +3 , i = 1, ..., n, we have y is located in some cube Q(cP , 2k−vP +4 ). Since α is log-H¨older continuous we can prove that  c (α)k 2 log if k < max(0, v − hn ) v(α(x)−α(y)) 2 . (α+ −α− )k 2 if k ≥ max(0, v − hn ), where c > 0 not depending on v and k. Therefore, (26) does not exceed Z ∞ X X (n−d+a)k+(v−k)n |λv,h |r χv,h (y)χQ(cP ,2k−vP +4 ) (y)dy 2vα(y)r c 2 k=0

.

∞ X

2

h∈Ωk

Q(x,2k−v+3 )

(n−d+a)k

MQ(x,2k−v+3 )

k=0

X

2

vα(·)r

h∈Ωk

 |λv,h | χv,h χQ(cP ,2k−vP +4 ) (x), r

where a = max (clog (α), α+ − α− ). To prove (25) we can distinguish two cases:

Case 1. |P | > 1. The left-hand side of (25) is bounded by 1 if and only if

∞ 

 X MQ(·,2k−v+3 ) (2−kn(r+1/t(·))τ (·) gv,k,vP ) q(·)/r r/q(·)

c 2 . 1, χP rτ (·) |P | p(·)/r v=0 k=0 ∞ X

̺k

with

gv,k,vP =

X

h∈Ωk

2v(α(·)+n/2)r |λv,h |r χv,h χQ(cP ,2k−vP +4 ) 27

(27)

and ̺ = n − d + a + nrτ + + n

τ+ t−

Let us prove that ∞ 

 X ω k MQ(·,2k−v+3) (2−kn(r+1/t(·))τ (·) gv,k,vP ) q(·)/r r/q(·)

.1 χP

rτ(·) |P | p(·)/r v=0

for any k, v ∈ N0 and any P ∈ Q, where ωk =

1 2

(w− nσ + )k t

+ 2kns

.

This is equivalent to
σ ∞  (·)  σr

 X ω k MQ(·,2k−v+3) (2−kn(r+1/t(·))τ (·) gv,k,vP ) τ (·)  q(·)τ rσ q(·)τ (·)

p(·)τ (·) . 1, |P |rσ rσ v=0 where σ > 0 such that τ + < σ
0 large enough and h is the same function in Lemma 3. Hence the term in k+3 ∞ X X (28), with in place of is bounded by v=0

v=0

c

k+3 X v=0

+

σ

M

Q(·,2k−v+3 ) (gv,k,vP ) τ (·)

p(·)τ (·)

nσ |Q(·, 2k−vP +3 )|rσ 2(w− t+ )k + 2kns rσ

1

c (k + 4)s nσ

2(w− t+ )k + 2kns

.

28

Since M is bounded in Lp(·)τ (·)/rσ the last norm is bounded by



M

σ σ

(g

τ (·) (gv,k,vP ) τ (·) 

v,k,vP ) . .

(·) k−vP +3 )|rσ p(·)τ (·) |Q(·, 2k−vP +3 )|rσ p(·)τ |Q(·, 2 rσ rσ

This term is bounded by 1 if and only if



(gv,k,vP )1/r

. 1, |Q(cP , 2k−vP +3 )|τ (·) p(·)

wich follows since |Q(·, 2k−vP +3 )| ≥ 1. Now, with w > n + clog (1/q) + clog (τ ), the term ∞ ∞ X X in (28), with in place of is bounded by v=0

v=k+4

σ ∞  (·)  σr

 X

η v,w ∗ (gv,k,vP ) τ (·)  q(·)τ rσ q(·)τ (·)

p(·)τ (·) nσ 2(w− t+ )k + 2kns v=k+4 |Q(·, 2k−vP +3 )|rσ rσ nσ

c 2(w− t+ )k

+

c 2kns

2

)k (w− nσ t+

+ 2kns

. 1, by Theorem 2. Therefore our estimate (27) follows by taking 0 < s < that d > n(r + 1)τ + + n + a + w.

w n

and the fact

Case 2. |P | < 1. We have |P |−τ (x) ≤ c |P |−τ (y) (1 + 2vP |x − y|)clog (τ ) ≤ c |P |−τ (y) (1 + 2v |x − y|)clog (τ ) for any x, y ∈ Rn . Hence, ηv,w ∗ gv,k,vP . η v,w−clog (τ ) ∗ |P |τ(·)



gv,k,vP |P |τ(·)



.

Therefore, we can use the similar arguments above to obtain the desired estimate, where we did not need to use Lemma 1, which could be used only to move |P |τ (·) inside the convolution and hence the proof is complete.

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