Relationship between Hardy Spaces Associated with Different Homogeneities and One-Parameter Hardy Spaces

Abstract

We prove that the Hardy spaces associated with different homogeneities 𝐻^𝑝_com, 0<𝑝<∞, are continuously embedded into the intersection of the isotropic Hardy spaces 𝐻^𝑝_iso and the nonisotropic Hardy spaces 𝐻^𝑝_non. As a consequence, we obtain that any operator bounded from either 𝐻^𝑝_iso or 𝐻^𝑝_non into 𝐿^𝑝 must be bounded from 𝐻^𝑝_com to 𝐿^𝑝.

1. Introduction

For 𝑥=(𝑥^′,𝑥_𝑛)∈ℝ^𝑛−1×ℝ and 𝛿>0, we consider two kinds of homogeneities on ℝ^𝑛: 𝛿∘_𝑒𝑥^′,𝑥_𝑛=𝛿𝑥^′,𝛿𝑥_𝑛,𝛿∘_ℎ𝑥^′,𝑥_𝑛=𝛿𝑥^′,𝛿²𝑥_𝑛. (1)

The first are the classical isotropic dilations occurring in the classical Calderón-Zygmund singular integrals, while the second are nonisotropic and related to the heat equations (also the Heisenberg groups). Let 𝑒(𝜉) and ℎ(𝜉) be functions on ℝ^𝑛 homogeneous of degree 0 in the isotropic sense and in the nonisotropic sense, respectively, and both smooth away from the origin. Then, it is well known that the Fourier multipliers 𝑇₁ defined by 𝑇₁(𝑓)(𝜉)=𝑒(𝜉)𝑓(𝜉) and 𝑇₂ given by 𝑇₂(𝑓)(𝜉)=ℎ(𝜉)𝑓(𝜉) are both bounded on 𝐿^𝑝 for 1<𝑝<∞, of weak-type (1,1), and bounded on the classical isotropic Hardy spaces 𝐻^𝑝_iso and, nonisotropic Hardy spaces 𝐻^𝑝_non, respectively. Riviere in [1] asked the question is the composition 𝑇₁∘𝑇₂ still of weak-type (1,1)? Phong and Stein in [2] answered this question and gave a necessary and sufficient condition for which 𝑇₁∘𝑇₂ is of weak-type (1,1). The operators Phong and Stein studied are in fact a composition of operators with different kinds of homogeneities which arise naturally in the 𝜕-Neumann problem. Recently, Han et al. [3] developed a theory of the Hardy spaces 𝐻^𝑝_com, 0<𝑝<∞, associated with the different homogeneities and proved that the composition of the two Calderón-Zygmund convolution operators with different homogeneities is bounded on 𝐻^𝑝_com(ℝ^𝑛). Weighted function spaces associated with different homogeneities and boundedness of composition of operators on them were recently investigated in [4–6].

The Hardy spaces 𝐻^𝑝_com introduced in [3] have surprising multiparameter structures which reflect the mixed homogeneities arising from the two operators under consideration. A natural question arises: Is there any relationship between 𝐻^𝑝_com and the two classical one-parameter Hardy spaces 𝐻^𝑝_iso and 𝐻^𝑝_non? The main purpose of this paper is to answer this question. We shall prove that 𝐻^𝑝_com are continuously embedded into the intersection of 𝐻^𝑝_iso and 𝐻^𝑝_non. As an application, we show that any operator boundedness from either 𝐻^𝑝_iso or 𝐻^𝑝_non into 𝐿^𝑝 must be bounded from 𝐻^𝑝_com into 𝐿^𝑝. Our methods are to use the partially discrete Calderón-type formula and the Littlewood-Paley theory in this context, which are appropriately developed.

Before stating the results more precisely, we first recall some notions and notations. For 𝑥=(𝑥^′,𝑥_𝑛)∈ℝ^𝑛−1×ℝ, we denote |𝑥|_𝑒=(|𝑥^′|²+|𝑥_𝑛|²)^1/2 and |𝑥|_ℎ=(|𝑥^′|²+|𝑥_𝑛|)^1/2. Let 𝜓⁽¹⁾∈𝒮(ℝ^𝑛) satisfy supp𝜓⁽¹⁾(𝜉)⊆1𝜉∶2<𝜉_𝑒≤2_𝑗∈ℤ,𝜓⁽¹⁾2^𝑗∘_𝑒𝜉=1∀𝜉∈ℝ^𝑛⧵{0}. (2) The isotropic discrete square function 𝑔^𝑑_iso(𝑓) is defined by 𝑔^𝑑_iso𝑓(𝑥)∶=_{𝑗∈ℤ𝑄∈𝒬𝑗iso}𝜓_𝑗⁽¹⁾∗𝑓𝑥_𝑄²𝜒_𝑄(𝑥)^1/2, (3) where 𝜓_𝑗⁽¹⁾(𝑥)=2^−𝑗𝑛𝜓⁽¹⁾(2^−𝑗∘_𝑒𝑥), 𝒬^𝑗_iso denotes the set of all dyadic cubes with sidelength 2^𝑗, and 𝑥_𝑄 denotes the left-lower corner of 𝑄. The isotropic Hardy spaces 𝐻^𝑝_iso(ℝ^𝑛), 0<𝑝<∞, are defined by 𝐻^𝑝_isoℝ^𝑛𝒮∶=𝑓∈^′𝒫∶𝑔^𝑑_iso𝑓_{𝐿𝑝(ℝ𝑛)}<∞. (4) Similarly, let 𝜓⁽²⁾∈𝒮(ℝ^𝑛) satisfy supp𝜓⁽²⁾(𝜉)⊆1𝜉∶2<𝜉_ℎ≤2_𝑘∈ℤ,𝜓⁽²⁾2^𝑘∘_ℎ𝜉=1∀𝜉∈ℝ^𝑛⧵{0}. (5) The nonisotropic discrete square function 𝑔^𝑑_non(𝑓) is defined by 𝑔^𝑑_non𝑓(𝑥)∶=_{𝑘∈ℤ𝑄∈𝒬𝑘non}𝜓_𝑘⁽²⁾∗𝑓𝑥_𝑄²𝜒_𝑄(𝑥)^1/2, (6) where 𝜓_𝑘⁽²⁾(𝑥)=2^{−𝑘(𝑛+1)}𝜓⁽²⁾(2^−𝑘∘_ℎ𝑥), 𝒬^𝑘_non is the set of dyadic nonisotropic “cubes” 𝑄=𝑄₁×𝑄₂⊂ℝ^𝑛−1×ℝ with sidelength ℓ(𝑄₁)=2^𝑘 and |𝑄₂|=2^2𝑘, and 𝑥_𝑄 is the left-lower corner of 𝑄. The nonisotropic Hardy spaces 𝐻^𝑝_non, 0<𝑝<∞, are defined by 𝐻^𝑝_nonℝ^𝑛𝒮∶=𝑓∈^′𝒫∶𝑔^𝑑_non𝑓_{𝐿𝑝(ℝ𝑛)}<∞. (7)

For 𝑗,𝑘∈ℤ, let 𝜓_𝑗,𝑘=𝜓_𝑗⁽¹⁾∗𝜓_𝑘⁽²⁾. The discrete square function 𝑔^𝑑_com associated with different homogeneities is given by 𝑔^𝑑_com𝑓(𝑥)∶=_{𝑗,𝑘∈ℤ𝑅∈ℛ𝑗,𝑘}𝜓_𝑗,𝑘∗𝑓𝑥_𝑅²𝜒_𝑅(𝑥)^1/2, (8) where ℛ^𝑗,𝑘 denote the set of dyadic rectangles 𝑅=𝑄₁×𝑄₂ in ℝ^𝑛−1×ℝ with sidelength ℓ(𝑄₁)=2^𝑗∨𝑘 and |𝑄₂|=2^𝑗∨2𝑘 and 𝑥_𝑅 is the left-lower corner of 𝑅. The Hardy spaces 𝐻^𝑝_com, 0<𝑝<∞, associated with different homogeneities are defined by 𝐻^𝑝_comℝ^𝑛𝒮∶=𝑓∈^′𝒫∶𝑔^𝑑_com𝑓_{𝐿𝑝(ℝ𝑛)}<∞. (9)

The main result of this paper is as follows.

Theorem 1. Let 0<𝑝<∞. One has 𝐻^𝑝_comℝ^𝑛↪𝐻^𝑝_isoℝ^𝑛∩𝐻^𝑝_nonℝ^𝑛. (10) More precisely, there is a constant 𝐶 depending on 𝑝 and 𝑛 such that, for all 𝑓∈𝐻^𝑝_com(ℝ^𝑛), 𝑓_𝐻^𝑝_iso(ℝ𝑛)+𝑓_𝐻^𝑝_non(ℝ𝑛)≤𝐶𝑓_𝐻^𝑝_com(ℝ𝑛). (11)

In [3], it was proved that the composition of the two Calderón-Zygmund convolution operators with different homogeneities is bounded on 𝐻^𝑝_com (see [3, Theorem 1.9]). This result can be improved by the following.

Corollary 2. Let 0<𝑝<∞. One has that(1)any operator 𝑇 bounded from either 𝐻^𝑝_iso(ℝ^𝑛) or 𝐻^𝑝_non(ℝ^𝑛) into 𝐿^𝑝(ℝ^𝑛) must be bounded from 𝐻^𝑝_com(ℝ^𝑛) to 𝐿^𝑝(ℝ^𝑛); (2)if 𝑇₁ is bounded from 𝐻^𝑝_iso(ℝ^𝑛) to 𝐿^𝑝(ℝ^𝑛) and 𝑇₂ is bounded from 𝐻^𝑝_non(ℝ^𝑛) to 𝐿^𝑝(ℝ^𝑛), then 𝑇₁∘𝑇₂ is bounded from 𝐻^𝑝_com(ℝ^𝑛) to 𝐿^𝑝(ℝ^𝑛).

2. New Square Function Characterizations of 𝐻^𝑝_com

Let 𝜙⁽¹⁾∈𝒮(ℝ^𝑛) satisfy supp𝜙⁽¹⁾⊆𝐵(0,1), _ℝ𝑛𝜙⁽¹⁾(𝑥)𝑥^𝛼𝑑𝑥=0∀|𝛼|≤𝑀_𝑝,_𝑗∈ℤ𝜙⁽¹⁾2^𝑗∘_𝑒𝜉=1∀𝜉∈ℝ^𝑛⧵{0}, (12) where 𝑀_𝑝 is a sufficiently large constant depending on 𝑝 and 𝑛. Similarly, let 𝜙⁽²⁾∈𝒮(ℝ^𝑛) satisfy supp 𝜙⁽²⁾⊆𝐵(0,1), _ℝ𝑛𝜙⁽²⁾(𝑥)𝑥^𝛽𝑑𝑥=0∀𝛽≤𝑀_𝑝,_𝑘∈ℤ𝜓⁽²⁾2^𝑘∘_ℎ𝜉=1∀𝜉∈ℝ^𝑛⧵{0}. (13) Let 𝜙_𝑗⁽¹⁾(𝑥)=2^−𝑗𝑛𝜙(2^−𝑗∘_𝑒𝑥), 𝜙_𝑘⁽²⁾(𝑥)=2^{−𝑘(𝑛+1)}𝜙(2^−𝑘∘_ℎ𝑥), and 𝜙_𝑗,𝑘=𝜙_𝑗⁽¹⁾∗𝜙_𝑘⁽²⁾. Then, 𝐻^𝑝_com can be characterized via the continuous square function 𝑔_com defined by 𝑔_com𝑓(𝑥)=_{𝑗,𝑘∈ℤ}𝜙_𝑗,𝑘∗𝑓(𝑥)^21/2. (14) The following Calderón-type identities are well known (cf. [7–9]).

Lemma 3. For 𝑓∈𝐿²(ℝ^𝑛), there is a sufficiently large integer 𝑁 such that 𝑓=_{𝑗∈ℤ𝑄∈𝒬𝑗−𝑁iso}𝜙_𝑗⁽¹⁾⋅−𝜗_𝑄𝜙_𝑗⁽¹⁾∗𝑓𝜗_𝑄=_{𝑘∈ℤ𝑄∈𝒬𝑘−𝑁non}𝜙_𝑘⁽²⁾⋅−𝜗_𝑄𝜙_𝑘⁽²⁾∗𝑓𝜗_𝑄, (15) where the two series converge in 𝐿²(ℝ^𝑛) and 𝜗_𝑄 denotes any fixed point of 𝑄.

The Calderón-type identities in Lemma 3 lead to the following square functions: 𝑔⁽¹⁾_com𝑓(𝑥)∶=_{𝑗,𝑘∈ℤ𝑄∈𝒬𝑗−𝑁iso}𝜙_𝑗,𝑘∗𝑓𝑥_𝑄²𝜒_𝑄(𝑥)^1/2,𝑔⁽²⁾_com𝑓(𝑥)∶=_{𝑗,𝑘∈ℤ𝑄∈𝒬𝑘−𝑁non}𝜙_𝑗,𝑘∗𝑓𝑥_𝑄²𝜒_𝑄(𝑥)^1/2, (16) where 𝑁 is the sufficiently large integer in Lemma 3.

The purpose of this section is to prove the following.

Theorem 4. For 𝑝∈(0,∞), one has that 𝑓_𝐻^𝑝_com∼𝑔⁽¹⁾_com𝑓_𝐿𝑝∼𝑔⁽²⁾_com𝑓_𝐿𝑝. (17)

Proof. We only prove that 𝑓_𝐻^𝑝_com∼𝑔⁽¹⁾_com𝑓_𝐿𝑝, (18) as ‖𝑓‖_𝐻^𝑝_com∼‖𝑔⁽²⁾_com(𝑓)‖_𝐿𝑝 can be obtained in the same manner. To prove (18), it suffices to show that _{𝑗,𝑘𝑄∈𝒬𝑗−𝑁iso}sup_𝑢∈𝑄𝜙_𝑗,𝑘∗𝑓(𝑢)²𝜒_𝑄^1/2_𝐿𝑝≲_{𝑗′,𝑘𝑄∈𝒬𝑗′−𝑁iso}inf_𝑢∈𝑄𝜙_𝑗′,𝑘∗𝑓(𝑢)²𝜒_𝑄^1/2_𝐿𝑝. (19)
To verify (19), we need the following lemma.

Lemma 5 (see [10]). Let 𝑄∈𝒬^𝑗−𝑁_iso and 𝑀>0. Then, for any 𝑢,𝜗_𝑄∈𝑄, 𝜗_𝑄′∈𝑄^′, and 𝛿∈(𝑛/(𝑛+𝑀),1], one has that_{𝑄′∈𝒬𝑗′−𝑁iso}2^{𝑀(𝑗∨𝑗′)}2^{𝑗∨𝑗′}+𝑢−𝜗_𝑄′^𝑛+𝑀𝑔_𝑗′𝜗_𝑄′≤𝐶_𝑁2^{𝑛|𝑗−𝑗′|((1/𝛿)−1)}×ℳ_{𝑄′∈𝒬𝑗′−𝑁iso}𝑔_𝑗′𝜗_𝑄′²𝜒_𝑄′^𝛿/2𝜗_𝑄^1/𝛿, (20)where ℳ denotes the classical isotropic Hardy-Littlewood maximal operator and 𝑎∨𝑏=max(𝑎,𝑏).

We first assume that 𝑓∈𝐿²∩𝐻^𝑝_com(ℝ^𝑛). Set 𝑓_𝑘=𝜙_𝑘⁽²⁾∗𝑓. Applying the discrete Calderón-type identity in (15), the classical almost orthogonality estimate, and Lemma 5, we deduce that for any 𝑢,𝜗_𝑄∈𝑄, 𝜗_𝑄′∈𝑄^′, and 𝛿∈(𝑛/(𝑛+𝑀),min(𝑝,1)), 𝜙_𝑗,𝑘(𝑢=∗𝑓)_{𝑗′∈ℤ𝑄′∈𝒬𝑗′−𝑁iso}𝜙_𝑗⁽¹⁾∗𝜙_𝑗⁽¹⁾_′𝑢−𝜗_𝑄′𝜙_𝑗⁽¹⁾_′∗𝑓_𝑘𝜗_𝑄′≲_{𝑗′∈ℤ}2^{−|𝑗−𝑗′|10𝑀}×_{𝑄′∈𝒬𝑗′−𝑁iso}2^{𝑀(𝑗∨𝑗′)}2^{𝑗∨𝑗′}+𝑢−𝜗_𝑄′^𝑛+𝑀𝜙_𝑗⁽¹⁾_′∗𝑓_𝑘𝜗_𝑄′≲_{𝑗′∈ℤ}2^{−|𝑗−𝑗′|𝑀}×ℳ_{𝑄′∈𝒬𝑗′−𝑁iso}𝜙_𝑗⁽¹⁾_′∗𝑓_𝑘𝜗_𝑄′²×𝜒_𝑄′^𝛿/2𝜗_𝑄^1/𝛿. (21) Since 𝑢 and 𝜗_𝑄′ are arbitrary points in 𝑄 and 𝑄^′, respectively, sup_𝑢∈𝑄𝜙_𝑗,𝑘∗𝑓(𝑢)≲_{𝑗′∈ℤ}2^{−|𝑗−𝑗′|𝑀}×ℳ_{𝑄′∈𝒬𝑗′−𝑁iso}inf_{𝑣∈𝑄′}𝜙_𝑗⁽¹⁾_′∗𝑓_𝑘(𝑣)²×𝜒_𝑄′^𝛿/2𝜗_𝑄^1/𝛿, (22) which, by the Cauchy-Schwarz inequality, implies that sup_𝑢∈𝑄𝜙_𝑗,𝑘∗𝑓(𝑢)²≲_{𝑗′∈ℤ}2^{−|𝑗−𝑗′|𝑀}×ℳ_{𝑄′∈𝒬𝑗′−𝑁iso}inf_{𝑣∈𝑄′}𝜙_𝑗⁽¹⁾_′∗𝑓_𝑘(𝑣)²×𝜒_𝑄′^𝛿/2𝜗_𝑄^2/𝛿, (23) where we have used ∑_{𝑗′∈ℤ}2^{−|𝑗−𝑗′|𝑀}≤𝐶. Multiplying 𝜒_𝑄 from both sides and summing over 𝑗,𝑘∈ℤ,𝑄∈𝒬^𝑗−𝑁_iso yield that, for any 𝜗_𝑄∈𝑄, _{𝑗,𝑘∈ℤ𝑄∈𝒬𝑗−𝑁iso}sup_𝑢∈𝑄𝜙_𝑗,𝑘∗𝑓(𝑢)²𝜒_𝑄(𝑥)≲_{𝑗,𝑘∈ℤ𝑗′∈ℤ}2^{−|𝑗−𝑗′|𝑀}×ℳ_{𝑄′∈𝒬𝑗′−𝑁iso}inf_{𝑣∈𝑄′}𝜙_𝑗⁽¹⁾_′∗𝑓_𝑘(𝑣)²×𝜒_𝑄′^𝛿/2𝜗_𝑄^2/𝛿≲_{𝑗′∈ℤ𝑘∈ℤ}×ℳ_{𝑄′∈𝒬𝑗′−𝑁iso}inf_{𝑣∈𝑄′}𝜙_𝑗⁽¹⁾_′∗𝑓_𝑘(𝑣)²×𝜒_𝑄′^𝛿/2𝜗_𝑄.^2/𝛿 (24) It follows that _{𝑗,𝑘𝑄∈𝒬𝑗−𝑁iso}sup_𝑢∈𝑄𝜙_𝑗,𝑘∗𝑓(𝑢)²𝜒_𝑄(𝑥)^1/2≲_{𝑗′∈ℤ𝑘∈ℤ}×ℳ_{𝑄′∈𝒬𝑗′−𝑁iso}inf_{𝑣∈𝑄′}𝜙_𝑗⁽¹⁾_′∗𝑓_𝑘(𝑣)²×𝜒_𝑄′^𝛿/2(𝑥)^2/𝛿1/2. (25)

Finally, taking the 𝐿^𝑝 norm on both sides and applying the Fefferman-Stein vector-valued inequality yield (19). Since 𝐿²∩𝐻^𝑝_com is dense in 𝐻^𝑝_com (see [3]), a limiting argument concludes the proof of Theorem 4.

3. Proofs of Theorem 1 and Corollary 2

We first give the following.

Proof of Theorem 1. For 1<𝑝<∞, Theorem 1 is trivial since 𝐻^𝑝_com=𝐻^𝑝_iso=𝐻^𝑝_non=𝐿^𝑝. We now assume that 0<𝑝≤1. We only prove ‖𝑓‖_𝐻^𝑝_non≲‖𝑓‖_𝐻^𝑝_com as the inequality ‖𝑓‖_𝐻^𝑝_iso≲‖𝑓‖_𝐻^𝑝_com can be proved similarly. Since 𝐿²∩𝐻^𝑝_com is dense in 𝐻^𝑝_com, we may assume that 𝑓∈𝐿²∩𝐻^𝑝_com. For 𝑖∈ℤ, set Ω_𝑖=𝑥∈ℝ^𝑛∶𝑔⁽¹⁾_com𝑓(𝑥)>2^𝑖. (26) Denote that ℬ_𝑖=𝑗,𝑄∶𝑄∈𝒬^𝑗−𝑁_iso,𝑄∩Ω_𝑖>12|𝑄|,𝑄∩Ω_𝑖+1≤12|𝑄|, (27) where 𝑁 is the sufficiently large integer in Lemma 3.
Applying the discrete Calderón-type reproducing formula in Lemma 3, 𝑓_𝑘(𝑥)=_{𝑖∈ℤ𝑗,𝑄∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾𝑥−𝑥_𝑄,𝑘∈ℤ, (28) where 𝑓_𝑘∶=𝜙_𝑘⁽²⁾∗𝑓 and the series converges in the 𝐿²(ℝ^𝑛) norm. We claim that _{𝑘∈ℤ𝑗,𝑄∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾⋅−𝑥_𝑄^21/2𝑝_𝑝≤𝐶2^𝑖𝑝Ω_𝑖. (29) Assume the claim for the moment. Then, by the continuous square function characterization of 𝐻^𝑝_non, 𝑓^𝑝_𝐻𝑝non∼_{ℝ𝑛𝑘∈ℤ𝑖∈ℤ(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾×𝑥−𝑥_𝑄^2𝑝/2≤𝑑𝑥_{ℝ𝑛𝑖∈ℤ𝑘∈ℤ(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾×𝑥−𝑥_𝑄^21/2𝑝𝑑𝑥≤_{𝑖∈ℤ𝑘∈ℤ(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾×𝑥−𝑥_𝑄^2𝑝/2≲𝑑𝑥_𝑖∈ℤ2^𝑖𝑝Ω_𝑖≲𝑔⁽¹⁾_com𝑓^𝑝_𝑝∼𝑓^𝑝_𝐻𝑝com, (30) where in the second inequality, we have used Minkowski's inequality, for ℓ² and in the third inequality used the inequality (∑_𝑖|𝑎_𝑖|)^𝑝≤∑_𝑖|𝑎_𝑖|^𝑝 for 0<𝑝≤1.
Thus, to finish the proof of Theorem 1, it suffices to verify claim (29). Since 𝜙⁽¹⁾ is supported in unit ball of ℝ^𝑛, for (𝑗,𝑄)∈ℬ_𝑖, {𝜙_𝑗⁽¹⁾(⋅−𝑥_𝑄)} are supported in Ω_𝑖=𝑥∶ℳ𝜒_Ω𝑖(𝑥)>12. (31) Thus, by Hölder's inequality and |Ω_𝑖|≤𝐶|Ω_𝑖| (which follows from 𝐿² boundedness of ℳ), _{𝑘(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾⋅−𝑥_𝑄^21/2𝑝_𝑝=Ω_{𝑖𝑘(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾×𝑥−𝑥_𝑄^2𝑝/2≤Ω𝑑𝑥_{𝑖1−𝑝/2}Ω_{𝑖𝑘(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾×𝑥−𝑥_𝑄²≲Ω𝑑𝑥_{𝑖1−𝑝/2𝑘(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾×⋅−𝑥_𝑄²₂. (32) We now estimate the last 𝐿² norm by duality argument. For 𝜁 with ‖𝜁‖₂≤1, applying the Cauchy-Schwarz inequality yields _{𝑗,𝑄∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾⋅−𝑥_𝑄=,𝜁_{(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾𝑥−𝑥_𝑄𝜁_𝑘(𝑥)≤𝑑𝑥_{𝑗,𝑄∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄^21/2×_{𝑗,𝑄∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝜁𝑥_𝑄^21/2, (33) where 𝜙_𝑗⁽¹⁾(𝑥)=𝜙_𝑗⁽¹⁾(−𝑥). Since _{(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝜁𝑥_𝑄²=_{ℝ𝑛(𝑗,𝑄)∈ℬ𝑖}𝜙_𝑗⁽¹⁾∗𝜁𝑥_𝑄²𝜒_𝑄(𝑥)^(1/2)⋅2≤𝑑𝑥_ℝ𝑛𝑔^𝑑_iso(𝜁)(𝑥)²𝑑𝑥≲𝜁²_𝐿2≤1, (34) we now have that _{𝑘(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄𝜙_𝑗⁽¹⁾⋅−𝑥_𝑄²_𝐿2≲_{𝑘(𝑗,𝑄)∈ℬ𝑖}|𝑄|𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄², (35) which is, in turn, bounded by 2_{𝑘(𝑗,𝑄)∈ℬ𝑖}𝜙_𝑗⁽¹⁾∗𝑓_𝑘𝑥_𝑄²𝑄∩Ω_𝑖−Ω_𝑖+1Ω=2_{𝑖⧵Ω𝑖+1𝑘(𝑗,𝑄)∈ℬ𝑖}𝜙_𝑗,𝑘∗𝑓𝑥_𝑄²𝜒_𝑄(𝑥)^(1/2)⋅2Ω𝑑𝑥≤2_{𝑖⧵Ω𝑖+1}𝑔⁽¹⁾_com𝑓(𝑥)²𝑑𝑥≲2^2𝑖Ω_𝑖, (36) where in the first inequality, we have used |𝑄|≤2|𝑄∩(Ω_𝑖⧵Ω_𝑖+1)| when (𝑗,𝑄)∈ℬ_𝑖. Substituting this estimate back to (32) verifies the claim (29), and hence, Theorem 1 follows.

Finally, we give the following.

Proof of Corollary 2. (1) follows directly from Theorem 1, while (2) follows from (1).

Acknowledgments

This research was supported by NNSF, China (Grant no. 11101423), and supported in part by NNSF, China (Grant no. 11171345). The author would like to express his deep gratitude to the referee for his/her valuable comments and suggestions.