Hardy Spaces Associated to Schrödinger Operators on Product Spaces

Abstract

Let 𝐿=−Δ+𝑉 be a Schrödinger operator on ℝ^𝑛, where 𝑉∈𝐿¹_loc(ℝ^𝑛) is a nonnegative function on ℝ^𝑛. In this article, we show that the Hardy spaces 𝐿 on product spaces can be characterized in terms of the Lusin area integral, atomic decomposition, and maximal functions.

1. Introduction

Let 𝑉∈𝐿^1loc(ℝ^𝑛) be a nonnegative function on ℝ^𝑛. The Schrödinger operator with potential 𝑉 is defined by 𝐿=−Δ+𝑉onℝ^𝑛.(1.1) The operator 𝐿 is a self-adjoint positive definite operator. From the Feynman-Kac formula, it is well known (see, e.g., [1, page 195]) that the kernel ℎ_𝑡(𝑥,𝑦) of the semigroup 𝑒^−𝑡𝐿 satisfies the estimate0≤ℎ_𝑡(𝑥,𝑦)≤1(4𝜋𝑡)^𝑛/2𝑒^{−|𝑥−𝑦|2/4𝑡}(1.2) for all 𝑡>0 and 𝑥,𝑦∈ℝ^𝑛.

Let us consider the Hardy space on product domains. We note that the usual Hardy space 𝐻¹(ℝ^𝑛×ℝ^𝑛) on the product domain is now well understood (see, e.g., [2–4]). In this paper we will be concerned with the space 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛) associated to 𝐿 as introduced in [5] (see [6, 7] for one-parameter theory). Firstly, we set𝐻²(ℝ^𝑛×ℝ^𝑛)=ℛ(𝐿⊗𝐿)=(𝐿⊗𝐿)𝑢∈𝐿²(ℝ^𝑛×ℝ^𝑛)∶𝑢∈𝐿²(ℝ^𝑛×ℝ^𝑛)(1.3) and note that𝐿²(ℝ^𝑛×ℝ^𝑛)=ℛ(𝐿⊗𝐿)⊕𝒩(𝐿⊗𝐿)=𝐻²(ℝ^𝑛×ℝ^𝑛)⊕𝒩(𝐿⊗𝐿),(1.4) where ℛ(𝐿⊗𝐿) (resp., 𝒩(𝐿⊗𝐿)) stands for the range (resp., the nullspace) of 𝐿⊗𝐿, and the sum is orthogonal. For a function 𝑓∈𝐿²(ℝ^𝑛×ℝ^𝑛), define𝑆_𝐿(𝑓)𝑥₁,𝑥₂=_{111||𝑥22||2|𝑥−𝑦|<𝑡−𝑦<𝑡}|||𝑄_𝑡21𝑡22𝑓𝑦₁,𝑦₂|||²𝑑𝑦₁𝑑𝑦₂𝑑𝑡₁𝑡^1𝑛+1𝑑𝑡₂𝑡^2𝑛+1^1/2,(1.5) where 𝑄_𝑡1𝑡2𝑓=−𝑡₁𝑑𝑑𝑡₁−𝑡²𝑑𝑑𝑡₂𝐻^𝑡1𝑡2𝐻𝑓,_𝑡1𝑡2𝑓𝑦₁,𝑦₂=_{ℝ𝑛×ℝ𝑛}ℎ_𝑡1𝑦₁,𝑧₁ℎ_𝑡2𝑦₂,𝑧₂𝑓𝑧₁,𝑧₂𝑑𝑧₁𝑑𝑧₂.(1.6) The space 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛) is defined as the completion of 𝐻²(ℝ^𝑛×ℝ^𝑛) in the norm given by‖𝑓‖_𝐻^1𝐿_(ℝ^𝑛_×ℝ^𝑛₎=‖‖𝑆_𝐿(𝑓)‖‖_{𝐿1(ℝ2𝑛)}.(1.7)

The main purpose of this article is to derive atomic characterizations and the maximal characterizations of 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛). Before stating our results, let us recall some necessary notations (see also [8, 9]). Suppose that Ω⊂ℝ^𝑛×ℝ^𝑛 is an open set with finite measure. Denote by 𝑚(Ω) the maximal dyadic subrectangles of Ω in the form of 𝑅=𝐼×𝐽, where 𝐼 and 𝐽 are cubes in ℝ^𝑛. Let 𝑅=𝐼×𝐽∈𝑚(Ω) and we denote by ℓ(𝐼) and ℓ(𝐽) the side lengths of 𝐼 and 𝐽, respectively. For given 𝜆>0, we will write 𝜆𝑅 for the 𝜆-fold dilation of 𝑅=𝐼×𝐽 with the same center.

An 𝐿-atom is a function 𝑎 on ℝ^2𝑛, together with an open set Ω of finite measure, which satisfies the following properties:(i)supp(𝑎)⊂Ω (see (2.4));(ii)𝑎(𝑥) can be further decomposed as 𝑎=∑_{𝑅∈𝑚(Ω)}𝑎_𝑅, where supp𝐿⁻¹^𝑖⊗𝐿⁻¹^𝑗𝑎_𝑅⊂10𝑅,_{𝑅∈𝑚(Ω)}‖‖ℓ(𝐼)⁻²𝐿⁻¹^𝑖⊗ℓ(𝐽)⁻²𝐿⁻¹^𝑗𝑎_𝑅‖‖^2𝐿_2(ℝ2𝑛)≤||Ω||⁻¹,𝑖,𝑗=0,1;(1.8)(iii)‖𝑎‖_𝐿²_(ℝ^2𝑛₎≤|Ω|⁻².

We can define the atomic Hardy space 𝐻¹_atom(ℝ^𝑛×ℝ^𝑛) by^∞_𝑗=0𝜆_𝑗𝛼_𝑗∶𝜆_𝑗^∞𝑗=0∈ℓ¹and𝛼_𝑗are𝐿-atoms(1.9) with the norm ‖𝑓‖_𝐻¹_atom(ℝ^𝑛_×ℝ^𝑛₎ given by the natural quotient norm:inf^∞_𝑗=0||𝜆_𝑗||∶𝑓=^∞_𝑗=0𝜆_𝑗𝛼_𝑗,where𝜆_𝑗^∞𝑗=0∈ℓ¹and𝛼_𝑗are𝐿-atoms.(1.10)

The first result of this paper is the following theorem.

Theorem 1.1. Let 𝐿 be the Schrödinger operator as (1.1). Then the spaces 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛) and 𝐻¹_atom(ℝ^𝑛×ℝ^𝑛) coincide. In other words, ‖𝑓‖_𝐻^1𝐿_(ℝ^𝑛_×ℝ^𝑛₎≈‖𝑓‖_𝐻¹_atom(ℝ^𝑛_×ℝ^𝑛₎,forevery𝑓∈𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛).(1.11)

Next we give the “maximal" characterizations of 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛). Given a function 𝑓∈𝐿¹(ℝ^2𝑛) we define two functions:𝒩𝑓^+ℎ(𝑥)∶=sup_{𝑡1,𝑡2>0}||𝐻_𝑡1,𝑡2𝑓(𝑥)_ℎ||,𝑓(𝑥)∶=sup_{|𝑥1−𝑦1|<𝑡1|𝑥2−𝑦2|<𝑡2}|||𝐻_{𝑡21,𝑡22}𝑓(𝑦)|||.(1.12) Similarly, one can consider the Poisson semigroup generated by the operator 𝐿 and the operators𝑆_𝑃𝑓(𝑥)∶=_{111||𝑥22||2|𝑥−𝑦|<𝑡−𝑦<𝑡}|||𝑡₁√𝐿₁𝑒^{−𝑡1√𝐿1}⊗𝑡₂√𝐿₂𝑒^{−𝑡2√𝐿2}𝑓(𝑦)|||²𝑑𝑦₁𝑑𝑦₂𝑑𝑡₁𝑑𝑡₂𝑡^1𝑛+1𝑡^2𝑛+1^1/2𝑓_+𝑃,(𝑥)∶=sup_{𝑡1𝑡>02>0}|||𝑒_{−𝑡1√𝐿1}𝒩⊗𝑒^{−𝑡2√𝐿2}𝑓(𝑥)_𝑃|||,𝑓(𝑥)∶=sup_{|𝑥1−𝑦1|<𝑡1|𝑥2−𝑦2|<𝑡2}|||𝑒_{−𝑡1√𝐿1}⊗𝑒^{−𝑡2√𝐿2}𝑓(𝑦)|||,(1.13) with 𝑥∈ℝ^𝑛×ℝ^𝑛 and 𝑓∈𝐿²(ℝ^2𝑛).

Define the spaces 𝐻^1max,ℎ(ℝ^𝑛×ℝ^𝑛), 𝐻^1𝒩_ℎ(ℝ^𝑛×ℝ^𝑛),𝐻^1𝑆_𝑃(ℝ^𝑛×ℝ^𝑛),𝐻^1max,𝑃(ℝ^𝑛×ℝ^𝑛), and 𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛) as the completion of 𝐻²(ℝ^𝑛×ℝ^𝑛) in the norms given by the 𝐿¹ norm of the corresponding square or maximal functions, respectively. For example,‖𝑓‖_𝐻^1max,ℎ_(ℝ^𝑛_×ℝ^𝑛₎∶=‖‖𝑓^+ℎ‖‖_{𝐿1(ℝ2𝑛)}.(1.14) By a similar manner, the norms of 𝐻^1𝒩_ℎ(ℝ^𝑛×ℝ^𝑛),𝐻^1𝑆_𝑃(ℝ^𝑛×ℝ^𝑛),𝐻^1max,𝑃(ℝ^𝑛×ℝ^𝑛), and 𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛) are defined. The second result of this paper is the following.

Theorem 1.2. Let 𝐿 be the Schrödinger operator as (1.1). Then the spaces 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛), 𝐻^1max,ℎ(ℝ^𝑛×ℝ^𝑛), 𝐻^1𝒩_ℎ(ℝ^𝑛×ℝ^𝑛),𝐻^1𝑆_𝑃(ℝ^𝑛×ℝ^𝑛),𝐻^1max,𝑃(ℝ^𝑛×ℝ^𝑛), and 𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛) coincide with equivalent norms.

This paper is organized as follows. In Section 2, we will give some preliminary results including the properties of Schrödinger operators and tent spaces on product spaces. The proofs of Theorems 1.1 and 1.2 will be given in Sections 3 and 4, respectively.

Throughout this paper, the letter “𝐶" or “𝑐" will denote (possibly different) constants that are independent of the essential variables.

2. Preliminaries

2.1. Tent Spaces on Product Domains

A theory of “tent spaces" was developed by Coifman et al. [10, 11]. These spaces are useful for the study of a variety of problems in harmonic analysis. In particular, we note that the tent spaces give a natural and simple approach to the atomic decomposition of functions in the classical Hardy space 𝐻^𝑝(ℝ^𝑛) by using the area integral functions and the theory of the Carleson measure. See also [6, 12].

Tent spaces have been studied by [13, 14] in connection with the theory of Carleson measures on product domains. Let ℝ^+𝑛+1 be the usual upper half-space in ℝ^𝑛+1. For any 𝛼>0, we set Γ_𝛼(𝑥)={(𝑦,𝑡)∈ℝ^+𝑛+1×ℝ^+𝑛+1,|𝑥₁−𝑦₁|<𝛼𝑡₁,and|𝑥₂−𝑦₂|<𝛼𝑡₂} as the standard cone (of aperture 𝛼) with vertex 𝑥∈ℝ^𝑛×ℝ^𝑛. In particular, we set Γ(𝑥)=Γ₁(𝑥). If (𝑥,𝑡)∈ℝ^+𝑛+1×ℝ^+𝑛+1, then 𝑅_𝑥,𝑡 denotes the rectangle centered at 𝑥∈ℝ^𝑛×ℝ^𝑛 whose side lengths are 𝑡₁ and 𝑡₂, respectively. For any open set Ω⊂ℝ^𝑛×ℝ^𝑛, the tent over Ω, 𝑇(Ω), is the set(𝑥,𝑡)∈ℝ^+𝑛+1×ℝ^+𝑛+1∶𝑅_𝑥,𝑡⊂Ω.(2.1) For any function 𝑓(𝑦,𝑡) defined on ℝ^+𝑛+1×ℝ^+𝑛+1 we will write𝒜(𝑓)(𝑥)=_Γ(𝑥)||𝑓(𝑦,𝑡)||²𝑑𝑦𝑑𝑡𝑡^1𝑛+1𝑡^2𝑛+1^1/2.(2.2) The tent space 𝑇^𝑝2 is then defined as the space of functions 𝑓 such that 𝒜𝑓∈𝐿^𝑝(ℝ^2𝑛) and is equipped with the norm, ‖𝑓‖_𝑇^𝑝2=‖𝒜(𝑓)‖_𝑝, 0<𝑝<∞.

We now introduce 𝑇¹²-atoms.

Definition 2.1. A function 𝐴(𝑥,𝑡)∶ℝ^+𝑛+1×ℝ^+𝑛+1→ℂ is called a 𝑇¹²-atom if there exists an open set Ω⊂ℝ^𝑛×ℝ^𝑛 of finite measure satisfying the following properties:(i)𝐴(𝑥,𝑡) can be further decomposed as 𝐴=∑_{𝑅∈𝑚(Ω)}𝐴_𝑅, where each 𝐴_𝑅 is supported in 𝑇(3𝑅), and 𝑅⊂Ω is a maximal dyadic subrectangle of Ω in the form of 𝑅=𝐼×𝐽, where 𝐼 and 𝐽 are cubes in ℝ^𝑛;(ii)‖𝐴‖_𝐿²_{(𝑑𝑦𝑑𝑡/𝑡1𝑡2)}≤|Ω|^−1/2 and ∑_{𝑅∈𝑚(Ω)}‖𝐴_𝑅‖^2𝐿_{2(𝑑𝑦𝑑𝑡/𝑡1𝑡2)}≤|Ω|⁻¹.

Proposition 2.2. Suppose 𝑓∈𝑇¹². Then 𝑓=∑_𝑘𝜆_𝑘𝐴_𝑘, where 𝐴_𝑘 are 𝑇¹²-atoms, 𝜆_𝑘∈ℂ, and ∑_𝑘|𝜆_𝑘|≤𝑐‖𝑓‖_𝑇¹² so that the sum converges in the 𝑇¹² norm. Moreover, if one assumes that 𝑓∈𝑇²², then the sum also converges in the 𝑇²² norm.

Proof. See [5, Proposition  3.3] for the proof.

2.2. Some Results on Product Spaces

We recall that the strong maximal function is defined as follows:𝑀_𝑠𝑓(𝑥)=sup_{𝑥∈𝑅=𝐼×𝐽}1||𝑅||_𝑅||𝑓(𝑥)||𝑑𝑥,(2.3) where 𝐼 and 𝐽 are cubes in ℝ^𝑛. It is well known that the operator 𝑀_𝑠 is bounded on 𝐿^𝑝(ℝ^2𝑛), for 1<𝑝<∞.

Now for any open set Ω⊂ℝ^𝑛×ℝ^𝑛 with finite measure, we setΩ=𝑥∣∈ℝ^𝑛×ℝ^𝑛,𝑀_𝑠𝜒_Ω(𝑥)>12^2𝑛.(2.4) By the strong maximal theorem, |Ω|≤𝑐|Ω|. Denote by 𝑚₁(Ω) the dyadic subrectangles 𝑅=𝐼×𝐽⊂Ω that are maximal in the 𝑥₁ direction, where 𝐼and𝐽 are dyadic cubes in ℝ^𝑛. Define 𝑚₂(Ω) similarly. It is well known that Journé’s covering lemma holds (see [8, 15]).

Lemma 2.3. Let Ω⊂ℝ^𝑛×ℝ^𝑛. For any 𝑅=𝐼×𝐽∈𝑚₂(Ω), one sets 𝛾₁(𝑅)=sup_{𝐼⊂𝑙𝑙×𝐽⊂}Ω|𝑙|/|𝐼|. Define 𝛾₂ similarly. Then for any 𝛿>0, one has _𝑅∈_𝑚2(Ω)||𝑅||𝛾^1−𝛿(𝑅)≤𝑐_𝛿||Ω||,_{𝑅∈𝑚1(Ω)}||𝑅||𝛾^2−𝛿(𝑅)≤𝑐_𝛿||Ω||,(2.5) where 𝑐_𝛿 is a constant depending only on 𝛿, but not on Ω.

The following lemma shows that in order to prove that an operator is bounded from 𝐻¹_atom(ℝ^𝑛×ℝ^𝑛) to 𝐿¹(ℝ^2𝑛), we just need to check that the operator is uniformly bounded on the 𝐿-atoms.

Lemma 2.4. Assume that 𝑇 is either a linear operator or a positive sublinear operator, bounded on 𝐿²(ℝ^2𝑛) and for every 𝐿-atom 𝑎, ‖𝑇(𝑎)‖_𝐿¹_(ℝ^2𝑛₎≤𝑐(2.6) with constant 𝑐 independent on 𝑎. Then 𝑇 can extend to a bounded operator from 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛) to 𝐿¹(ℝ^2𝑛), and ‖𝑇(𝑓)‖_𝐿¹_(ℝ^2𝑛₎≤𝑐‖𝑓‖_𝐻^1𝐿_(ℝ^𝑛_×ℝ^𝑛₎.(2.7)

Proof. Its proof is similar to that of [16, Lemma  3.3] and we omit it here. See also [17].

2.3. Some Properties of the Schrödinger Operator 𝐿 on ℝ^𝑛

Let 𝐿 be the Schrödinger operator as (1.1), and let ℎ_𝑡(𝑥,𝑦) be the kernels of the operators of semigroup {𝑒^−𝑡𝐿}.

First we note that, for each 𝑘∈ℕ, there exist two positive constants 𝐶_𝑘 and 𝑐_𝑘 such that the time derivatives of ℎ_𝑡 satisfy||||𝜕^𝑘𝜕𝑡^𝑘ℎ_𝑡(𝑥,𝑦)||||≤𝐶_𝑘𝑡^{−(𝑛+2𝑘)/2}exp−||𝑥−𝑦||²𝑐_𝑘𝑡(2.8) for all 𝑡>0 and almost all 𝑥,𝑦∈ℝ^𝑛. For the proof, see, for example, [18, 19].

Next, for 𝑠>0, we define𝔽(𝑠)∶=𝜓∶ℂ→ℂmeasurable∶||𝜓(𝑧)||≤𝐶|𝑧|^𝑠1+|𝑧|^2𝑠.(2.9) Then for any nonzero function 𝜓∈𝔽(𝑠), we have that {∫^∞0|𝜓(𝑡)|²𝑑𝑡/𝑡}^1/2<∞. Denote 𝜓_𝑡(𝑧)=𝜓(𝑡𝑧). It follows from the spectral theory in [20] that for any 𝑓∈𝐿²(ℝ^𝑛),^∞0‖‖𝜓𝑡√𝐿𝑓‖‖^2𝐿_2(𝑋)𝑑𝑡𝑡^1/2=^∞0𝜓𝑡√𝐿𝜓𝑡√𝐿𝑓,𝑓𝑑𝑡𝑡^1/2=^∞0||𝜓||²𝑡√𝐿𝑑𝑡𝑡𝑓,𝑓^1/2=𝜅‖𝑓‖_𝐿²_(ℝ^𝑛₎,(2.10) where 𝜅={∫^∞0|𝜓(𝑡)|²𝑑𝑡/𝑡}^1/2. As an application, we have ‖𝑆_𝐿(𝑓)‖₂=𝑐‖𝑓‖₂ for some constant 𝑐.

Lemma 2.5. Let 𝑓∈𝐿²(ℝ^𝑛) and 𝑢=𝑒^{√−𝑡𝐿}𝑓. Then for any 𝑝>0, there exists a constant 𝐶=𝐶(𝑛,𝑝)>0 such that sup_{𝐵((𝑥0,𝑡0),𝑟)}||𝑢(𝑥,𝑡)||≤𝐶1𝑟^𝑛+1_{𝐵((𝑥0,𝑡0),2𝑟)}||𝑢(𝑥,𝑡)||^𝑝𝑑𝑥𝑑𝑡^1/𝑝,(2.11) where 𝐵((𝑥₀,𝑡₀),𝑟)=𝐵_𝑟(𝑥₀)×[𝑡₀−𝑐𝑟,𝑡₀] with 𝑡₀>2𝑐𝑟 and 𝑐>0.

Proof. For the proof, we refer to [7, Lemma  8.4].

Recall that if 𝐿 is a nonnegative, self-adjoint operator on 𝐿²(ℝ^𝑛), and 𝐸_𝐿(𝜆) denotes its spectral decomposition, then for every bounded Borel function 𝐹∶[0,∞)→ℂ, one defines the operator 𝐹(𝐿)∶𝐿²(ℝ^𝑛)→𝐿²(ℝ^𝑛) by the formula𝐹(𝐿)∶=^∞0𝐹(𝜆)𝑑𝐸_𝐿(𝜆).(2.12) In particular, the operator cos(𝑡√𝐿) is then well defined on 𝐿²(ℝ^𝑛). Moreover, it follows from [21, Theorem  3] that the Schwartz kernel 𝐾_{√cos(𝑡𝐿)}(𝑥,𝑦) of cos(𝑡√𝐿) satisfiessupp𝐾_{√cos(𝑡𝐿)}(𝑥,𝑦)⊂(𝑥,𝑦)∈ℝ^𝑛×ℝ^𝑛∶||𝑥−𝑦||≤𝑡.(2.13) See also [22]. By the Fourier inversion formula, whenever 𝐹 is an even bounded Borel function with the Fourier transform of 𝐹, 𝐹∈𝐿¹(ℝ), we can write 𝐹(√𝐿) in terms of cos(𝑡√𝐿). In fact, using (2.12) we have𝐹√𝐿=(2𝜋)⁻¹^∞−∞𝐹(𝑡)cos𝑡√𝐿𝑑𝑡,(2.14) which, when combined with (2.13), gives𝐾_{√𝐹(𝐿)}(𝑥,𝑦)=(2𝜋)⁻¹_{|||||𝑡|≥𝑥−𝑦}𝐹(𝑡)𝐾_{√cos(𝑡L)}(𝑥,𝑦)𝑑𝑡.(2.15)

Lemma 2.6. Let 𝜙∈𝐶^∞0(ℝ) be even, 𝜙≥0, ‖𝜙‖_𝐿¹_(ℝ)=1,supp𝜙⊆[−1,1], and set 𝜑(𝑥)=2𝜙(2𝑥)−𝜙(𝑥) and 𝜑_𝑡(𝑥)=(1/𝑡)𝜑(𝑥/𝑡) for 𝑡>0. Let Φ and Φ_𝑡 denote the Fourier transform of 𝜑 and 𝜑_𝑡,respectively. Then, the kernels of the operators Φ_𝑡(√𝐿) and 𝑡²𝐿Φ_𝑡(√𝐿) have supports contained in {(𝑥,𝑦)∈ℝ^𝑛×ℝ^𝑛∶|𝑥−𝑦|≤𝑡}.

Proof. For the proof, we refer the reader to [7, Lemma  3.5].

Lemma 2.7. Let Ψ(𝑥)=𝑥²Φ(𝑥) as in Lemma 2.6; then the operator 𝜋_𝐿(𝑓)(𝑥)=^∞0Ψ𝑡₁√𝐿⊗Ψ𝑡₂√𝐿(𝑓(⋅,𝑡))(𝑥)𝑑𝑡¹𝑑𝑡₂𝑡₁𝑡₂(2.16) is bounded from 𝑇^𝑝2 to 𝐿^𝑝, if 1<𝑝<∞.

Proof. For the proof, we refer to [5, Lemma  3.4].

3. Proof of Theorem 1.1

3.1. The Inclusion of 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛)⊂𝐻¹_atom(ℝ^𝑛×ℝ^𝑛)

Let 𝑓∈𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛)∩𝐻²(ℝ^𝑛×ℝ^𝑛). Then 𝐹=𝑡²¹𝐿𝑒^{−𝑡21𝐿}⊗𝑡²²𝐿𝑒^{−𝑡22𝐿}𝑓∈𝑇¹²∩𝑇²². We start with a suitable version of the Calderón reproducing formula. Let Φ be as in Lemma 2.6, Ψ(𝑥)∶=𝑥²Φ(𝑥), and let 𝑐_Ψ be a constant such that 𝑐_Ψ∫^∞0𝑡Ψ(𝑡)𝑒^−𝑡2𝑑𝑡=1. By 𝐿²-functional calculus [23], one can write𝑓(𝑥)=𝑐^2Ψ^∞0Ψ𝑡₁√𝐿𝑡²¹𝐿𝑒^{−𝑡21𝐿}⊗Ψ𝑡₂√𝐿𝑡²²𝐿𝑒^{−𝑡22𝐿}𝑓(𝑥)𝑑𝑡¹𝑑𝑡₂𝑡₁𝑡₂,(3.1) where the integral converges in 𝐿²(ℝ^2𝑛). By Proposition 2.2, 𝐹 has a 𝑇¹²-atomic decomposition: 𝐹=∑^∞𝑘=0𝜆_𝑘𝐴_𝑘, where ∑^∞𝑘=0|𝜆_𝑘|≤𝐶‖𝐹‖_𝑇¹²≤𝐶‖𝑓‖_𝐻^1𝐿_(ℝ^𝑛_×ℝ^𝑛₎, and 𝐴_𝑘 are 𝑇¹²-atoms associated to an open set Ω_𝑘. It is easy to see that the sum converges in 𝑇¹² and 𝑇²². We have𝑓(𝑥)=𝑐^2Ψ^∞0Ψ𝑡₁√𝐿𝑡²¹𝐿𝑒^{−𝑡21𝐿}⊗Ψ𝑡₂√𝐿𝑡²²𝐿𝑒^{−𝑡22𝐿}𝑓(𝑥)𝑑𝑡¹𝑑𝑡₂𝑡₁𝑡₂=^∞_𝑘=0𝑐^2Ψ^∞0Ψ𝑡₁√𝐿⊗Ψ𝑡₂√𝐿𝜆_𝑘𝐴_𝑘(⋅,𝑡)𝑑𝑡¹𝑑𝑡₂𝑡₁𝑡₂=∶^∞_𝑘=0𝜇_𝑘𝑎_𝑘(𝑥),(3.2) where 𝜇_𝑘=𝑐₀𝜆_𝑘 and 𝑎_𝑘(𝑥)=𝑐⁰⁻¹∬^∞0Ψ(𝑡₁√𝐿)⊗Ψ(𝑡₂√𝐿)(𝐴_𝑘(⋅,𝑡))(𝑥)𝑑𝑡₁𝑑𝑡₂/𝑡₁𝑡₂. Here 𝑐₀ is a large constant determined later. Using Lemma 2.7, we can show that the sum (3.2) converges in 𝐿²(ℝ^2𝑛).

It is known that ∑^∞𝑘=0|𝜇_𝑘|≤𝐶‖𝑓‖_𝐻^1𝑆𝐿_(ℝ^𝑛_×ℝ^𝑛₎. Now we turn to check that 𝑎_𝑘 are 𝐿-atoms associated to open sets Ω_𝑘. Since 𝐴_𝑘 are 𝑇¹²-atoms, then 𝐴_𝑘=∑_{𝑅∈𝑚(Ω𝑘)}𝐴_𝑘,𝑅. Thus𝑎_𝑘(𝑥)=_{Ω𝑅∈𝑚𝑘}𝑐⁰⁻¹^∞0Ψ𝑡₁√𝐿⊗Ψ𝑡₂√𝐿𝐴_𝑘,𝑅(⋅,𝑡)(𝑥)𝑑𝑡¹𝑑𝑡₂𝑡₁𝑡₂=∶_{Ω𝑅∈𝑚𝑘}𝑎_𝑘,𝑅(𝑥).(3.3) Lemma 2.6 implies that for 𝑖,𝑗=0,1, supp((𝐿⁻¹)^𝑖⊗(𝐿⁻¹)^𝑗𝑎_𝑘)⊂Ω_𝑘 and supp((𝐿⁻¹)^𝑖⊗(𝐿⁻¹)^𝑗𝑎_𝑘,𝑅)⊂10𝑅.

To continue, we write_𝑅∈_𝑚(Ω𝑘)‖‖(ℓ𝐼)⁻²𝐿⁻¹^𝑖⊗(ℓ𝐽)⁻²𝐿⁻¹^𝑗𝑎_𝑘,𝑅‖‖²²≤_{𝑅∈𝑚(Ω𝑘)}sup_{‖ℎ‖2≤1}||||_ℝ2𝑛ℓ(𝐼)⁻²𝐿⁻¹^𝑖⊗ℓ(𝐽)⁻²𝐿⁻¹^𝑗𝑎_𝑘,𝑅(𝑥)ℎ(𝑥)𝑑𝑥||||²≤𝑐⁰⁻¹_𝑅∈_𝑚(Ω𝑘)sup_{‖ℎ‖2≤1}||||_{ℝ+𝑛+1×ℝ+𝑛+1}×ℓ(𝐼)₋₂𝐴_𝑘,𝑅(𝑦,𝑡)𝐿⁻¹^𝑖Ψ_𝑡1√𝐿⊗ℓ(𝐽)⁻²𝐿⁻¹^𝑗Ψ_𝑡2√𝐿ℎ(𝑦)𝑑𝑦𝑑𝑡𝑡₁𝑡₂||||²≤𝑐⁰⁻¹_𝑅∈_𝑚(Ω𝑘)‖‖𝐴_𝑘,𝑅‖‖^2𝐿_{2(𝑑𝑦𝑑𝑡/𝑡1𝑡2)}≤||Ω_𝑘||⁻¹,(3.4) if we choose 𝑐₀ large enough. By a similar argument, we have‖‖ℓ(𝐼)⁻²𝐿⁻¹^𝑖⊗ℓ(𝐽)⁻²𝐿⁻¹^𝑗𝑎_𝑘‖‖₂≤||Ω_𝑘||^−1/2.(3.5)

For general 𝑓∈𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛), we just need a standard argument. The inclusion of 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛)⊂𝐻¹_atom(ℝ^𝑛×ℝ^𝑛) is finished.

3.2. The Inclusion of 𝐻¹_Atom(ℝ^𝑛×ℝ^𝑛)⊂𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛)

By Lemma 2.4, it is enough to show that ‖𝑆_𝐿(𝑎)‖₁≤𝐶 for any 𝐿-atom 𝑎 associated to an open set Ω. For 𝑅=𝐼×𝐽∈𝑚(Ω), we let 𝑙 be the maximal dyadic cube such that 𝐼⊆𝑙 and 𝑙×𝐽⊆Ω. Also we let 𝑆 be the maximal dyadic cube such that 𝐽⊆𝑆 and 𝑙×𝑆⊆Ω. Let 𝑅^′=100𝑙×100𝑆. We can see that |∪_{𝑅∈𝑚(Ω)}𝑅′|≤𝐶|Ω|, since_𝑅∈_𝑚(Ω)𝑅^⊆𝑥∣∈ℝ^𝑛×ℝ^𝑛,𝑀_𝑠𝜒^Ω(𝑥)>2010^2𝑛.(3.6) Due to Hölders inequality one has_{⋃𝑅∈𝑚(Ω)𝑅′}𝑆_𝐿(𝑎)(𝑥)𝑑𝑥≤|||||_{𝑅∈𝑚(Ω)}𝑅^|||||^1/2_ℝ2𝑛𝑆^2𝐿(𝑎)(𝑥)𝑑𝑥^1/2≤𝐶||Ω||^1/2‖𝑎‖₂≤𝐶.(3.7) Let us prove ∫_{(⋃𝑅∈𝑚(Ω)𝑅′)𝑐}𝑆_𝐿(𝑎)(𝑥)𝑑𝑥≤𝐶. One can write_{(⋃𝑅∈𝑚(Ω)𝑅′)𝑐}𝑆_𝐿(𝑎)(𝑥)𝑑𝑥≤_{𝑅∈𝑚(Ω)}_{(100𝑙×100𝑆)𝑐}𝑆_𝐿𝑎_𝑅𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂≤_{𝑅∈𝑚(Ω)}_{𝑥1∉100𝑙}+_{𝑥2∉100𝑆}𝑆^𝐿𝑎_𝑅𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂=∶I+II.(3.8) We only estimate the term I since the proof of the term II is similar. Observe thatI=_{𝑅∈𝑚(Ω)}_{𝑥1∉100𝑙}_{𝑥2∈100𝐽}+_{𝑥2∉100𝐽}𝑆^𝐿𝑎_𝑅𝑥₁,𝑥₂𝑑𝑥₂𝑑𝑥₁=∶I₁+I₂.(3.9) Consider term I₁. By Hölder’s inequality, we obtainI₁≤_{𝑅∈𝑚(Ω)}||𝐽||^1/2_{𝑥1∉100𝑙}_{𝑥2∈100𝐽}𝑆^2𝐿𝑎_𝑅𝑥₁,𝑥₂𝑑𝑥₂^1/2𝑑𝑥₁≤𝐶_{𝑅∈𝑚(Ω)}||𝐽||^1/2_{𝑥1∉100𝑙}_{|𝑥1−𝑦1|<𝑡1}_ℝ𝑛|||𝑡₂₁𝐿𝑒^{−𝑡21𝐿}𝑎_𝑅⋅,𝑥₂𝑦₁|||²𝑑𝑥₂𝑑𝑦₁𝑑𝑡₁𝑡^1𝑛+1^1/2𝑑𝑥₁≤𝐶_{𝑅∈𝑚(Ω)}||𝐽||^1/2_{𝑥1∉100𝑙}^0ℓ(𝐼)_{𝐵(𝑥1,𝑡1)}_ℝ𝑛|||𝑡₂₁𝐿𝑒^{−𝑡21𝐿}𝑎_𝑅⋅,𝑥₂𝑦₁|||²𝑑𝑥₂𝑑𝑦₁𝑑𝑡₁𝑡^1𝑛+1^1/2𝑑𝑥₁+𝐶_{𝑅∈𝑚(Ω)}||𝐽||^1/2_{𝑥1∉100𝑙}^∞ℓ(𝐼)_{𝐵(𝑥1,𝑡1)}_ℝ𝑛|||𝑡₂₁𝐿𝑒^{−𝑡21𝐿}𝑎_𝑅⋅,𝑥₂𝑦₁|||²𝑑𝑥₂𝑑𝑦₁𝑑𝑡₁𝑡^1𝑛+1^1/2𝑑𝑥₁=∶I₁₁+I₁₂.(3.10) Consider the term I₁₁. It follows from estimate (2.8) thatI₁₁≤𝐶_{𝑅∈𝑚(Ω)}||𝐽||^1/2×_{𝑥1∉100𝑙}⎧⎪⎨⎪⎩^0ℓ(𝐼)_{𝐵(𝑥1,𝑡1)}_ℝ𝑛_ℝ^𝑛𝑡₁𝑡₁+||𝑦₁−𝑧₁||^𝑛+1||𝑎_𝑅𝑧₁,𝑥₂||𝑑𝑧₁²𝑑𝑥₂𝑑𝑦₁𝑑𝑡₁𝑡^1𝑛+1⎫⎪⎬⎪⎭^1/2𝑑𝑥₁.(3.11) Let 𝑥_𝐼 denote the center of cube 𝐼. Note that 𝑥₁∉100𝑙 and |𝑥₁−𝑦₁|<𝑡₁<𝑙(𝐼). We use Hölder’s inequality to obtainI₁₁≤𝐶_{𝑅∈𝑚(Ω)}||𝐽||^1/2||𝐼||^1/2_{𝑥1∉100𝑙}^0ℓ(𝐼)𝑡₁||𝑥₁−𝑥_𝐼||^2𝑛+2𝑑𝑡₁^1/2𝑑𝑥₁⋅‖‖𝑎_𝑅‖‖_{𝐿2(ℝ2𝑛)}≤𝐶_{𝑅∈𝑚(Ω)}||𝑅||^1/2‖‖𝑎_𝑅‖‖_{𝐿2(ℝ2𝑛)}𝛾₁(𝑅)^−1/𝑛≤𝐶_{𝑅∈𝑚(Ω)}‖‖𝑎_𝑅‖‖^2𝐿_{2ℝ2𝑛}^1/2_{𝑅∈𝑚(Ω)}||𝑅||𝛾₁(𝑅)^−2/𝑛^1/2≤𝐶,(3.12) where in the last inequality we use Lemma 2.3.

For the term I₁₂, we apply the definition of 𝐿-atom to obtainI₁₂=𝐶_{𝑅∈𝑚(Ω)}||𝐽||^1/2ℓ(𝐼)²×_{𝑥1∉100𝑙}^∞ℓ(𝐼)_{𝐵(𝑥1,𝑡1)}_ℝ𝑛|||𝑡₄₁𝐿²𝑒^{−𝑡21𝐿}ℓ(𝐼)⁻²𝐿⁻¹𝑎_𝑅⋅,𝑥₂𝑦₁|||²𝑑𝑥₂𝑑𝑦₁𝑑𝑡₁𝑡^1𝑛+5^1/2𝑑𝑥₁≤𝐶_{𝑅∈𝑚(Ω)}||𝐽||^1/2ℓ(𝐼)²×_{𝑥1∉100𝑙}⎧⎪⎨⎪⎩^∞ℓ(𝐼)_{𝐵(𝑥1,𝑡1)}_ℝ𝑛_ℝ^𝑛𝑡₁𝑡₁+||𝑦₁−𝑧₁||^𝑛+1||ℓ(𝐼)⁻²𝐿⁻¹𝑎_𝑅⋅,𝑥₂𝑧₁||𝑑𝑧₁²𝑑𝑥₂×𝑑𝑦¹𝑑𝑡₁𝑡^1𝑛+5^1/2𝑑𝑥₁≤𝐶_{𝑅∈𝑚(Ω)}||𝐽||^1/2||𝐼||^1/2ℓ(𝐼)²×_{𝑥1∉100𝑙}^∞ℓ(𝐼)𝑡¹⁻³𝑡₁+||𝑥₁−𝑥_𝐼||^2𝑛+2𝑑𝑡₁^1/2𝑑𝑥₁‖‖ℓ(𝐼)⁻²𝐿⁻¹⊗𝐼𝑎_𝑅‖‖_{𝐿2(ℝ2𝑛)}≤𝐶_{𝑅∈𝑚(Ω)}||𝑅||^1/2_{𝑥1∉100𝑙}ℓ(𝐼)||𝑥₁−𝑥_𝐼||^𝑛+1𝑑𝑥₁‖‖ℓ(𝐼)⁻²𝐿⁻¹⊗𝐼𝑎_𝑅‖‖_{𝐿2(ℝ2𝑛)}≤𝐶_{𝑅∈𝑚(Ω)}||𝑅||^1/2𝛾₁(𝑅)^−1/𝑛‖‖ℓ(𝐼)⁻²𝐿⁻¹⊗𝐼𝑎_𝑅‖‖_{𝐿2(ℝ2𝑛)}≤𝐶,(3.13) which, together with estimate of I₁₁, show that I₁≤𝐶. By a similar argument as mentioned previously, we can show I₂≤𝐶. We have obtained the required estimate ‖𝑆_𝐿(𝑎)‖₁≤𝐶. This completes the proof of Theorem 1.1.

4. Proof of Theorem 1.2

In this section, we will give the proof of Theorem 1.2 in the following routine: 𝐻𝐻¹_atom(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝒩_ℎ(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1max,ℎ(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1max,𝑃(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝑆_𝑃(ℝ^𝑛×ℝ^𝑛)⊆_1atom(ℝ^𝑛×ℝ^𝑛). The main idea comes from [9, 16].

Step I. 𝐻¹_atom(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝒩_ℎ(ℝ^𝑛×ℝ^𝑛). It is similar to the proof of 𝐻¹_atom(ℝ^𝑛×ℝ^𝑛)⊂𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛). Here we omit the details.

Step II. 𝐻^1𝒩_ℎ(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1max,ℎ(ℝ^𝑛×ℝ^𝑛) because 𝑓^+ℎ(𝑥)≤𝒩_ℎ(𝑓)(𝑥).

Step III. 𝐻^1max,ℎ(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1max,𝑃(ℝ^𝑛×ℝ^𝑛). By 𝐿²-functional calculus ([23]), we have𝑒=^{−𝑡1√𝐿}⊗1𝑒^{−𝑡2√𝐿}𝑓𝑥₁,𝑥₂√𝜋√1𝜋^∞0𝑒^−𝑢1√𝑢₁𝑒^−𝑢2√𝑢₂𝑒^{−(𝑡21/4𝑢1)𝐿}⊗𝑒^{−(𝑡22/4𝑢2)𝐿}𝑓𝑥₁,𝑥₂𝑑𝑢₁𝑑𝑢₂.(4.1) Therefore,‖‖𝑓^+𝑃‖‖₁≤𝐶1√𝜋√1𝜋^∞0𝑒^−𝑢1√𝑢₁𝑒^−𝑢2√𝑢₂‖‖𝑓^+ℎ‖‖₁𝑑𝑢₁𝑑𝑢₂≤𝐶‖‖𝑓^+ℎ‖‖₁,(4.2) which completes the proof of 𝐻^1max,ℎ(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1max,𝑃(ℝ^𝑛×ℝ^𝑛).

Step IV. 𝐻^1max,𝑃(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛), for any (𝑦₁,𝑦₂,𝑡₁,𝑡₂)∈Γ(𝑥), and 𝑥∈ℝ^𝑛×ℝ^𝑛. Applying Lemma 2.5 with 0<𝑝<1, we obtain|||𝑒_{−𝑡1√𝐿}⊗𝑒^{−𝑡2√𝐿}𝑓𝑦₁,𝑦₂|||^𝑝≤𝐶𝑡^1𝑛+1𝑡^2𝑛+1^𝑡1_𝑡1/2_{𝐵(𝑥1,2𝑡1)}^𝑡2_𝑡2/2_{𝐵(𝑥2,2𝑡2)}|||𝑒_{−𝑠1√𝐿}⊗𝑒^{−𝑠2√𝐿}𝑓𝑧₁,𝑧₂|||^𝑝𝑑𝑧₂𝑑𝑠₂𝑑𝑧₁𝑑𝑠₁≤𝐶𝑡^𝑛1𝑡^𝑛2_{𝐵(𝑥1,2𝑡1)}_{𝐵(𝑥2,2𝑡2)}𝑓^+𝑃^𝑝≤𝐶𝑀_𝑠(𝑧)𝑑𝑧||𝑓^+𝑃||^𝑝(𝑥).(4.3) Therefore,‖‖𝒩_ℎ𝑓‖‖_{𝐿1(ℝ2𝑛)}≤‖‖𝑀_𝑠||𝑓^+𝑃||^𝑝^1/𝑝‖‖_{𝐿1(ℝ2𝑛)}≤𝐶‖‖𝑓^+𝑃‖‖_{𝐿1(ℝ2𝑛)}.(4.4) This completes the proof of 𝐻^1max,𝑃(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛).

Lemma 4.1. Let 𝑓 and 𝑔 be the functions of 𝐿²(ℝ^𝑛), and suppose that 𝜙∈𝐶^∞0(ℝ^𝑛) is radial and supp(𝜙)⊆𝐵(0,1). Let 𝑢(𝑥,𝑡)=𝑒^{√−𝑡𝐿}𝑓(𝑥). Then one has _ℝ+𝑛+1|||𝜕𝑢𝜕𝑡(𝑥,𝑡)|||²≤𝐶_ℝ𝑛||𝑔∗𝜙_𝑡(𝑥)||²𝑡𝑑𝑥𝑑𝑡||𝑓(𝑥)||²||𝑔(𝑥)||²𝑑𝑥+_ℝ+𝑛+1||𝑢(𝑥,𝑡)||²||𝑔∗Ψ_𝑡(𝑥)||²𝑑𝑥𝑑𝑡𝑡,(4.5) where Ψ∈𝐶^∞0(ℝ^𝑛) with ∫Ψ=0 is a vector-value function independent on 𝑓 and 𝑔 and supp(Ψ)⊆𝐵(0,1).

Proof. We note that Δ+𝜕𝜕𝑡²²|𝑢|²=Δ+𝜕𝜕𝑡²²𝑢⋅𝑢+Δ+𝜕²𝜕𝑡²𝑢⋅𝑢+2||∇𝑢||²+2|||𝜕𝑢𝜕𝑡|||²=2𝑉|𝑢|²+2||∇𝑢||²+2|||𝜕𝑢𝜕𝑡|||²≥2|||𝜕𝑢𝜕𝑡|||².(4.6) Following the steps in [24], we obtain _ℝ+𝑛+1Δ+𝜕𝜕𝑡²²|𝑢|²(𝑥,𝑡)||𝑔∗𝜙_𝑡≤𝐶_ℝ𝑛(𝑥)||²𝑡𝑑𝑥𝑑𝑡||𝑓(𝑥)||²||𝑔(𝑥)||²𝑑𝑥+_ℝ+𝑛+1||𝑢(𝑥,𝑡)||²||𝑔∗Ψ_𝑡(𝑥)||²𝑑𝑥𝑑𝑡𝑡.(4.7) Then the lemma follows readily.

Applying Lemma 4.1, we can obtain the following lemma.

Lemma 4.2. Suppose that 𝑓,𝑔∈𝐿²(ℝ^𝑛×ℝ^𝑛). Let 𝜙 and Ψ be functions as in Lemma 4.1. Then one has _{ℝ+𝑛+1×ℝ+𝑛+1}|||𝑡₁√𝐿𝑒^{−𝑡1√𝐿}⊗𝑡₂√𝐿𝑒^{−𝑡2√𝐿}𝑓(𝑥)|||²||𝑔∗𝜙_𝑡1⊗𝜙_𝑡2(𝑥)||²𝑑𝑥𝑑𝑡𝑡₁𝑡₂≤𝐶_{ℝ+𝑛+1×ℝ+𝑛+1}|||𝑒_{−𝑡1√𝐿}⊗𝑒^{−𝑡2√𝐿}𝑓(𝑥)|||²||𝑔∗Ψ_𝑡1⊗Ψ_𝑡2(𝑥)||²𝑑𝑥𝑑𝑡𝑡₁𝑡₂+_ℝ𝑛_ℝ+𝑛+1|||𝐼⊗𝑒_{−𝑡2√𝐿}𝑓(𝑥)|||²||𝑔𝑥₁,⋅∗Ψ_𝑡2𝑥₂||²𝑑𝑥₂𝑑𝑡₂𝑡₂𝑑𝑥₁+_ℝ𝑛_ℝ+𝑛+1|||𝑒_{−𝑡1√𝐿}⊗𝐼𝑓(𝑥)|||²||𝑔⋅,𝑥₂∗Ψ_𝑡1𝑥₁||²𝑑𝑥₁𝑑𝑡₁𝑡₁𝑑𝑥₂+_ℝ2𝑛||𝑓(𝑥)||²||𝑔(𝑥)||²𝑑𝑥.(4.8)

Proof. The proof of Lemma 4.2 can be obtained by iterating Lemma 4.1.

We begin to show 𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝑆_𝑃(ℝ^𝑛×ℝ^𝑛) by following the idea of [9] (see also [25, pages 107–109]).

Set 𝑢(𝑥,𝑡)=𝑒^{−𝑡1√𝐿}⊗𝑒^{−𝑡2√𝐿}𝑓(𝑥) and 𝑢^∗(𝑥)=sup_{(𝑦,𝑡)∈Γ(𝑥)}|𝑢(𝑦,𝑡)|.

We claim that||𝑥∣𝑆_𝑃(𝑓)(𝑥)>𝛼||≤𝐶||𝑥∣𝑢^∗>𝛼||+𝛼⁻²_{𝑢∗≤𝛼}||𝑢^∗(𝑥)||²𝑑𝑥.(4.9) Once the claim holds, we integrate 𝛼 from 0 to ∞ to complete the proof of 𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝑆_𝑃(ℝ^𝑛×ℝ^𝑛). Now we turn to prove the claim.

The boundedness of the strong maximal operator 𝑀_𝑠 on 𝐿²(ℝ^2𝑛) implies||||𝑥∣𝑀_𝑠𝜒_{𝑢^∗_>𝛼}(𝑥)≥110^2𝑛||||≤𝐶_ℝ2𝑛≤𝐶_ℝ2𝑛𝑀^2𝑠𝜒_{𝑢^∗_>𝛼}(𝑥)𝑑𝑥𝜒^2{𝑢_∗>𝛼}(𝑥)𝑑𝑥=𝐶||𝑢^∗>𝛼||.(4.10) To prove (4.9), one just need to estimate||||𝑥∣𝑆_𝑃(𝑓)(𝑥)>𝛼,𝑀_𝑠𝜒_{𝑢^∗_>𝛼}<110^2𝑛||||.(4.11) Observe that_{𝑀𝑠(𝜒∗{𝑢>𝛼})<1/102𝑛}≤𝑆^2𝑃(𝑓)(𝑥)𝑑𝑥_{ℝ+𝑛+1×ℝ+𝑛+1}|||𝑡₁√𝐿𝑒^{−𝑡1√𝐿}⊗𝑡₂√𝐿𝑒^{−𝑡2√𝐿}𝑓(𝑦)|||²||||𝑅(𝑦,𝑡)∩𝑀_𝑠𝜒_{u^∗_>𝛼}<110^2𝑛||||𝑡𝑑𝑦^1𝑛+1𝑡𝑑𝑡^2𝑛+1≤_𝑅∗|||𝑡₁√𝐿𝑒^{−𝑡1√𝐿}⊗𝑡₂√𝐿𝑒^{−𝑡2√𝐿}𝑓(𝑦)|||²𝑑𝑦𝑑𝑡𝑡₁𝑡₂,(4.12) where 𝑅^∗={(𝑦,𝑡)|∈ℝ^+𝑛+1×ℝ^+𝑛+1,|𝑅(𝑦,𝑡)∩{𝑢^∗>𝛼}|<|𝑅(𝑦,𝑡)|/10^2𝑛} and 𝑅(𝑦,𝑡)={(𝑥₁,𝑥₂)|∈ℝ^𝑛×ℝ^𝑛∶|𝑥₁−𝑦₁|<𝑡₁,|𝑥₂−𝑦₂|<𝑡₂}. Now we choose a radial nonnegative function 𝜙∈𝐶^∞0(ℝ^𝑛) such that 𝜙(𝑥)=1 if |𝑥|≤1/3 and 𝜙(𝑥)=0 if |𝑥|>1. Also set 𝑔(𝑥)=𝜒_{{𝑢∗≤𝛼}}(𝑥). We can easily check that if (𝑦,𝑡)∈𝑅^∗, then 𝑔∗(𝜙_𝑡1⊗𝜙_𝑡2)(𝑦)≥𝐶, for some positive constant 𝐶. Applying Lemma 4.2, we have_{𝑀𝑠(𝜒𝑢∗>𝛼)<1/102𝑛}≤𝐶𝑆^2𝑃(𝑓)_{ℝ+𝑛+1×ℝ+𝑛+1}(𝑥)𝑑𝑥|||𝑒_{−𝑡1√𝐿}⊗𝑒^{−𝑡2√𝐿}𝑓(𝑦)|||²||𝑔∗Ψ_𝑡1⊗Ψ_𝑡2(𝑦)||²𝑑𝑦𝑑𝑡𝑡₁𝑡₂+_ℝ𝑛_ℝ+𝑛+1|||𝑒_{−𝑡2√𝐿}𝑓𝑥₁,⋅𝑦₂|||²||𝑔𝑥₁,⋅∗Ψ_𝑡2𝑦₂||²𝑑𝑦₂𝑑𝑡₂𝑡₂𝑑𝑥₁+_ℝ𝑛_ℝ+𝑛+1|||𝑒_{−𝑡1√𝐿}𝑓⋅,𝑥₂𝑦₁|||²||𝑔⋅,𝑥₂∗Ψ_𝑡1𝑦₁||²𝑑𝑦₁𝑑𝑡₁𝑡₁𝑑𝑥₂=∶I++II+_ℝ2𝑛||I𝑓II(𝑥+)||I²V||𝑔.(𝑥)||²𝑑𝑥(4.13)

We firstly consider the term I. If 𝑔∗(Ψ_𝑡1⊗Ψ_𝑡2)(𝑦)≠0, then 𝑢^∗(𝑥)≤𝛼, for some 𝑥∈𝑅(𝑦,𝑡). As a consequence, |𝑢(𝑦,𝑡)|≤𝛼. ThusI≤𝐶𝛼²_{ℝ+𝑛+1×ℝ+𝑛+1}||𝑔∗Ψ_𝑡1⊗Ψ_𝑡2(𝑦)||²𝑑𝑦𝑑𝑡𝑡₁𝑡₂=𝐶𝛼²_{ℝ+𝑛+1×ℝ+𝑛+1}||(1−𝑔)∗Ψ_𝑡1⊗Ψ_𝑡2(𝑦)||²𝑑𝑦𝑑𝑡𝑡₁𝑡₂≤𝐶𝛼²‖1−𝑔‖²²≤𝐶𝛼²||𝑢^∗>𝛼||.(4.14) Consider the term II. If 𝑔(𝑥₁,⋅)∗Ψ_𝑡2(𝑦₂)≠0, then there exists 𝑥₂, such that 𝑢^∗(𝑥₁,𝑥₂)≤𝛼 and |𝑥₂−𝑦₂|<𝑡₂. Thus, 𝑒^{−𝑡2√𝐿}𝑓(𝑥₁,⋅)(𝑦₂)≤𝛼. We obtainII≤𝐶𝛼²_ℝ𝑛_ℝ+𝑛+1||𝑔𝑥₁,⋅∗Ψ_𝑡2𝑦₂||²𝑑𝑦₂𝑑𝑡₂𝑡₂𝑑𝑥₁=𝐶𝛼²_ℝ𝑛_ℝ+𝑛+1||1−𝑔𝑥₁,⋅∗Ψ_𝑡2𝑦₂||²𝑑𝑦₂𝑑𝑡₂𝑡₂𝑑𝑥₁≤𝐶𝛼²‖1−𝑔‖²²≤𝐶𝛼²||𝑢^∗>𝛼||.(4.15) Similarly, III≤𝐶𝛼²|{𝑢^∗>𝛼}|. For the term IV, it follows from the fact 𝑓(𝑥)≤𝑢^∗(𝑥) thatIV≤_{𝑢∗(𝑥)≤𝛼}||𝑓(𝑥)||²𝑑𝑥≤_{𝑢∗(𝑥)≤𝛼}||𝑢^∗(𝑥)||²𝑑𝑥.(4.16)

Summarizing the estimates aforementioned, we have_{𝑀𝑠(𝜒𝑢∗>𝛼)<1/102𝑛}𝑆^2𝑃(𝑓)(𝑥)𝑑𝑥≤𝐶𝛼²||𝑢^∗>𝛼||+_{𝑢∗≤𝛼}||𝑢^∗(𝑥)||²,(4.17) which completes the proof of 𝐻^1𝒩_𝑃(ℝ^𝑛×ℝ^𝑛)⊆𝐻^1𝑆_𝑃(ℝ^𝑛×ℝ^𝑛).

Step V. 𝐻^1𝑆_𝑃(ℝ^𝑛×ℝ^𝑛)⊆𝐻¹_atom(ℝ^𝑛×ℝ^𝑛). It is similar to the proof of 𝐻^1𝐿(ℝ^𝑛×ℝ^𝑛)⊂𝐻¹_atom(ℝ^𝑛×ℝ^𝑛). We omit the details.

Acknowledgments

The authors thank Y. S. Han and L. X. Yan for helpful suggestions. L. Song is supported by NNSF of China (no. 11001276) and the Fundamental Research Funds for the Central Universities (no. 11lgpy79). C. Q. Tan is supported by Specialized Research Fund for the Doctoral Program of Higher Education (no. 20104402120002) and NSF of Guangdong (no. 10451503101006384).