The Calderón Reproducing Formula Associated with the Heisenberg Group H^𝑑

Abstract

We establish the Calderón reproducing formula for functions in 𝐿² on the Heisenberg group H^𝑑. Also, we develop this formula in 𝐿^𝑝(H^𝑑) with 1<𝑝<∞.

1. Introduction

The classical Calderón reproducing formula reads𝑓=₀^+∞𝑓∗𝜙_𝑡∗𝜓_𝑡𝑑𝑡𝑡, (1.1) where 𝜙_𝑡(𝑥)=𝑡⁻¹𝜙(𝑥/𝑡), 𝜓_𝑡(𝑥)=𝑡⁻¹𝜓(𝑥/𝑡), and ∗ denotes the convolution on 𝐑. The Calderón reproducing formula is a useful tool in pure and applied mathematics (see [1–4]), particularly in wavelet theory (see [5, 6]). We always call (1.1) an inverse formula of wavelet transform. In [7], the authors generalized (1.1) to 𝐑^𝑛 when 𝜙 and 𝜓 are sufficiently nice normalized radial wavelet functions. The generalization of (1.1) involving nonradial wavelets 𝜙 and 𝜓 can be written in the following form:𝑓=_{𝑆𝑂(𝑛)}𝑑𝛾₀^+∞𝑓∗𝜙_𝛾,𝑡∗𝜓_𝛾,𝑡𝑑𝑡𝑡, (1.2) where 𝜙_𝛾,𝑡 and 𝜓_𝛾,𝑡 are rotated versions of 𝜙 and 𝜓 on 𝐑^𝑛. The authors in [8, 9] established (1.1) for 𝑓∈𝐿^𝑝(𝐑). Holschneider [10] studied the formula (1.2) in case 𝑓∈𝐿^𝑝(𝐑²) and gave an inversion formula of the Radon transform in 𝐿^𝑝-space by using wavelets. Furthermore, Rubin [4] developed the Calderón reproducing formula, windowed X-ray transforms, the Radon transforms, and 𝑘-plane transforms in 𝐿^𝑝-spaces on 𝐑^𝑛.

It is a remarkable fact that the Heisenberg group, denoted by 𝐇^𝑑, arises in two fundamental but different setting in analysis. On the one hand, it can be realized as the boundary of the unit ball in several complex variables. On the other hand, an important aspect of the study of the Heisenberg group is the background of physics, namely, the mathematical ideas connected with the fundamental nations of quantum mechanics. In other words, there is its genesis in the context of quantum mechanics which emphasizes its symplectic role in connection with the Fourier transform, pseudodifferential operators, and related matters (see [11]). Due to this reason, many interesting works were devoted to the theory of harmonic analysis on 𝐇^𝑑 in [11–13] and the references therein. Also, the researches of wavelet analysis on 𝐇^𝑑 are concerned increasingly; for this we refer readers to [14–16]. And the inversion formula of the Radon transform by using inverse wavelet transform on 𝐇^𝑑 was established in [17]. Our goal of the present article is to study the Calderón reproducing formula on the Heisenberg group in 𝐿^𝑝-space with 1<𝑝<∞. In the sequel we will develop the theory of inverse Radon transform on 𝐇^𝑑.

The Heisenberg group 𝐇^𝑑 is a Lie group with the underlying manifold 𝐂^𝑑×𝐑, and the multiplication law is given by(𝑧,𝑡)(𝑧^′,𝑡^′)=(𝑧+𝑧^′,𝑡+𝑡^′+2Im𝑧𝑧^′), (1.3) where 𝑧𝑧^′=∑^𝑑_𝑗=1𝑧_𝑗𝑧^′_𝑗. The dilation of 𝐇^𝑑 is defined by 𝜌(𝑧,𝑡)=(𝜌𝑧,𝜌²𝑡) with 𝜌>0. For (𝑧,𝑡)∈𝐇^𝑑, the homogeneous norm of (𝑧,𝑡) is given by (see [11]) |(𝑧,𝑡)|=|(𝑧,𝑡)⁻¹|=max{|𝑧|,|𝑡|^1/2}. (1.4) Notice that |𝜌(𝑧,𝑡)|=max{|𝜌𝑧|,|𝜌²𝑡|^1/2}=𝜌|(𝑧,𝑡)|. In addition, |⋅| satisfies the quasitriangle inequality:|(𝑧,𝑡)(𝑧^′,𝑡^′)|≤|(𝑧,𝑡)|+|(𝑧^′,𝑡^′)|. (1.5) The homogeneous dimension of 𝐇^𝑑 is 2𝑑+2, and the volume of a ball ℬ((𝑧,𝑡),𝑟)={(𝑧^′,𝑡^′)∈𝐇^𝑑∶|(𝑧,𝑡)⁻¹(𝑧^′,𝑡^′)|≤𝑟} is 𝑐^′𝑟^2𝑑+2, where 𝑐^′ is a constant.

Let 𝐏={(𝑧,𝑡,𝜌)∶(𝑧,𝑡)∈𝐇^𝑑,𝜌>0}; then 𝐏 is a locally compact nonunimodular group with the group law (𝑧,𝑡,𝜌)(𝑧^′,𝑡^′,𝜌^′)=(𝑧+𝜌𝑧^′,𝑡+𝜌²𝑡^′+2𝜌Im𝑧𝑧^′,𝜌𝜌^′). (1.6) The left and right Haar measures on 𝐏 are given by 𝑑𝜇_𝑙(𝑧,𝑡,𝜌)=𝑑𝑧𝑑𝑡𝑑𝜌𝜌^2𝑑+3,𝑑𝜇_𝑟(𝑧,𝑡,𝜌)=𝑑𝑧𝑑𝑡𝑑𝜌𝜌, (1.7) where 𝑑𝑧 denotes the Lebesgue measure on 𝐂^𝑑.

Let 𝐿^𝑝(𝐇^𝑑) be the space of measurable functions 𝑓 on 𝐇^𝑑, such that ‖𝑓‖_{𝐿𝑝(𝐇𝑑)}=(_𝐇𝑑|𝑓(𝑧,𝑡)|^𝑝𝑑𝑧𝑑𝑡)^1/𝑝<+∞,if𝑝∈[1,+∞),‖𝑓‖_{𝐿𝑝(𝐇𝑑)}=esssup_{(𝑧,𝑡)∈𝐇𝑑}|𝑓(𝑧,𝑡)|<+∞,if𝑝=+∞. (1.8)

Let 𝐙⁺={0,1,2,…} and 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑑)∈(𝐙⁺)^𝑑. The Fock space ℋ is the space of holomorphic functions 𝐹 on 𝐂^𝑑 such that ‖𝐹‖²_ℋ2=(𝜋)^𝑑_𝐂𝑑|𝐹(𝜁)|²𝑒^−2|𝜁|2𝑑𝜁<+∞. (1.9) From [18] we know that {𝐸_𝛼(𝜁)=(√2𝜁)^𝛼/√𝛼!∶𝛼∈(𝐙⁺)^𝑑} is an orthonormal basis of the Hilbert space ℋ. For 𝜆∈𝐑⧵{0}, let 𝜋_𝜆 be the Bargmann-Fock representation of 𝐇^𝑑 which acts on ℋ by 𝜋_𝜆𝑒(𝑧,𝑡)𝐹(𝜁)=^{−𝑖𝜆𝑡−𝜆|𝑧|2√+2𝜆𝜁𝑧}𝐹(𝜁−√𝜆𝑧),if𝑒𝜆>0,_{−𝑖𝜆𝑡+𝜆|𝑧|2−2|𝜆|𝜁𝑧}𝐹(𝜁+√𝜆𝑧),if𝜆<0. (1.10) The group Fourier transform of a function 𝑓∈𝐿¹(𝐇^𝑑) is defined by𝑓(𝜆)=_𝐇𝑑𝑓(𝑧,𝑡)𝜋_𝜆(𝑧,𝑡)𝑑𝑧𝑑𝑡. (1.11)

Let 𝑆_𝑝(ℋ)(1≤𝑝<+∞) be the classes of Schatten-von Neumann operators on Hilbert space ℋ, and let 𝑆_∞(ℋ) denote the algebra of all bounded operators, that is, 𝑆_∞(ℋ)=𝐵(ℋ). For 𝑇∈𝑆_𝑝(ℋ), let ‖𝑇‖_𝑝=(tr(𝑇^∗𝑇)^1/𝑝)^𝑝/2 denote the 𝑆_𝑝-norm of 𝑇. If 𝑝=2,‖𝑇‖₂ is just the Hilbert-Schmit norm of 𝑇, that denotes ‖𝑇‖_HS. Let ‖𝑇‖_∞ denote the usual operator norm of 𝑇 in 𝑆_∞(ℋ). For 1≤𝑝≤+∞, let ℒ^𝑝 be the Banach space consisting of all weak measurable operator value functions 𝐹, which also satisfy 𝐹(𝜆)∈𝑆_𝑝(ℋ_|𝜆|), a.e. 𝜆∈𝐑⧵{0}, and ‖𝐹‖_ℒ^𝑝1=(𝜋^𝑑+1_𝐑⧵{0}‖𝐹(𝜆)‖^𝑝_𝑝|𝜆|^𝑑𝑑𝜆)^1/𝑝<+∞,if1≤𝑝<+∞,‖𝐹‖_ℒ^𝑝=esssup_{𝜆∈𝐑⧵{0}}‖𝐹(𝜆)‖_∞<+∞,if𝑝=∞. (1.12) For 𝑓,𝑔∈𝐿²(𝐇^𝑑), the Parseval formula is⟨𝑓,𝑔⟩_{𝐿2(𝐇𝑑)}=2^𝑑−1𝜋^𝑑+1+∞_−∞tr(𝑔(𝜆)^∗𝑓(𝜆))|𝜆|^𝑑𝑑𝜆, (1.13) where 𝑔(𝜆)^∗ denotes the adjoint of 𝑔(𝜆). The Plancherel formula is‖𝑓‖²_{𝐿2(𝐇𝑑)}=2^𝑑−1𝜋^𝑑+1+∞_−∞‖𝑓(𝜆)‖²_HS|𝜆|^𝑑𝑑𝜆. (1.14) As a consequence of (1.13), one has the inversion of the Fourier transform:2𝑓(𝑧,𝑡)=^𝑑−1𝜋^𝑑+1+∞_−∞tr(𝜋_𝜆(𝑧,𝑡)^∗𝑓(𝜆))|𝜆|^𝑑𝑑𝜆. (1.15) Suppose 𝜌>0, and let𝑓_𝜌(𝑧,𝑡)=𝜌^−(2𝑑+2)𝑓(𝜌⁻¹(𝑧,𝑡))=𝜌^−(2𝑑+2)𝑧𝑓(𝜌,𝑡𝜌²). (1.16) By a direct computation, we have𝑓_𝜌(𝜆)=𝑓(𝜌𝜆). (1.17) Let 𝑓∗𝑔 be the convolution of 𝑓 and 𝑔, that is,𝑓∗𝑔(𝑧,𝑡)=_𝐇𝑑𝑓(𝑧^′,𝑡^′)𝑔((𝑧^′,𝑡^′)⁻¹(𝑧,𝑡))𝑑𝑧^′𝑑𝑡^′. (1.18) Then𝑓∗𝑔(𝜆)=𝑓(𝜆)𝑔(𝜆). (1.19) We should notice the following facts: if 𝑓(𝑧,𝑡)=𝑓((𝑧,𝑡)⁻¹)=𝑓(−𝑧,−𝑡), then𝑓(𝜆)=𝑓(𝜆)^∗. (1.20) And if 𝑔(𝑧,𝑡)=𝑓(−𝑧,−𝑡), then𝑔(𝜆)=𝑓(−𝜆). (1.21) The further detail of harmonic analysis on 𝐇^𝑑 can be found in [11, 12].

2. Calderón Reproducing Formula

The authors in [14, 15, 18] studied the theory of continuous wavelet associated with the concept of square integrable group representations. The unitary representation of 𝐏 on 𝐿²(𝐇^𝑑) is defined by 𝑈_{(𝑧,𝑡,𝜌)}𝑓(𝑧^′,𝑡^′)=𝜌^−(𝑑+1)𝑧𝑓(^′−𝑧𝜌,𝑡^′−𝑡−2Im𝑧𝑧^′𝜌²). (2.1) Let 𝐑⁺ denote the set of all positive real numbers, 𝐑⁻=−𝐑⁺. Let 𝑃_𝛼(𝛼∈(𝑍⁺)^𝑑) be the projection from 𝐿²(𝐑^𝑑) to 1-dimensional subspace spanned by 𝐸_𝛼, and let 𝜎=+or−,𝐻^𝜎_𝛼={𝑓∈𝐿²(𝐇^𝑑)∶𝑓(𝜆)=𝑓(𝜆)𝑃_𝛼,and𝑓(𝜆)=0if𝜆∉𝐑^𝜎}. (2.2) From [15, Theorem 1], we have𝐿²(𝐇^𝑑)=_{𝛼∈(𝑍+)𝑑}(𝐻⁺_𝛼⊕𝐻⁻_𝛼). (2.3)

Let 𝛼∈(𝐙⁺)^𝑑 and 𝜎=+or−, 𝜙∈𝐻^𝜎_𝛼; if 𝜙≠0 and satisfies𝐶_𝜙=⟨_𝐑𝜎𝜙(𝜆)^∗𝜙(𝜆)𝑑𝜆𝐸|𝜆|^𝛼,𝐸_𝛼⟩_ℋ<+∞, (2.4) then we call 𝜙 an admissible wavelet and write 𝜙∈AW^𝜎_𝛼. Let 𝜙∈AW^𝜎_𝛼,𝑓∈𝐻^𝜎_𝛼; the continuous wavelet transform of 𝑓 with respect to 𝜙 is defined by𝑊_𝜙𝑓(𝑧,𝑡,𝜌)=⟨𝑓,𝑈_{(𝑧,𝑡,𝜌)}𝜙⟩_{𝐿2(𝐇𝑑)}. (2.5) And the following Calderón reproducing formula holds in the weak sense:𝑓(𝑧^′,𝑡^′)=𝐶_𝜙⁻¹₀^+∞_𝐇𝑑𝑊_𝜙𝑓(𝑧,𝑡,𝜌)𝑈_{(𝑧,𝑡,𝜌)}𝜙(𝑧^′,𝑡^′)𝑑𝑧𝑑𝑡𝑑𝜌𝜌^2𝑑+3. (2.6)

2.1. Calderón Reproducing Formula in 𝐿²(𝐇^𝑑)

By (1.16) and (1.18), we can rewrite (2.5) and (2.6) as follows:𝑊_𝜙𝑓(𝑧,𝑡,𝜌)=𝜌^𝑑+1𝑓∗𝜙_𝜌(𝑧,𝑡),𝑓(𝑧,𝑡)=𝐶_𝜙⁻¹₀^+∞𝑓∗𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)𝑑𝜌𝜌. (2.7)(2.8)

For 0<𝜀<𝜂<+∞, let𝑓_𝜀,𝜂(𝑧,𝑡)=𝐶_𝜙^−1𝜂_𝜀𝑓∗𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)𝑑𝜌𝜌,Φ_𝜀,𝜂(𝑧,𝑡)=𝐶_𝜙^−1𝜂_𝜀𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)𝑑𝜌𝜌, (2.9)(2.10) then 𝑓_𝜀,𝜂(𝑧,𝑡)=𝑓∗Φ_𝜀,𝜂(𝑧,𝑡). We are now in a position to show that 𝑓_𝜀,𝜂 converges to 𝑓 in 𝐿²-space when 𝜀→0 and 𝜂→∞. The result in this paper is an extension of that of Mourou and Trimèche [19].

Lemma 2.1. Suppose that 𝜙∈AW^𝜎_𝛼 and 𝜙∈𝐻^𝜎_𝛼 satisfies 𝜙(𝜆)∈𝑆_∞(ℋ). Let Φ_𝜀,𝜂 be defined by (2.10). Then one has Φ_𝜀,𝜂∈𝐿²(𝐇^𝑑).

Proof. By Hölder's inequality, we have |Φ_𝜀,𝜂|²≤𝐶_𝜙^−2𝜂_𝜀|𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)|²𝑑𝜌𝜌^𝜂_𝜀𝑑𝜌𝜌. (2.11) Thus _𝐇𝑑|Φ_𝜀,𝜂|²𝑑𝑧𝑑𝑡≤𝐶_𝜙^−2𝜂_𝜀𝐇𝑑|𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)|²𝑑𝑧𝑑𝑡𝑑𝜌𝜌^𝜂_𝜀𝑑𝜌𝜌. (2.12) By (1.14), (1.19), and (1.20), we have _𝐇𝑑|𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)|²2𝑑𝑧𝑑𝑡=^𝑑−1𝜋^𝑑+1+∞_−∞‖𝜙(𝜌𝜆)𝜙(𝜌𝜆)^∗‖²_HS|𝜆|^𝑑=2𝑑𝜆_𝑑−1𝜋^𝑑+1+∞_−∞tr(𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆)𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆))|𝜆|^𝑑𝑑𝜆. (2.13) Noticing that tr(𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆)𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆))=_𝛼⟨𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆)𝐸_𝛼,𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆)𝐸_𝛼⟩_ℋ, (2.14) we obtain tr(𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆)𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆))≤‖𝜙(𝜌𝜆)‖²_∞tr(𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆)). (2.15) Thus we get _𝐇𝑑|𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)|²2𝑑𝑧𝑑𝑡≤^𝑑−1𝜋^𝑑+1+∞_−∞‖𝜙‖²_ℒ^∞tr(𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆))|𝜆|^𝑑=2𝑑𝜆_𝑑−1𝜋^𝑑+1‖𝜙‖²_ℒ^∞+∞_−∞tr(𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆))|𝜆|^𝑑𝑑𝜆=‖𝜙‖²_ℒ^∞‖𝜙_𝜌‖²_{𝐿2(𝐇𝑑)}. (2.16) Therefore, _𝐇𝑑|Φ_𝜀,𝜂|²𝑑𝑧𝑑𝑡≤𝐶_𝜙^−2𝜂_𝜀‖𝜙‖²_ℒ^∞‖𝜙_𝜌‖²_{𝐿2(𝐇𝑑)}𝑑𝜌𝜌^𝜂_𝜀𝑑𝜌𝜌=𝐶_𝜙⁻²‖𝜙‖²_ℒ^∞‖𝜙_𝜌‖²_{𝐿2(𝐇𝑑)}^𝜂_𝜀𝜌^−2(𝑑+1)𝑑𝜌𝜌^𝜂_𝜀𝑑𝜌𝜌<+∞. (2.17) Then we complete the proof of this lemma.

Theorem 2.2. Let 𝜙∈AW^𝜎_𝛼 and 𝜙∈𝐻^𝜎_𝛼 satisfy 𝜙(𝜆)∈𝑆_∞(ℋ). Then for 𝑓∈𝐻^𝜎_𝛼, one has 𝑓_𝜀,𝜂∈𝐿²(𝐇^𝑑) and lim_{𝜀→0,𝜂→∞}𝑓_𝜀,𝜂=𝑓.

Proof. Notice that 𝑓_𝜀,𝜂(𝑧,𝑡)=𝑓∗Φ_𝜀,𝜂(𝑧,𝑡), (2.18) and by (1.19) and Lemma 2.1 we deduce 𝑓_𝜀,𝜂∈𝐿²(𝐇^𝑑). Then by (1.17) and (1.19), we have lim_{𝜀→0𝜂→∞}⟨_𝐇𝑑𝑓_𝜀,𝜂(𝑧,𝑡)𝜋_𝜆(𝑧,𝑡)𝑑𝑧𝑑𝑡𝐸_𝛼,𝐸_𝛼⟩_ℋ=lim_{𝜀→0𝜂→∞}⟨_𝐇𝑑𝑓∗Φ_𝜀,𝜂(𝑧,𝑡)𝜋_𝜆(𝑧,𝑡)𝑑𝑧𝑑𝑡𝐸_𝛼,𝐸_𝛼⟩_ℋ=lim_{𝜀→0𝜂→∞}⟨𝐶_𝜙^−1𝜂_𝜀𝑓(𝜆)𝜙(𝜌𝜆)^∗𝜙(𝜌𝜆)𝑑𝜌𝜌𝐸_𝛼,𝐸_𝛼⟩_ℋ=lim_{𝜀→0𝜂→∞}𝐶_𝜙⁻¹⟨^𝜂_𝜀𝜙^𝜎_𝛼(𝜆^′)^∗𝜙^𝜎_𝛼(𝜆^′)𝑑𝜆^′|𝜆^′|𝐸_𝛼,𝐸_𝛼⟩_ℋ⟨𝑓^𝜎_𝛼(𝜆)𝐸_𝛼,𝐸_𝛼⟩_ℋ=⟨𝑓(𝜆)𝐸_𝛼,𝐸_𝛼⟩_ℋ, (2.19) where 𝜎=+ if 𝜆>0, otherwise 𝜎=−. By (1.11) we get the desired result.

2.2. Calderón Reproducing Formula in 𝐿^𝑝(𝐇^𝑑) with 1<𝑝<∞

For 𝑓∈𝐿^𝑝(𝐇^𝑑) with 1<𝑝<∞, the continuous wavelet transform of 𝑓 with respect to a wavelet 𝜙 can be defined by formula (2.7) under certain conditions on 𝜙. In this part we will show that 𝑓_𝜀,𝜂 converges to 𝑓.

Let 𝑓 be a measurable function on 𝐇^𝑑; for (𝑧,𝑡)∈𝐇^𝑑, define𝑓^∗(𝑧,𝑡)=esssup{|𝑓(𝑧^′,𝑡^′)|∶(𝑧^′,𝑡^′)∈𝐇^𝑑,|(𝑧^′,𝑡^′)|>|(𝑧,𝑡)|}. (2.20) It is easy to see that 𝑓^∗ is nonnegative and radially decreasing, that is, 0≤𝑓^∗(𝑧₁,𝑡₁)≤𝑓^∗(𝑧₂,𝑡₂)onlyif|(𝑧₁,𝑡₁)|>|(𝑧₂,𝑡₂)|, (2.21) and 𝑓^∗≥|𝑓| a.e. on 𝐇^𝑑.

Let 𝑔∈𝐿¹(𝐑⁺). Then for any (𝑧,𝑡)∈𝐇^𝑑, we define𝐺^∗(𝑧,𝑡)=sup{𝜌^−(2𝑑+2)|_𝜌^+∞𝑔(𝑠)𝑑𝑠|∶𝜌>|(𝑧,𝑡)|}, (2.22) and thus we have the following lemma.

Lemma 2.3. Let 𝑔∈𝐿¹(𝐑⁺) and 𝐺^∗(𝑧,𝑡) be defined by (2.22). Then one has _{𝐇𝑑⧵ℬ}𝐺^∗(𝑧,𝑡)𝑑𝑧𝑑𝑡≤𝑐₁^+∞|𝑔(𝑠)|ln𝑠𝑑𝑠, (2.23) where 𝑐 is a positive constant, ℬ=ℬ((0,0),1).

Proof. First we let 𝐺(𝑧,𝑡)=|(𝑧,𝑡)|^{−(2𝑑+2)+∞}_{|(𝑧,𝑡)|}|𝑔(𝑠)|𝑑𝑠. (2.24) It is obvious that 𝐺^∗(𝑧,𝑡)≤𝐺(𝑧,𝑡) for any (𝑧,𝑡)∈𝐇^𝑑⧵{(0,0)}. Since 𝐺(𝑧,𝑡) is nonnegative and radially decreasing, from [11, page 542], we know that there exists a positive constant 𝑐 such that _𝐇𝑑𝐺(𝑧,𝑡)𝑑𝑧𝑑𝑡=𝑐₀^+∞𝐺(𝑟)𝑟^2𝑑+1𝑑𝑟. (2.25) Thus, _{𝐇𝑑⧵ℬ}𝐺^∗(𝑧,𝑡)𝑑𝑧𝑑𝑡≤_{𝐇𝑑⧵ℬ}𝐺(𝑧,𝑡)𝑑𝑧𝑑𝑡=𝑐₁^+∞𝐺(𝑟)𝑟^2𝑑+1𝑑𝑟=𝑐₁^+∞𝑟^−(2𝑑+2)_𝑟^+∞|𝑔(𝑠)|𝑑𝑠𝑟^2𝑑+1𝑑𝑟=𝑐₁^+∞𝑠₁|𝑔(𝑠)|𝑑𝑠𝑑𝑟𝑟=𝑐₁^+∞|𝑔(𝑠)|ln𝑠𝑑𝑠. (2.26) This completes the proof.

Let 𝑘 be a radial function in 𝐿^𝑝(𝐇^𝑑) (1<𝑝<∞) and let𝐾(𝑧,𝑡)=|(𝑧,𝑡)|^−(2𝑑+2)₀^{|(𝑧,𝑡)|}𝑘(𝑠𝜔)𝑠^2𝑑+1𝑑𝑠 (2.27) for any (𝑧,𝑡)∈𝐇^𝑑⧵{(0,0)}, where 𝜔 is a unit vector of 𝐇^𝑑. Then we have the following lemma.

Lemma 2.4. Let 𝑘 be a radial function in 𝐿^𝑝(𝐇^𝑑)(1<𝑝<∞) and let 𝐾(𝑧,𝑡) be defined by 𝑘 in (2.27). If 𝑘 satisfies ∫_𝐇𝑑𝑘(𝑧,𝑡)𝑑𝑧𝑑𝑡=0, and 𝑘(𝑧,𝑡)ln|(𝑧,𝑡)|∈𝐿¹(𝐇^𝑑), then 𝐾∈𝐿¹(𝐇^𝑑).

Proof. Analogous to (2.20) we define 𝐾^∗(𝑧,𝑡)=esssup{|𝐾(𝑧^′,𝑡^′)|∶(𝑧^′,𝑡^′)∈𝐇^𝑑,|(𝑧^′,𝑡^′)|>|(𝑧,𝑡)|}, (2.28) and then 𝐾^∗≥|𝐾| a.e. on 𝐇^𝑑.
Let (𝑧,𝑡),(𝑧^′,𝑡^′)∈𝐇^𝑑 and 0<|(𝑧,𝑡)|<|(𝑧^′,𝑡^′)|; by (2.25) we have |𝐾(𝑧^′,𝑡^′1)|=|(𝑧^′,𝑡^′)|^(2𝑑+2)|^{|(𝑧′,𝑡′}₀^)|𝑘(𝑠𝜔)𝑠^2𝑑+1≤1𝑑𝑠||(𝑧^′,𝑡^′)|^{(2𝑑+2)|(𝑧′,𝑡′0)|}|𝑘(𝑠𝜔)|𝑠^2𝑑+1=1𝑑𝑠𝑐|(𝑧^′,𝑡^′)|^{(2𝑑+2)ℬ((0,0),|(𝑧′,𝑡′)|)}|𝑘(𝑧^′′,𝑡^′′)|^′𝑑𝑧^′′𝑑𝑡^′′≤2^2𝑑+2𝑐(|(𝑧,𝑡)|+|(𝑧^′,𝑡^′)|)^{(2𝑑+2)ℬ((𝑧,𝑡),|(𝑧,𝑡)|+|(𝑧′,𝑡′)|)}|𝑘(𝑧^′′,𝑡^′′)|𝑑𝑧^′′𝑑𝑡^′′≤2^2𝑑+2(𝑀𝑘)(𝑧,𝑡), (2.29) where 𝑀𝑘 is the Hardy-Littlewood maximal function of 𝑘. From the definition of 𝐾^∗, we have 𝐾^∗≤2^2𝑑+2(𝑀𝑘). Because 𝑘∈𝐿^𝑝(𝐇^𝑑), we get 𝐾^∗∈𝐿^𝑝(𝐇^𝑑).
By the hypothesis 𝑘 is radial and ∫_𝐇𝑑𝑘(𝑧,𝑡)𝑑𝑧𝑑𝑡=0; together with the definition of 𝐾, we have 1|𝐾(𝑧,𝑡)|=|(𝑧,𝑡)|^(2𝑑+2)|^+∞_{|(𝑧,𝑡)|}𝑘(𝑠𝜔)𝑠^2𝑑+1𝑑𝑠|. (2.30) Since 𝑘(𝑧,𝑡)(ln|(𝑧,𝑡)|)∈𝐿¹(𝐇^𝑑), it follows from Lemma 2.3 that ∫_{𝐇𝑑⧵ℬ}𝐾^∗(𝑧,𝑡)𝑑𝑧𝑑𝑡<+∞, that is, 𝐾^∗∈𝐿¹_loc(𝐇^𝑑). On the other hand, 𝐾^∗∈𝐿^𝑝(𝐇^𝑑)⊂𝐿¹_loc(𝐇^𝑑), and thus 𝐾^∗∈𝐿¹(𝐇^𝑑), which implies that 𝐾∈𝐿¹(𝐇^𝑑).

Without loss of generality, we assume that 𝐶_𝜙=1; then (2.9) and (2.10) can be written asΦ_𝜀,𝜂(𝑧,𝑡)=^𝜂_𝜀𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)𝑑𝜌𝜌,𝑓_𝜀,𝜂(𝑧,𝑡)=^𝜂_𝜀𝑓∗𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡)𝑑𝜌𝜌. (2.31)(2.32)

In fact, Φ_𝜀,𝜂 is always stated under conditions on 𝑘∶=𝜙∗𝜙 rather than under conditions on 𝜙 for convenience (see [4, 10]). By Lemma 2.4 we have the following theorem.

Theorem 2.5. Let 𝑘 be in the conditions of Lemma 2.4 and let 𝐾 be defined by (2.27). Suppose 𝜙∗𝜙=𝑘 and ∫_𝐇𝑑𝐾(𝑧,𝑡)𝑑𝑧𝑑𝑡=1. Then for 𝑓∈𝐿^𝑝(𝐇^𝑑), one has lim_{𝜀→0,𝜂→∞}𝑓_𝜀,𝜂=𝑓.

Proof. From (2.31) we have Φ_𝜀,𝜂(𝑧,𝑡)=Φ_𝜀,∞(𝑧,𝑡)−Φ_𝜂,∞(𝑧,𝑡). (2.33) Since 𝑘(𝑧,𝑡)=𝜙∗𝜙(𝑧,𝑡), and 𝜙(𝑧,𝑡)=𝜙(−𝑧,−𝑡), we deduce 𝑘_𝜌(𝑧,𝑡)=𝜌^−(2𝑑+2)_𝐇𝑑𝜙(𝑧^′,𝑡^′)𝜙((𝜌⁻¹𝑧,𝜌⁻²𝑡)⁻¹(𝑧^′,𝑡^′))𝑑𝑧^′𝑑𝑡^′=𝜌^−(4𝑑+4)_𝐇𝑑𝜙(𝑧^′,𝑡^′)𝜙(𝜌⁻¹((𝑧,𝑡)⁻¹(𝜌𝑧^′,𝜌²𝑡^′)))𝑑(𝜌𝑧^′)𝑑(𝜌²𝑡^′)=𝜌^−(4𝑑+4)_𝐇𝑑𝜙(𝜌⁻¹(𝑧^′,𝑡^′))𝜙(𝜌⁻¹((𝑧,𝑡)⁻¹(𝑧^′,𝑡^′)))𝑑𝑧^′𝑑𝑡^′=𝜙_𝜌∗𝜙_𝜌(𝑧,𝑡). (2.34) Then we have Φ_𝜀,∞(𝑧,𝑡)=_𝜀^+∞𝑘_𝜌(𝑧,𝑡)𝑑𝜌𝜌=_𝜀^+∞𝑘((𝑧,𝑡)𝜌)𝑑𝜌𝜌^2𝑑+3=1|(𝑧,𝑡)|^{2𝑑+2+∞𝜀/|𝑧,𝑡|}𝑘((𝑧,𝑡))𝜌|(𝑧,𝑡)|𝑑𝜌𝜌^2𝑑+3=1|(𝑧,𝑡)|^{2𝑑+20|𝑧,𝑡|/𝜀}𝑘(𝜌(𝑧,𝑡)|(𝑧,𝑡)|)𝜌^2𝑑+1=𝜀𝑑𝜌_−2(𝑑+1)|(𝑧,𝑡)/𝜀|^{2𝑑+20|(𝑧,𝑡)|/𝜀}𝑘(𝜌𝜔)𝜌^2𝑑+1𝑑𝜌=𝐾_𝜀(𝑧,𝑡). (2.35) By Lemma 2.4 together with the approximation of the identity, we have lim_{𝜀→0𝜂→∞}𝑓_𝜀,𝜂(𝑧,𝑡)=lim_{𝜀→0𝜂→∞}Φ_𝜀,𝜂∗𝑓(𝑧,𝑡)=lim_𝜀→0𝐾_𝜀∗𝑓(𝑧,t)−lim_𝜂→∞𝐾_𝜂∗𝑓(𝑧,𝑡)=𝑓(𝑧,𝑡). (2.36) Then we complete the proof of this theorem.

Acknowledgments

The work for this paper is supported by the National Natural Science Foundation of China (no.s 10671041 and 10971039) and the Doctoral Program Foundation of the Ministry of Education of China (no. 200810780002).