EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 1, 2018, 138-149 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Littlewood-Paley g-function and Radon transform on the Heisenberg group Zheng Fang1, Jianxun He1,∗ 1 School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China Abstract. In this paper, we consider Radon transform on the Heisenberg group Hn, and obtain new inversion formulas via dual Radon transforms and Poisson integrals. We prove that the Radon transform is a unitary operator from Sobelov space W into L2(Hn). Moreover, we use the Radon transform to define the Littlewood-Paley g-function on a hyperplane and obtain the Littlewood- Paley theory. 2010 Mathematics Subject Classifications: 44A12; 43A15 Key Words and Phrases: Heisenberg group, Little-wood Pelay g-function, Radon transform 1. Introduction We identify any point (x, y) in R2n with the point z = x + iy in Cn and denote the symplectic form [·, ·] on Cn by [z, w] = 1 2 Im〈z, w〉 for z, w ∈ Cn. We define the multiplication on Cn × R by (z, t)(z′, t′) = ( z + z′, t+ t′ + 1 2 Im〈z, z′〉 ) (1) for all (z, t) and (z′, t′) in Cn × R. The group Cn × R with respect to the multiplication defined by (1) is denoted by Hn and is called the Heisenberg group. It is well known that the Heisenberg group plays an important role in several branches of mathematics. There are, therefore, several ways of realising the group due to the widely application of the Heisenberg group (see [2, 17]). In 1917, Radon proved that a smooth function in R3 is completely determined by its integrals over all the planes. This leads in a more general setting to the consideration of the Radon transform. The research of Radon transform has made important influence due to its wide applications to partial differential equations, X-ray technology, radio astronomy and so on. The basic theory and some new results can be found in [9]. Geller-Stein [5] and Strichartz [16] introduced the Heisenberg-Radon ∗Corresponding author. Email addresses: fangzheng@e.gzhu.edu.cn (Z. Fang), hejianxun@gzhu.edu.cn (J. He) http://www.ejpam.com 138 c© 2018 EJPAM All rights reserved. Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 139 transform on Hn. He-Liu considered inversion formulas of the Radon transform on Hn and Siegel type Lie group in [6, 7, 8] by using the continuous wavelet transforms. The combination of Radon transform and wavelet transform has proved to be very useful both in pure mathematics and applied science. Therefore, it is very meaningful to give the inversion formula the Radon transform by using various ways. In addition, because of great application of the Heisenberg group, many scholars have studied Littlewood-Paley theory on the Heisenberg group in recent years. Thangavelu [17] studied the g-function connected with the semigroup generated by the sub-Laplacian. Liu-Ma [12] investigated the g-function related to a class of radial functions in which the characterization of the Lp(Hn)-norm of a function on Hn was obtained. This paper is organized as follows. In section 2, we recall some notations, definitions and preliminary fact. Motivated by [5, 9, 10], we mainly introduce singular convolution operators on Hn and obtain inverse formulas of Radon transform in section 3. Section 4 is to define Littlewood-Paley g-function in some hyperplanes and establishes Littlewood- Paley theory, which generalizes the results in [12]. At last, in section 5 we show that the Radon transform and the Poisson integral for the Šliov boundary are equivalent, and the inverse formula is given by the classical Schwarz theorem. 2. Preliminaries We denote by O the space all holomorphic functions on Cn, the Fock space with λ ∈ (0, ∞) is defined by Hλ := { f ∈ O : ‖f‖2 := ∫ Cn |f(w)|2e−πλ|w|2λndw <∞ } , and H−λ := {f : f ∈Hλ}. The scalar product is given by 〈f, g〉 := ∫ Cn f(w)g(w)e−πλ|w| 2 λndw. Now, an arbitrary complete orthonormal system of functions ψλ,α ∈ Hλ on Cn is repre- sented by ψλ,α(w) = λ|α|/2wα (2|α|α!)1/2 = wα1 1 ((2/λ)α1α1!)1/2 · · · wαn n ((2/λ)αnαn!)1/2 (2) which satisfies 〈ψλ,α, ψλ,β〉 = δαβ, where δαβ denotes the Kronecker symbol and α = (α1, ..., αn) ∈ Nn. It is natural that f ∈Hλ is represented by a series f(w) = ∑ α∈Nn 〈f, ψλ,α〉ψλ,α. Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 140 We define the Bargmann-Fock representation πλ from Hn into the group G of all unitary operators on Hλ by, for any ψ ∈H±λ and any (z, t) ∈ Hn, πλ(z, t)ψ(w) = e−2πiλt+πλ〈w,z〉−πλ|z| 2/2ψ(w − z), and π−λ(z, t)ψ(w) = e2πiλt+πλ〈z,w〉−πλ|z| 2/2ψ(w − z), for any λ ∈ R∗ := R \ {0}. It is easy to see that πλ : Hn → G is a group homomorphism and πλ(z, t)ψ → ψ in Hλ as (z, t)→ (0, 0). Then the unitary representation πλ of Hn on Hλ is irreducible in the sense that the only closed subspaces of Hλ that are invariant under all the operators πλ(z, t), (z, t) ∈ Hn, are {0} and Hλ . Two unitary representations πλ and πµ of Hn are unitarily equivalent if and only if λ = µ. Now, we are able to introduce the following notion of the group Fourier transform on Hn. Let f ∈ L1(Hn). Then the group Fourier transform πλ on f is defined by πλ(f) = ∫ Hn f(z, t)πλ(z, t)dzdt. (3) Thus, the Plancherel theorem states as follows. Lemma 1. Let f ∈ L2(Hn). Then ‖f‖2L2(Hn) = ∫ R∗ ‖πλ(f)‖2HS |λ|ndλ. And the inversion formula is valid: f(z, t) = ∫ R∗ tr(π∗λ(f)πλ(z, t))|λ|ndλ. 3. The inverse formulas of Radon transforms In this section, our purpose is to get the inverse formulas of Radon transforms. In order to do this we first recall the Abel Fourier transform. The Fourier transform for t and z-variable, respectively, are given by F2f(z, t) = ∫ R f(z, t′)e−2πit ′tdt′, (4) and F1f(z, t) = ∫ Cn f(z′, t)e−2πiRe〈z,z′〉dz′. The symplectic Fourier transform on Cn is defined by, for any g ∈ L2(Cn), Fsg(z) = ∫ Cn g(z′)eπiIm〈z,z ′〉dz′, (5) Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 141 and then we also obtain Fsg(z) = F1g (iz/2) . (6) Let πλ(f) be as in (3) and πλ(z, 0) := πλ(z). Then we define πλ(f) = ∫ Cn F2f(z, λ)πλ(z)dz. Therefore, the integral operator wλ for g is defined by wλ(g) = ∫ Cn g(z)πλ(z)dz, (7) and Plancherel formula is given by ‖wλ(g)‖2HS = λ−n ∫ Cn |g(z)|2dz. (8) Next, to introduce the notion of the singular convolution operator on Hn, we need the following notation. Let Tn = ( ∂ ∂t )n . We consider hyperplanes E(z,t) := {( z, t+ 1 2 Im〈z, z′〉 ) for any z′ ∈ Cn } and E ∗(z′,t′) := {( z′, t′ − 1 2 Im〈z, z′〉 ) for any z ∈ Cn } . As we know, Tn is studied in fractional differential equations due to its wide applications (see [1, 14, 15]). In this paper, Combining with these hyperplanes and Tn we have the following definition. Definition 1. The singular convolution operator Rn and the dual singular convolution operator Rtn for function f, φ, respectively, are defined by Rn(f)(z, t) = ∫ Cn Tnf ( z′, t+ 1 2 Im〈z, z′〉 ) dz′ (9) and Rtnφ(z′, t′) = ∫ Cn Tnφ ( z, t′ − 1 2 Im〈z, z′〉 ) dz. (10) In particular, R0 is the Heisenberg Radon transform when n = 0 (see [16]). It is easy to see that Rtnφ = Rnφ (also see [4]). We denote by S (Hn) the space all Schwartz functions on Hn. Using (9) and (10), we have the following identity∫ Hn f(z, t)Rtnφ(z, t)dzdt = ∫ Hn Rnf(z, t)φ(z, t)dzdt for any f, φ ∈ S (Hn). Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 142 Proposition 1. Let f ∈ S (Hn). Then, for any λ ∈ R∗, we have πλ(Rnf(z, t)) = (2πiλ)nwλ(Ff(iλz/2, λ)). Proof. Let ϕλ,α be as in (2) and differential operator Tn. As we know, F2(T nf)(λ) = (2πiλ)nF2(f)(λ). (11) According to the (7), (11), together with (4) and (5), we have [πλ(Rnf)ϕλ,α] = [wλF2(Rnf(z, λ))ϕλ,α(w)] = ∫ Cn F2(Rnf(z, λ))[πλ(z)ϕλ,α(w)]dz = ∫ Cn ∫ R Rnf(z, t)e−2πiλt[πλ(z)ϕλ,α(w)]dzdt = (2πiλ)n ∫ Cn ∫ R ∫ Cn f(z′, t)e−2πiλteπiλIm〈z,z ′〉[πλ(z)ϕλ,α]dz′dzdt = (2πiλ)n ∫ Cn ∫ Cn F2f(z′, λ)eπiλIm〈z,z ′〉[πλ(z)ϕλ,α(w)]dz′dz = (2πiλ)n ∫ Cn F2Fsf(λz, λ)[πλ(z)ϕλ,α(w)]dz. In addition, we need the full Euclidean Fourier transform Ff(z, t) = ∫ R ∫ Cn f(z′, t′)e−2πitt ′ e−2πiRe〈z,z′〉dz′dt′. By this and (5), we find that [πλ(Rnf)ϕλ,α(w)] = (2πiλ)n ∫ Cn Ff (iλz/2, λ) [πλ(z)ϕλ,α(w)]dz = (2πiλ)n [wλ (Ff (iλz/2, λ))ϕλ,α(w)] . The proof is completed. Proposition 2. Let φ ∈ S (Hn). Then (8π)n(iλ)−nF2φ (z/λ, λ) = FsF2[R t nφ(z, λ)] = F [Rtnφ(iz/2, λ)]. Proof. Let φ ∈ S (Hn), By (10), (11) and (6), we deduce that FsF2[R t nφ(z, λ)] = (2πiλ)n ∫ Cn ∫ R ∫ Cn φ ( w, t− 1 2 Im〈w, z′〉 ) e−2πiλteπiIm〈z,z ′〉dwdz′dt = (2πiλ)n ∫ Cn ∫ Cn F2φ(w, λ)eπiIm〈z,z ′〉e−πλiIm〈w,z ′〉dwdz′ Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 143 = (2πiλ)n ∫ Cn ∫ Cn F2φ(w, λ)eπiIm〈z,z ′〉eπλiIm〈z ′,w〉dwdz′ = (2πiλ)n ∫ Cn F2Fsφ(λz′, λ)eπiIm〈z,z ′〉dz′ = (2πiλ)n ∫ Cn F1F2φ(iλz′/2, t)eπiIm〈z,z ′〉dz′ = (2πiλ)n ∫ Cn F1F2φ(iλz′/2, λ)e2πiRe〈z/λ,iλz′/2〉dz′ = (8π)n(iλ)−nF2φ (z/λ, λ) . This finishes the proof of Proposition 2. Theorem 1. Let f ∈ S (Hn)∩L2(Hn) and φ(z, t) := Rnf(z, t) ∈ S (Hn). Then f(z, t) = (4π)−2nRtnφ(z, t) holds in L2(Hn). Proof. According to the inversion formula in Lemma 1 and (8), we obtain∫ Hn |Rnf(z, t)|2dzdt = ∫ R∗ ‖πλ(Rnf)‖2HS |λ|ndλ = ∫ R∗ ‖(2πiλ)nwλ(Ff(iλz/2, λ))‖2HS |λ|ndλ = ∫ R∗ ∫ Cn (4π)2n(λ)−n|Ff(z, λ)|2dz|λ|n|dλ = ∫ R∗ (4π)2n‖wλ(Ff(z, λ))‖2HS |λ|ndλ = ∫ R∗ (4π)2n‖πλF1f(·, λ)‖2HS |λ|ndλ = ∫ Hn (4π)2n|F1f(z, t)|2dzdt, which implies that, (4π)nF1f(z, t) = ∫ R∗ tr(π∗λ(Rnf)πλ(z, t))|λ|ndλ = Rnf(z, t) and hence f(z, t) = (4π)−nF−11 Rnf(z, t). (12) From (6), it follows that∫ Hn |F1F2R t nφ(iz/2, λ)|2dzdλ = 4n ∫ Hn |F1F2R t nφ(z, λ)|2dzdλ. By Proposition 2, we find that 4n ∫ Hn |F1F2R t nφ(z, λ)|2dzdλ = (8π)2n(iλ)−2n ∫ Hn |F2φ (z/λ, λ) |2dzdλ Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 144 = (8π)2n ∫ Hn |F2φ(z, λ)|2dzdλ, which implies that, F1F2R t nφ(z, λ) = (4π)nF2φ(z, λ) = (4π)nF2Rnf(z, λ) (13) holds in L2(Hn). Combining with (12) and (13), we have f(z, t) = (4π)−2nRtnφ(z, t). We complete the proof of Theorem 1. The Sobolev space W on Hn is defined by W := { f ∈ L2(Hn) : ∫ Hn |Tnf(z, t)|2dzdt <∞ } . Theorem 2. Let f ∈ W . Then there exists a constant Cn such that TnRt0T nR0f = Cnf holds in L2(Hn). Proof. Let f ∈W ∩ L2(Hn), it is easy to verify that R0(T n(f)) = Tn(R0(f)). (14) By Lemma 1 and the proof of Theorem 1, we find that∫ Cn |R0T nf(z, t)|2dzdt = ∫ R∗ ‖πλ(Rn(f))‖2HS |λ|ndλ = (4π)2n‖f‖2L2(Hn). Together with the density argument, we obtain that (14) holds in W ∩S (Hn). Similarly, we also have Rt0T nφ = TnRt0φ. (15) Obviously, CnR0T n(f) = φ ∈W is well defined. According to (15) and (14), we see that Rt0T nφ = TnRt0φ = CnT nRt0T nR0f = CnR̃ t 0R0f = Cnf. We remark that the inverse formula of Radon transform is obtained via Propositions 1, 2 and Theorems 1, 2. 4. Littlewood-Paley g-function Let Xj = ∂ ∂xj + 1 2yj ∂ ∂t , Yj = ∂ ∂yj − 1 2xj ∂ ∂t , j = 1, . . . , n. Xj , Yj be left invariant vector fields on Hn. The gradient operator on Hn is given by ∇ = (X1, . . . , Xn, Y1, . . . , Yn). Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 145 The sub-Laplacian operator of Hn is defined by L = n∑ j=1 ( X2 j + Y 2 j ) . It is known that πλ(L f)(λ) = πλ(f) ∑ α∈Nn (2|α|+ n)|λ|φλ, α. L is a positive self-adjoint operator. For a suitable function ψ defined on (0, ∞), the operator ψ(L ) can be defined in terms of the spectral expansion of L . Then πλ(ψ(L )f)(λ) = πλ(f) ∑ α∈Nn ψ(2|α|+ n)|λ|φλ, α. Let φ denote the kernel function of ψ(L ). Then φ(λ) = ∑ α∈Nn ψ(2|α|+ n)|λ|φλ, α. We denote by R(Hn) the set of all these functions φ. Let φ ∈ R(Hn), the Littlewood-Paley g-function on Hn is defined by g(Rnf, u) = [∫ ∞ 0 |Rnf ∗ φρ(u)|2 dρ ρ ]1/2 , where Rnf is as in (9) and φρ(u) = ρ−n−1φ( u√ ρ) for all ρ > 0. The homogeneous norm on the Heisenberg group is given by |u| = |(z, t)| = (|z|4 + t2)1/4, which satisfies the trigonometric inequality |uv| ≤ |u|+ |v|. Theorem 3. If φ ∈ R(Hn) is a nonzero function on Hn such that L − 2n+2 4 φ ∈ L2(Hn) and |∇φ(u)| ≤ C(1 + |u|)−2n−3−ε, where constants C, ε > 0, then for p = 2n+2 2n+1 and q = 2n+ 2, there exist constants Aq, Bp > 0, such that Aq‖Rn(f)‖Lq(Hn) ≤ ‖g(Rnf)‖Lq(Hn) ≤ Bp‖f‖Lp(Hn) for any f ∈ Lp(Hn). To prove Theorem 3, we need the following several technique Lemmas. Firstly, we consider the form of Fourier transform of |x|α−n, which is used to develop the boundedness of the singular convolution operator from Lp(Hn) to Lq(Hn), where 0 < α < n. Now, we show the Fourier transform of a Guassian function. For ε > 0, denote by gε the Gaussian function on Rn given by gε(x) = exp[−π|x|2ε] for x ∈ Rn. Then ĝε(k) = ε−n/2 exp[−π|k|2/ε]. (16) By (16), we obtain the following lemma. Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 146 Lemma 2. Let cα := π−α/2Γ(α/2) (0 < α < n). Then, for any s ∈ Rn, Γ(α/2) ∫ Rn |s|−αe2πiλsds = πα/2cn−α|λ|α−n. (17) Proof. Our starting point is the elementary formula cα|s|−α = ∫ ∞ 0 exp[−π|s|2ε]εα/2−1dε. (18) By Fubini’s theorem, together with (18) and (16), we have, Γ(α/2) ∫ R |s|−αe2πiλsds = πα/2 ∫ ∞ 0 ∫ R exp[−π|s|2ε]εα/2−1e2πiλsdsdε = πα/2 ∫ ∞ 0 εα/2−1 ∫ R exp[−π|s|2ε]e2πiλsdsdε = πα/2 ∫ ∞ 0 ε−n/2εα/2−1 exp[−π|λ|2/ε]dε = πα/2cn−α|λ|α−n. Lemma 3. The estimate ‖Rnf‖q ≤ c‖f‖p holds if and only if p = 2n+2 2n+1 and q = 2n+ 2. Proof. To show that the estimate ‖Rnf‖q ≤ c‖f‖p holds we use an analytic families interpolation argument. We let Tα,nf(z, t) = Γ(α/2) ∫ Hn Tnf ( w, t+ s+ 1 2 Im〈z, w〉 ) |s|−αdsdw, where α is complex parameter in the strip 0 ≤ Reα ≤ 1. It is obvious that on the line Reα = 0 the operator Tα,n is bounded from L1 to L∞, while on the line Reα = 1 a simple computation with (17) shows F2Tα,nf(z, λ) = πα/2c1−α|λ|α−1(2πiλ)nFf(iλz/2, λ), and a modification of the proof of Theorem 1 shows that Tα,n is bounded from L2 to L2. The various Γ-factors are innocuous, so the Stein interpolation theorem yields the boundedness of Rn from Lp to Lq for exactly p = 2n+2 2n+1 , q = 2n+ 2. The necessity of proof is similar to that of [16, p. 387], the details being omitted. The following lemma is just [12]. Lemma 4. If φ ∈ S (Hn) is a nonzero function on Hn such that L − 2n+2 4 φ ∈ L2(Hn) and |∇φ(u)| ≤ C(1 + |u|)−2n−3−ε, where constants C, ε > 0, there exist constants Ap, Bp > 0, such that Ap‖f‖Lp(Hn) ≤ ‖g(f)‖Lp(Hn) ≤ Bp‖f‖Lp(Hn) for any f ∈ Lp(Hn). Z. Fang, H. Jianxun He / Eur. J. Pure Appl. Math, 11 (1) (2018), 138-149 147 Proof. [Proof of Theorem 3] Let f ∈ Lp(Hn). For p = 2n+2 2n+1 and q = 2n+2, by Lemma 3, we know that ‖Rnf‖Lq(Hn) ≤ c‖f‖Lp(Hn). From Lemma 4, it follows that Aq‖Rnf‖Lq(Hn) ≤ ‖g(Rnf)‖Lq(Hn) ≤ B̃q‖Rnf‖Lq(Hn) ≤ Bp‖f‖Lp(Hn). The proof is completed. 5. The Poisson integral as a Radon transform on Šilov boundary ∂Un+1 Let Un+1 be the generalized upper half-plane in Cn+1, Un+1 = { (z1, z̃) ∈ Cn+1 : Imz1 > |z̃|2 } , where z̃ = (z2, · · ·, zn+1) ∈ Cn, |z̃|2 = n+1∑ j=2 |zj |2; see, for example, [13]. For any α = (α1, α̃), β = (β1, β̃) ∈ Un+1, we consider the almost analytic extension of ρ(α, β) = i 2 (β1 − α1)− n+1∑ k=2 αkβk. In particular, when α = β, ρ(α, α) = ρ(α) = Imα1 − |α̃|2. For f holomorphic on the Siegel upper half-space Un+1, we define ‖f‖H2 = sup ρ>0 (∫ ∫ |f(z̃, z1 + i|z̃|2 + iρ)|2d|z̃|d|z1| ) 1 2 . Then we set H2(Un+1) = {f : f is holomorphic on Un+1, ‖f‖H2 <∞}, where ρ is introduced on Un+1. For F ∈ H2(Un+1), we let Fρ(α̃, t) = F ( α̃, t+ i|α̃|2 + iρ ) . (19) Definition 2. The Poisson-Szegö kernel P is defined by P (α, β) := |S(α, β)|2 S(α, α) , (20) where S(α, β) = (n+1)! 4πn+1 · 1 ρ(α,β)n+1 is Szegö kernel. REFERENCES 148 Let the Poisson kernel P be as in (20). 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