International Journal of Analysis and Applications Volume 18, Number 5 (2020), 718-723 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-718 SPHERICAL-RADIAL MULTIPLIERS ON THE HEISENBERG GROUP M.E. EGWE∗ Department of Mathematics, University of Ibadan, Ibadan, Nigeria ∗Corresponding author: murphy.egwe@ui.edu.ng Abstract. Let IHn be the (2n + 1)-dimensional Heisenberg group. We consider a radial Fourier multiplier which is a spherical function on IHn and show that it is a Herz-Schur multiplier. 1. Introduction The Theory of multipliers has grown over the years to yield several results and applications in virtually all aspects of Analysis and Mathematics in general. Its use in Harmonic Analysis has assumed an enormous dimension. The theory was introduced on the Heisenberg group by G. Mauceri [13] and several other authors. Recently, Bagchi [2] revisited Fourier multipiers on the Hesisenberg group showing some variance of the results of [14] and [15]. A transference result of Fourier multipliers from SU(2) to the Heisenberg group was considered by F.Ricci [15]. The spherical functions form a large subject matter on this group [3], [1]. A construction of spherical radial functions on the Heisenberg group was given in [5], [6] and [7]. The concept of Schur multipliers or completely bounded functions has attained an exciting peak in Harmonic Analysis. However, the version of the result we shall consider in this work can be seen in [4]. Received March 25th, 2020; accepted April 16th, 2020; published June 17th, 2020. 2010 Mathematics Subject Classification. 43A90, 42A45, 22E46. Key words and phrases. spherical-radial multipliers; Herz-Schur; Heisenberg group. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 718 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-718 Int. J. Anal. Appl. 18 (5) (2020) 719 2. Main Result Definition 2.0.1 [2]: Given a bounded measurable function m(η) on IRn, we can define a transformation Tm by setting (T̂mf) = m(η)f̂(η), f ∈ L2(IRn). (1) By Placherel Theorem, Tm is a bounded operator on L 2(IRn). Definition 2.0.1: Let p ∈ [0,∞), if m is a continuous function on IRn such that ∀ � > 0, the operators (M̂�f) = m(� −1n)f̂(n) (2) are uniformly bounded multiplier operator on Lp(TTn), then m defines a bounded multiplier operator on Lp(IRn). When Tm extends to L p(IRn) as a bounded operator, we say that m (or equivalently Tm) is a Fourier multiplier for Lp(IRn). Theorem 2.0.3 (Hormander’s Multiplier Theorem): Let k = [n 2 ] + 1 and m be of class Ck away from the origin. If for any β ∈ INn satisfying |β| < k, we have sup R R|β|− n 2 (∫ IRn |Dβm(η)|2χ{R<|η|<2R}(η)dη )1/2 < ∞, (3) then m is a Fourier multiplier for Lp(IRn) for 1 < p < ∞. In particular, if |Dβm(η)| ≤ c|η|−|β|, then m is an Lp-multiplier, 1 < p < ∞. 2.1 The Heisenberg Group (IHn) Define the Heisenberg group of dimension (2n + 1) by IHn = IC × IR equipped with the group law (z,t)(z′, t′) = (z + z′, t + t′ + 1 2 =z.z′), z.z′ = n∑ j=1 zj.z̄′j t ∈ IR, z ∈ IC. (4) This gives a two-step nilpotent Lie group with centre given by Z = {(0, t) : t ∈ IR}. (5) Full details on the ubiquity of this group can be found in [12] [9], [17], [14]. For each µ > 0, we have two non-equivalent irreducible representations of IHn on the Fock space F µ consisting of the entire functions F on ICn such that ‖F‖2Fµ = µ π ∫ IC |F(w)|2e−µ|w| 2 dw < ∞. (6) Int. J. Anal. Appl. 18 (5) (2020) 720 These representations have the form [15] (ρµ(γ,t)F)(w) = eµ(it+γw+ 1 2 |γ|2)F(w + γ) (ρµ(γ,t)F)(w) = eµ(−it−γw+ 1 2 |γ|2)F(w − γ̄).   (7) The monomials ηλj (w) = ( µj j! )1/2 wj form a orthonormal basis for Fµ and the matrix entries corresponding to the representations with respect to the monomials is given by τ ±µ ij (µ,t) = 〈ρ ±µ(γ,t)η µ i ,η µ j 〉. (8) Now, let du = ( 1 2π2 ) dzdt denote the normalized Haar measure on IHn. Then, given an integrable function f on IHn and a nonzero real number µ, we have a countably infinite matrix with (i,j) entry given by f̂(µ,i,j) = ∫ IHn f(u)τ µ ij(u)du (9) With this normalisation and matrix entries, we obtain the Plancherel formula given by∫ IHn |f(u)|2du = ∫ ∞ ∞ ∞∑ i,j=0 |f̂(µ,i,j)|2|µ|dµ. (10) We now give the following definition following Hormander’s theorem. Definition 2.1.1: Let µ 6= 0 and m(µ) a countably infinite matrix with entries m(µ,i,j) that are measurable in µ for each i,j. We say that this induces a bounded multiplier on Lp(IHn) if ‖Mf‖p ≤ c‖f‖, (11) where (̂Mf)(µ) = f̂(µ)m(µ) for some f in some dense subspace of Lp(IHn). In what follows, we shall construct the spherical radial multipliers following [5], [6]. Let ϕKλ be a K-spherical function on IHn. That is the distinguished spherical function restricted to L1(K \G/K) where (K,G) is a Gelfand pair, K a compact subgroup of Aut(IHn). In this case, G may be taken as a semi-direct product of K and IHn denoted as G := K o IHn [1]. Now, recall that the Heisenberg group heat equation defined on IHn × IR+ is given by ∂tU(u,t) = 4U(u,t), U(u,t) ∈ IHn × IR+. (12) The fundamental solution of (12) is given in [16] as Kt(x,u,ξ) = cn ∫ IR eλEe−tλ 2 ( λ sinh λt ) e 1 4 λ(coth tλ)(x.x+u.u)dλ, (13) Int. J. Anal. Appl. 18 (5) (2020) 721 where cn = (4π) −n, λ ∈ IR∗ := IR\{0}. By a unique transformation of Kt(x,u,ξ) given explicitly in [6], we obtain that Kt(u) = cnt −n/2ϕKλ (u)δ −2 r (u)e |u|2 4t . (14) This gives a representations in (8) of IHn with respect to the dilations on the group. Thus, (9) becomes f̂(µ,i,j) = ∫ IHn f(u)Kλij(u)du, (15) where (from (14) we have) Kλij = 〈ϕ K λ (ξ,t)η λ i ,η λ j 〉. (16) The spherical transforms of a function on IHn are then obtained and given as [1], [5]: f̃(λ,t) = ∫ IHn f(z,t)ϕKλ (z,t)dzdt (17) and f̃(0,ρ) = ∫ IHn f(z,t)Jρ0 (z)dzdt, (18) where ϕKλ = e 2πiλte−2π|λ||z| 2 n∏ j=1 L0k(4π|λ||zj| 2), λ ∈ IR∗,k ∈ (Z+)n (19) and Jρ0 = n∏ j=1 J0(ρj · |zj|), ρ ∈ (IR+)n. (20) Here, L0k is the Laguerre polynomial of degree k and J0 is the Bessel function (of first kind) of index 0. Definition 2.1.2: Let M = {M(λ) ∈ B(L2(IRn)) : λ ∈ IR∗} be a family of operators. Suppose that TM is the corresponding group Fourier multiplier. Also, let ϕKM (λ)be the spherical Fourier multiplier associated with the parameter and operator M(λ). Then, it becomes clear that TMf(z,t) = ∫ IR e−iλtϕK M(λ) (λ)fλ(t)dλ, for all f ∈ L1 ∩L2(IHn). This implies that TMf(z,t) = ∫ IR Kλjkf(t)e −iλtdλ, f ∈ S (IHn) [1]. (21) Definition 2.1.3: We shall say a matrix-valued function M(µ) = (M(µ,i,j))i,j∈N is a bounded Fourier multiplier for IHn if m(., i,j) ∈ L∞(IR) for every i,j ∈ N, and if ‖M‖∞ = ess. sup‖M(µ)‖L(`2) < ∞. This definition together with 2.1.1 yield the following theorem as seen in [14]. Theorem 2.1.4: If M is a bounded Fourier multiplier on IHn, the requirement that T̂Mf(µ) = f̂(µ)M(µ) Int. J. Anal. Appl. 18 (5) (2020) 722 defines a bounded left invariant operator TM on L 2(IHn), with ‖TM‖L(L2(IHn)) = ‖M‖∞. Conversely, for any bounded left-invariant operator T on L2(IHn), there is a bounded Fourier multiplier M such that T = TM. Definition 2.1.5: A function f : IRn −→ IR is said to be radial if there is a function φ defined on [0,∞) such that f(x) = φ(|x|) for almost every x ∈ IRn. Simple and classical examples of radial functions and their properties can be seen in for example [10], [3], [6] and [11]. Thus, given a K-spherical function, ϕKλ restricted to L 1(K \G/K) where (K,IHn) is a Gelfand pair, K a compact subgroup of Aut(IHn), then ϕ K λ is a unique radial function since it is a radial eigenfunction of 4IHn [6], [7]. Thus, ϕKλ (u) = ψ(|u|) This forces forces (15) to become ϕKλ (u) = cnψ(e −iθ|u|, t) = cnK λ ij(|u|, t). This establishes (21). In fact, we have the following result [14]. Prposition 2.1.6: Kλjk is a unique radial function in span{ϕ K λ : |λ| ∈ ∑ }, where ∑ is the Heisenberg fan, up to scalar multiples. Definition 2.1.7: Let IHn be the 2n + 1-dimensional Heisenberg group. Then f on IHn is said to be Herz-Schur, f ∈ B2(IHn) if there exist u,ν ∈ IHn such that f(u−1ν) = 〈ρµ1 (u), ρ µ 2 (ν)〉, (22) where ρ µ 1 and ρ µ 2 (ν) are irreducible unitary representations of IHn on L 2(IR). Here, we assume that sup u∈IHn ‖ρµ1 (u)‖ < ∞ and sup ν∈IHn ‖ρµ2 (ν)‖ < ∞, where ‖ · ‖ is the Fourier multiplier norm equivalent to the Koranyi norm [8]. Theorem 2.1.8 Let TM be the group Fourier multiplier on IHn acting on a K-bounded spherical function, f ∈ S (IHn). Then, TM is a Herz-Schur multiplier on IHn. Proof: Following [4], any f can be expressed in the form given in (22). Thus, if we consider (21) above, we readily see that up to scalar multiples, K µ ij is the unique radial function with ‖K µ ij‖≤ 1. Thus, |TMf(z,t)| = ∣∣∣∣ ∫ IR K µ ijf(t)e −iµt dµ ∣∣∣∣ ≤ ∫ IR ∣∣∣〈ϕKλ (ξ,t)ηµi ,ηµj 〉f(t)e−iµtdµ∣∣∣ ≤ sup |ϕKλ (ξ,t)| ∫ ∣∣∣f(t)e−iµtdµ∣∣∣ = MK,λ‖f‖S (IHn). Int. J. Anal. Appl. 18 (5) (2020) 723 Since the representations of IHn are uniformly bounded on L 2 and TM is acting on K-bounded spherical functions, then the last expression shows that TM ∈ B2(IHn) and therefore Herz-Schur. � Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] B. Astengo, D.B. Blasio, and F. Ricci. Gelfand Pairs on the Heisenberg Group and Schwartz Functions. J. Funct. 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