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Harmonic Analysis Associated with a Discrete Laplacian

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Harmonic Analysis Associated with a Discrete Laplacian HARMONIC ANALYSIS ASSOCIATED WITH A DISCRETE LAPLACIAN OSCAR´ CIAURRI, T. ALASTAIR GILLESPIE, LUZ RONCAL, JOSE´ L. TORREA, AND JUAN LUIS VARONA Abstract. It is well-known that the fundamental solution of ut(n; t) = u(n + 1; t) − 2u(n; t) + u(n − 1; t); n 2 Z; −2t with u(n; 0) = δnm for every fixed m 2 Z, is given by u(n; t) = e In−m(2t), where Ik(t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series X −2t Wtf(n) = e In−m(2t)f(m): m2Z By using semigroup theory, this formula allows us to analyze some op- erators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted p ` (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essen- tially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz trans- forms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel func- tions. 1. Introduction The purpose of this paper is the analysis of several operators associated with the discrete Laplacian ∆df(n) = f(n + 1) − 2f(n) + f(n − 1); n 2 Z; in a similar way to the analysis of classical operators associated with the @2 Euclidean Laplacian ∆ = @x2 . Among the operators that we study are the (discrete) fractional Laplacian, maximal heat and Poisson semigroups, This paper has been published in: J. Anal. Math. 132 (2017), 109{131. 2010 Mathematics Subject Classification. Primary: 39A12. Secondary: 26A33, 35K08, 39A14, 42A38, 42A50. Key words and phrases. Discrete Laplacian, heat semigroup, fractional Laplacian, square functions, Riesz transforms, conjugate harmonic functions, modified Bessel functions. Research partially supported by grants MTM2015-65888-C4-4-P and MTM2015-66157- C2-1-P MINECO/FEDER from the Spanish Government. 1 2 O.´ CIAURRI, T. A. GILLESPIE, L. RONCAL, J. L. TORREA, AND J. L. VARONA the square function operator, Riesz transforms, and the conjugate harmonic function operator. In order to define and study these operators we shall use the heat semi- t∆ group Wt = e d as a fundamental tool. This is not difficult because the fundamental solution of ut(n; t) = u(n + 1; t) − 2u(n; t) + u(n − 1; t); n 2 Z; −2t with u(n; 0) = δnm for every fixed m 2 Z, is given by u(n; t) = e In−m(2t) (see [6] and [7]), where Ik(t) is the Bessel function of imaginary argument (for these functions, see [11, Chapter 5]). Consequently, the heat semigroup is given by the formal series X −2t (1.1) Wtf(n) = e In−m(2t)f(m): m2Z Then, the function u(n; t) = Wtf(n) defined in (1.1) is the solution to the discrete heat equation ( @ u(n; t) = u(n + 1; t) − 2u(n; t) + u(n − 1; t); (1.2) @t u(n; 0) = f(n); where u is the unknown function and the sequence f = ff(n)gn2Z is the initial datum at time t = 0. Other second order differential operators and the associated discrete heat kernels arise when dealing with equations connected with physics, namely the Toda lattice, see [7, 8, 10]. We first shall see that the heat semigroup is a positive, Markovian, diffu- sion semigroup, see Section 2. Then, a maximum principle for the fractional Laplacian is proved, see Theorem 1 below. We can apply the general theory p p developed by E. M. Stein [18] to obtain the boundedness on ` := ` (Z), 1 < p < 1, of the maximal heat and Poisson operators and the square function. However this general theory does not cover the boundedness on the space `1. By analyzing the kernel of these operators we shall get the p p results in ` (w) := ` (Z; w), 1 ≤ p < 1; where w will be an appropriate weight, see Theorem 2. Consider the “first” order difference operators (1.3) Df(n) = f(n + 1) − f(n) and Dfe (n) = f(n) − f(n − 1); that allow factorization of the discrete Laplacian as ∆d = DDe . Having a factorization L = XXe for a general positive laplacian L, the \Riesz trans- forms" are usually defined as X(L)−1=2 and Xe(L)−1=2, see [22, 23]. In our −1=2 case, the operator (−∆d) is not well defined, so we overcome this diffi- culty by defining the following \Riesz transforms": −α −α (1.4) R = lim D(−∆d) and Re = lim De(−∆d) : α!(1=2)− α!(1=2)− It turns out that these operators are convolution operators with the kernels 1 1 f π(n+1=2) gn2Z and f π(n−1=2) gn2Z. A close analysis of the semigroup and HARMONIC ANALYSIS ASSOCIATED WITH A DISCRETE LAPLACIAN 3 its kernel allows us to define the harmonic conjugate functions and to see that some Cauchy{Riemann equations are satisfied, see Theorem 3 below. Moreover, it will be shown that the Riesz transforms are the limits of the conjugate harmonic functions. We provide definitions of these operators and we shall prove that they are effectively bounded on `2 by means of the Fourier transform and some abstract results on discrete distributions. We refer the reader to the books by R. E. Edwards [4] and by E. M. Stein and G. Weiss [21] for details. These are the operators that we consider in the paper: σ (i) The fractional Laplacians (−∆d) , 0 < σ < 1. ∗ (ii) The maximal heat semigroup W f = supt≥0 jWtfj with Wtf = t∆ ∗ e d f, and the maximal Poisson semigroup P f = sup jPtfj with p t≥0 −t −∆ Ptf = e d f. (iii) The square function Z 1 dt1=2 t∆d 2 g(f) = jt@te fj : 0 t (iv) The Riesz transforms, as defined in (1.4). (v) The conjugate harmonic functions Qtf = RPtf and Qetf = RePtf; t ≥ 0: The main results of this paper are collected in Theorems 1, 2 and 3 below. The first one contains maximum and comparison principles for the fractional Laplacian. Theorem 1. Let 0 < σ < 1. 2 (i) Let f 2 ` . Assume that f ≥ 0 and that f(n0) = 0 for some n0. Then σ (−∆d) f(n0) ≤ 0: σ Moreover, (−∆d) f(n0) = 0 only if f(n) = 0 for all n 2 Z: 2 (ii) Let f; g 2 ` be such that f ≥ g and f(n0) = g(n0) at some n0 2 Z: Then σ σ (−∆d) f(n0) ≤ (−∆d) g(n0): σ σ Moreover (−∆d) f(n0) = (−∆d) g(n0) only if f(n) = g(n) for all n 2 Z: In order to get mapping properties in `1 or `p(w) spaces, we shall use the vector-valued theory of Calder´on{Zygmund operators in spaces of homoge- neous type (Z; µ, j · j). Here, µ(A) is the counting measure, for a set A ⊆ Z, and j · j is the absolute value function in Z. A weight on Z is a sequence w = fw(n)gn2Z of nonnegative numbers. We recall that w is a discrete p Muckenhoupt weight for ` , and we write w 2 Ap, if there is a constant C < 1 such that, for every pair of integers M, N, with M ≤ N, N N X X 1=(p−1) w(k) w(k)−1=(p−1) ≤ C(N − M + 1)p; 1 < p < 1; k=M k=M 4 O.´ CIAURRI, T. A. GILLESPIE, L. RONCAL, J. L. TORREA, AND J. L. VARONA and N X w(k) sup w(k)−1 ≤ C(N − M + 1); p = 1; k=M k2[M;N] see [9, Section 8]. The second result concerns mapping `p(w) properties for the maximal heat and Poisson semigroups and square functions. ∗ ∗ Theorem 2. Let w 2 Ap, 1 ≤ p < 1. Then the operators W , P , and g are norms of vector-valued Calder´on{Zygmundoperators in the sense of the space of homogeneous type (Z; µ, j · j). Therefore, all these operators are bounded from `p(w) into itself for 1 < p < 1 and also from `1(w) into weak-`1(w). Finally, Riesz transforms can be seen as limits of conjugate harmonic func- tions; the latter are bounded on `p(w) and satisfy some Cauchy{Riemann equations. p Theorem 3. Let f 2 ` (w), 1 ≤ p < 1, with w 2 Ap. Then ∗ ∗ (i) The operators Q f = supt≥0 Qtf and Qe f = supt≥0 Qetf are bounded from `p(w); 1 < p < 1, into itself and from `1(w) into weak-`1(w): (ii) The operators Qt, Qet and Pt satisfy the Cauchy{Riemann type equa- tions ( ( @t(Qtf) = −D(Ptf); @t(Qetf) = −De(Ptf); De(Qtf) = @t(Ptf); D(Qetf) = @t(Ptf): 2 2 Moreover, @ttQtf(n)+∆dQtf(n) = 0 and @ttQetf(n)+∆dQetf(n) = 0. (iii) We have lim Qtf(n) = Rf(n) and lim Qetf(n) = Ref(n); t!0 t!0 p for n 2 Z. The limits also holds in ` (w) sense for 1 < p < 1. The study on `p of discrete operators of harmonic analysis was initiated p by M. Riesz [15]. Apart from the L (R) boundedness of the Hilbert trans- form, he showed the `p boundedness of its discrete analogue. Later, A. P. p d Calder´onand A. Zygmund [3] noticed that L (R ) boundedness of singu- p d lar integrals implied the ` (Z ) boundedness of their discrete analogues.
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