L p Norms of Certain Kernels on the N-Dimensional Torus

Abstract

In this paper we study a class of kernels FR which generalize the Bochner-Riesz kernels on the N-dimensional torus. Our main result consists in upper estimates for the LP norms of FR as R tends to infinity. As a consequence we prove a convergence theorem for means of functions belonging to suitable Besov spaces. 1. Throughout this paper we identify the N-dimensional torus TN (N > 2) with the N-dimensional cube QN = {x E R N: 1/2 0 we form the means FR* g(t) = f(R -m)g(m)exp(2'7imt) (1) m where t E TN, m ranges on the integer lattice ZN, and * denotes the convolution in TN. In several cases one is able to estimate the L 1( TN) norm of the kernel FR(t) = E f(R 1m)exp(2'srimt) (2) m and (if 0 E S and f(O) = 1) deduce convergence results for the means (1) (as R -x oc) when g belongs to certain classes of periodic functions (see e.g. [1]-[3], [7], [8], [12]). For instance, if f(x) = (1 -I X12)8 when lxl 1, then FR = K,A is the familiar Bochner-Riesz kernel. In this case K. I. Babenko (cf. [1]) has shown that IIKRIIl is of the same order as R(N-1)/2 as R tends to infinity (N > 2, 0 < 8 < (N 1)/2), while Stein proved the estimate IIKR Il log R if 8 = (N 1)/2 [9]. Recently Yudin [12] obtained the estimate lF = O(R (N1)/2) if f is the characteristic function of a closed balanced set C, whose boundary has finite upper Minkowski measure. The method of Yudin (which also yields estimates for the LP( TN) norms of FR) uses Jackson approximation theorem, which in tum involves estimates for the L2(R N) modulus of continuity of f. Such a method can be adapted to a more general situation, as we show in this paper. In ??2-4 we consider kernels of the type (2) associated with functionsf whose derivatives of certain orders are controlled by (not necessarily positive) Received by the editors May 13, 1980. 1980 Mathematics Subject Classification. Primary 42B99; Secondary 46E35.