Weak type estimates of square functions associated with quasiradial Bochner-Riesz means on certain Hardy spaces

Abstract

Let rho(d) is an element of C-infinity (R-n\{0}) be a non-radial homogeneous distance function of degree d is an element of N satisfying rho(d) (t xi) = t(d) rho(d) (xi). For f is an element of G(R-n), we define square functions G(rho d)(delta) f(x) associated with quasiradial Bochner-Riesz means R-rho d,t(delta) f of index delta by [GRAPHICS] where R-rho d,t(delta) f(x) = F-1 [1-rho(d)/t(d))(delta) + (f) over cap](x). If {xi is an element of R-n : rho(d)(xi) = 1} is a smooth convex hypersurface of finite type, then we prove in an extremely easy way that G(rho d)(delta) is well-defined on H-p(R-n) when delta = n(1/p-1/2)-1/2 and 0 < p < 1; moreover, it is a bounded operator from H-p(R-n) into L-p,L-infinity (R-n) . In addition, if rho(d) is an element of C-infinity (R-n\{0}), then we aslo prove that G(rho d)(delta) is a bounded operator from H-p(R-n) into L-p(R-n) when delta > n (1/p-1/2)-1/2 and 0 < p < 1. (c) 2007 Elsevier Inc. All rights reserved.