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Journal of the London Mathematical Society 1991 s2-43(3):465-484; doi:10.1112/jlms/s2-43.3.465
© 1991 by London Mathematical Society
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© Oxford University Press

Hp AND L2 Weighted Estimates for Convolutions with Singular and Oscillating Kernels

G. Sampson

Division of Mathematics, Foundations Analysis and Topology, Auburn University 218 Parker Hall, Alabama 36849-5310, USA

In this paper we prove weighted estimates for convolutions with singular and oscillating kernels in Rn. Set T1f(x) = {Omega}*fw; one problem is to determine those w, w complex valued, for which Formula with 0 < p ≤ 1. In some cases, we settle this problem completely.

Also we consider T2f{x) = {Omega}*f, where {Omega}(t) = (1 +boxvtboxvn)–bexp(iboxvtboxva)a > 1, and b ≥ 1–(1/2a). A second problem is to determine non-negative weights w, v such that ||T2f||2,w≤c||f||2,v where the norm ||.||2,u is Formula, with u in place of w or v.


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