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H p AND L 2 Weighted Estimates for Convolutions with Singular and Oscillating Kernels
Author(s) -
Sampson G.
Publication year - 1991
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-43.3.465
Subject(s) - mathematics , combinatorics , norm (philosophy) , physics , mathematical analysis , political science , law
In this paper we prove weighted estimates for convolutions with singular and oscillating kernels in R n . Set T 1 f ( x ) = Ω∗ fw ; one problem is to determine those w, w complex valued, for which‖ T 1 f ‖ p ⩽ c‖ f ‖H μ pwith 0 < p ⩽ 1. In some cases, we settle this problem completely. Also we consider T 2 f { x ) = Ω∗ f , where Ω( t ) = (1 +│ t │ n ) − b exp( i │ t │ a ) a > 1, and b ⩾ 1−(½ a ). A second problem is to determine non‐negative weights w, v such that ∥ T 2 f ∥ 2, w ⩽ c ∥ f ∥ 2, v where the norm ∥.∥ 2, u is( ∫| . | 2 u ( t ) d t )1 2, with u in place of w or v .