Vol. 96, No. 2, 1981

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On weak restricted estimates and endpoint problems for convolutions with oscillating kernels. I

Wolfgang B. Jurkat and Gary Sampson

Vol. 96 (1981), No. 2, 405–418
Abstract

Throughout we consider K(t) = ei|t|a |t|b, a > 0, a1, b < 1 and t R. Here we consider for fixed λ,μ > 0 the function B(λ,μ;K) = B(λ,μ) = sup
χλχμ χλ(x)K χμ(x)dx where the sup is taken over all “characteristic” functions χλ, χμ with complex signs (i.e., χμ is a measurable function for which |χμ| = 1 on E, |χμ| = 0 off E and |E|μ(μ > 0)). We estimate B(λ,μ;K) within constant factors from above and below. This settles the endpoint problems for these kernels, at least in the weak restricted sense.

Mathematical Subject Classification 2000
Primary: 47G05, 47G05
Secondary: 44A35
Milestones
Received: 16 February 1979
Revised: 15 December 1980
Published: 1 October 1981
Authors
Wolfgang B. Jurkat
Gary Sampson