On L^p, L^q multipliers of Fourier transforms

Richard Julian Bagby

Vol. 68 (1977), No. 1, 1–12

DOI: 10.2140/pjm.1977.68.1

Abstract

Let m be a tempered distribution on Rⁿ. We say m is an L^p, L^q multiplier (more briefly: m ∈ M_p^q) if, for each ϕ ∈𝒮, the inverse Fourier transform of mϕ is in L^q, and there is a constant C such that ∥ℱ⁻¹(mϕ)∥_q ≦ C∥ϕ∥_p for all such ϕ. The basic problem we shall consider is that of establishing sufficient conditions that a locally integrable function m ∈ M_p^q in the case 1 < p < q < ∞.

Mathematical Subject Classification

Primary: 42A18, 42A18

Milestones

Received: 12 February 1976

Published: 1 January 1977