Vol. 111, No. 1, 1984

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A Marcinkiewicz criterion for Lp-multipliers

Henry Dappa

Vol. 111 (1984), No. 1, 9–21
Abstract

Suppose m is a bounded measurable function on the n-dimensional Euclidean space Rn. Define a linear operator Tm by (Tmf)ˆ= mfˆ , where f L2 Lp(Rn), 1 p ≤∞, and fˆ denotes the Fourier transform of f:

 ˆ     ∫      −ixξ            ∑n
f (ξ) :=   f(x)e   dx    (xξ :=   xjξj).
j=1

(We omit the domain of integration if it is the whole Rn.) If Tm is bounded from Lp(Rn) to Lp(Rn), then m is called an Lp-(Fourier) multiplier, denoted m Mp(Rn). The norm of m coincides with the operator norm of Tm.

Theorem 1. Let m and mbe locally absolutely continuous on (0,) and

                ∫ 2j+1
B := ∥m ∥∞ + sup      r|m ′′(r)|dr < ∞.
j∈Z  2j

Then m(|ξ|) Mp(Rn) for all p with 1 2n∕(n + 3) < p < 2n∕(n 3) ≤∞; in particular, mMp(Rn) cB with c independent of m.

Mathematical Subject Classification 2000
Primary: 42B15
Secondary: 42B25
Milestones
Received: 26 March 1981
Revised: 6 January 1982
Published: 1 March 1984
Authors
Henry Dappa