Vol. 38, No. 1, 1971

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A sufficient condition for Lp-multipliers

Satoru Igari and Shigehiko Kuratsubo

Vol. 38 (1971), No. 1, 85–88

Suppose 1 p . For a bounded measurable function ϕ on the n-dimensional euclidean space Rn define a transformation Tϕ by (Tϕf) = ϕf, where f L2 Lp(Rn) and f is the Fourier transform of f:

fˆ(ξ) = fˆ√-1-   f(x)e−iξxdx.
2πn  Rn

If Tϕ is a bounded transform of Lp(Rn) to Lp(Rn), ϕ is said to be Lp-multiplier and the norm of ϕ is defined as the operator norm of Tϕ.

Theorem 1. Let 2n∕(n + 1) < p < 2n∕(n1) and ϕ be a radial function on Rn, so that, it does not depend on the arguments and may be denoted by ϕ(r),0 r < . If ϕ(r) is absolutely continuous and

                  ∫ 2R
M = ∥ϕ∥∞ + (sup R    |-dϕ(r)|2 dr)1∕2 < ∞,
R>0    R  dr

then ϕ is an Lp-multiplier and its norm is dominated by a constant multiple of M.

Mathematical Subject Classification
Primary: 42A18
Received: 12 October 1970
Published: 1 July 1971
Satoru Igari
Shigehiko Kuratsubo