Vol. 37, No. 1, 1971

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An inequality for the Hilbert transform

Raymond Moos Redheffer

Vol. 37 (1971), No. 1, 181–211

The purpose of this paper is to give a general inequality for the Hilbert transform that clearly distinguishes conditions of global and local integrability. The former conditions are associated with a certain product [f,g]m, the latter with another product {f,g}p. The resulting statement contains, as corollaries, a number of inequalities for the Hilbert transform that have not been hitherto noted. Presentation of these is a second objective here. It turns out that the general theorem also includes, in sharpened form, several classical inequalities of Hardy and Littlewood, Babenko, and others. Proof of these sharpened forms is a third objective.

By means of the theory of Calderón and Zygmund results similar to those of this paper can be established for Hilbert transforms in n dimensions and for singular integrals of more general types. However, this is not done here.

Mathematical Subject Classification 2000
Primary: 44A15
Received: 26 July 1967
Revised: 11 September 1970
Published: 1 April 1971
Raymond Moos Redheffer