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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.
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