Author:
Paley R. E. A. C.,Zygmund A.
Abstract
1. Letbe a function regular for |z| < 1. We say that u belongs to the class Lp (p > 0) ifIt has been proved by M. Riesz that, for p > 1, if u(r, θ) belongs to Lp, so does v (r, θ). Littlewood and later Hardy and Littlewood have shown that for 0 < p < 1 the theorem is no longer true: there exists an f(z) such that u(r, θ) belongs to every L1−ε and v(r, θ) belongs to no Lε(0 < ε < 1). The proof was based on the theorem (due to F. Riesz) that if, for an ε > 0, we havethen f(reiθ) exists for almost every θ.
Publisher
Cambridge University Press (CUP)
Reference4 articles.
1. Some properties of conjugate functions;Hardy;Journal für Math.,1932
2. On inequalities in the theory of functions;Littlewood;Proc. London Math. Soc.,1924
3. On some series of functions, (3)
4. On the convergence of lacunary trigonometric series
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