A note on analytic functions in the unit circle
- 1 July 1932
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 28 (3) , 266-272
- https://doi.org/10.1017/s0305004100010112
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.Keywords
This publication has 3 references indexed in Scilit:
- On some series of functions, (3)Mathematical Proceedings of the Cambridge Philosophical Society, 1932
- On the convergence of lacunary trigonometric seriesFundamenta Mathematicae, 1930
- On Inequalities in the Theory of FunctionsProceedings of the London Mathematical Society, 1925