On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz

Abstract

In their work on Fourier series Littlewood and Paley [5], introduced the function g as follows:is the function which is analytic in |z| <1, and whose real part has boundary value f(B).One of their main results is('):(A) \\g(j)\\P ^ Av\\j\l, Kp<*>.Accompanying this, they also proved the following "converse" result:where it is assumed that f(0)dO = 0. o In an earlier study on boundary values of analytic functions, Lusin [6] introduced the function (1.2) 5(*)(fl) = (// |*'|2^ .Here, £1( 6) is a standard "triangular" domain inside the unit circle whose vertex is at the point 6; dw is the Euclidean element of area(2).Marcinkiewicz and Zygmund [9], proved that ||S($)||, =g AV\\<S>\\V, 0 </><<*>.From this it follows, by a well-known theorem of M. Riesz, that:(B) ||S(*)||, ^ 4JI/II,, 1<*<».