A maximal theorem

Abstract

Let X denote the unit circle and LP, 1 <p < oo, the usual Lebesgue space. Given fCLP there is a harmonic function u in the unit disc with LP boundary value f. Set f*(x) =supr<, I u(r, x) j. The HardyLittlewood Maximal Theorem2 asserts that 'there exists a constant BP such that jff*jfp<Bpjjfjjp. A similar theorem is given in higher dimensions by H. E. Rauch [2] and K. T. Smith [3] where X is now the unit sphere in n-space. These results are obtained by first proving a maximal ergodic theorem and then passing over to the maximal theorem. The purpose of this note is to remark that the maximal theorem is a trivial deduction from a maximal ergodic theorem which is itself completely standard, so that, in effect, there is very little to prove. Before presenting the general procedure, I give an example which illustrates everything. Let X be the real line and take fELP. The harmonic function in the upper half plane with boundary values f is