Maximal Inequalities as Necessary Conditions for Almost Everywhere Convergence

Abstract

Consider the inequality

$$\mu \left( {T*f >\lambda } \right) \leq K\mathop \smallint \limits_\Omega |f|pd\mu /\lambda p.$$

Here (Ω д μ) is a σ-finite positive measure space $1 \leq p < \infty,f \in {L_p}\left( {\Omega,\wp,\mu } \right),T*f(\omega) = \sup |{T_n}f\left( \omega \right)|$where each T _n is a bounded linear operator in L _p X> 0, $1 \leq n < \infty$

This research was supported by the National Science Foundation under grant G 21507.