On restricted weak type $(1, 1)$

Abstract:

Let ${\{ {S_k}\} _{k \geqq 1}}$ be a sequence of linear operators defined on ${L^1}({R^n})$ such that for every $f \in {L^1}({R^n}),{S_k}f = f \ast {g_k}$ for some ${g_k} \in {L^1}({R^n}),k = 1,2, \cdots$, and $Tf(x) = {\sup _{k \geqq 1}}|{S_k}f(x)|$. Then the inequality $m\{ x \in {R^n};Tf(x) > y\} \leqq C{y^{ - 1}}\smallint _{{R^n}} {|f(t)|dt}$ holds for characteristic functions f (T is of restricted weak type (1, 1)) if and only if it holds for all functions $f \in {L^1}({R^n})$ (T is of weak type (1, 1)). In particular, if ${S_k}f$ is the kth partial sum of Fourier series of f, this theorem implies that the maximal operator T related to ${S_k}$ is not of restricted weak type (1, 1).