On restricted weak type (1,1)

Abstract

Let { S k } k ≧ 1 {\{ {S_k}\} _{k \geqq 1}} be a sequence of linear operators defined on L 1 ( R n ) {L^1}({R^n}) such that for every f ∈ L 1 ( R n ) , S k f = f ∗ g k f \in {L^1}({R^n}),{S_k}f = f \ast {g_k} for some g k ∈ L 1 ( R n ) , k = 1 , 2 , ⋯ {g_k} \in {L^1}({R^n}),k = 1,2, \cdots , and T f ( x ) = sup k ≧ 1 | S k f ( x ) | Tf(x) = {\sup _{k \geqq 1}}|{S_k}f(x)| . Then the inequality m { x ∈ R n ; T f ( x ) > y } ≦ C y − 1 ∫ R n | f ( t ) | d t 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 ∈ L 1 ( R n ) f \in {L^1}({R^n}) (T is of weak type (1, 1)). In particular, if S k f {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 {S_k} is not of restricted weak type (1, 1).