A characterization for the boundedness of positive operators in a   filtered measure space
  In terms of Sawyer type checking condition, a complete characterization is established for which the positive operator in a filtered measure space is bounded from $L^p(d\mu)$ to $L^q(d\mu)$ with $1<p\le q<\infty$. 
Hitosh Tanaka and Yutaka Terasawa
math.CA math.PR