Convexity Inequalities for Positive Operators

Abstract

We prove pointwise convexity (Jensen-type) inequalities of the form F(Tf) ≤ T(F(f)) where F is a convex function defined on a convex subset of some Banach space X and T is the X-valued extension of a positive operator on some function space. Examples include the pointwise Holder inequality T(fg) ≤ (Tf p ) 1/p (Tf q ) 1/q for a positive sublinear operator T. As applications we con- sider vector-valued conditional expectation and a "real" proof of the Riesz- Thorin theorem for positive operators. Mathematics Subject Classification (2000). 39B52; 47A50; 47B38; 47B65.