Multivariate rearrangements and Banach function spaces with mixed norms

Abstract

Multivariate nonincreasing rearrangement and averaging functions are defined for functions defined over product spaces. An investigation is made of Banach function spaces with mixed norms and using multivariate rearrangements. Particular emphasis is given to the L ( P , Q ; ∗ ) L(P,Q;\ast ) spaces. These are Banach function spaces which are in terms of mixed norms, multivariate rearrangements and the Lorentz L ( p , g ) L(p,g) spaces. Embedding theorems are given for the various function spaces. Several well-known theorems are extended to the L ( P , Q ; ∗ ) L(P,Q;\ast ) spaces. Principal among these are the Strong Type (Riesz-Thorin) Interpolation Theorem and the Convolution (Young’s inequality) Theorem.