Weighted Norm Inequalities for Classical Operators

Abstract

The principle problem to be discussed here is the determination of all pairs of non-negative functions U(x), V(x) such that

$$\int_{ - \infty }^\infty {\left| {{\text{Sf}}\left( {\text{x}} \right)} \right|^{\text{P}} {\text{U}}\left( {\text{x}} \right){\text{dx}} \leqslant {\text{C}}\int_{ - \infty }^\infty {\left| {{\text{Tf}}\left( {\text{x}} \right)} \right|^{\text{P}} {\text{V}}\left( {\text{x}} \right){\text{dx}}} } $$

((1.1))

where 1≤p<∞ S and T are two given operators and C is a constant independent of f. Results described here will be primarily on the real line; there are periodic analogues of most and almost all have been or can be generalized to Rⁿ. There are also closely related H_p results valid for all positive p.

Supported in part by N.S.F. grant 38540.