INTEGRAL-INEQUALITIES AND EQUALITIES FOR THE REARRANGEMENT OF HARDY AND LITTLEWOOD

[摘要] The main purpose of this paper is to prove the following result about the Hardy-Littlewood decreasing rearrangement f* of a function f. If f(x) is a.e. differentiable on [0, 1], PHI is a nonnegative Borel function on R, and PSI: [0, infinity) --> R is increasing then intergral-1/0 PHI (f*) PSI(\f*)'\less-than-or-equal-to intergral-1/0 PHI(f) PSI(\f'\). Furthermore, if PHI is strictly positive, PSI is strictly increasing, f is absolutely continuous, and there is equality above with both integrals being finite then f must be monotone. (C) 1994 Academic Press, Inc.

[发布日期] 1994-05-01