A sharp rearrangement inequality for the fractional maximal operator
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Bibliographic record
Abstract
. We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of f , M fl f , by an expression involving the nonincreasing rearrangement of f . This estimate is used to obtain necessary and sufficient conditions for the boundedness of M fl between classical Lorentz spaces. 1. Introduction and statement of main results For n 2 N and fl 2 [0; n), the fractional maximal operator M fl is defined at f 2 L 1 loc (R n ) by (M fl f)(x) = sup Q3x jQj fl n \\Gamma1 Z Q jf(y)j dy; x 2 R n ; where the supremum is extended over all cubes Q ae R n with sides parallel to the coordinate axes and jEj denotes the n-dimensional Lebesgue measure of a measurable subset E of R n . For the classical Hardy--Littlewood maximal operator M := M 0 , the rearrangement inequality (1.1) cf (t) (Mf) (t) Cf (t); t 2 (0; 1); holds, where f (t) = inf n ? 0; jfx 2 R n ; jf(x)j ? gj t o is the nonincreasing rearrangement of f , f (t) = t...
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.033 | 0.000 |
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