Endpoint estimates of fractional integral operators when α > n

IF 1.3 3区数学 Q2 MATHEMATICS, APPLIED

Bulletin des Sciences Mathematiques Pub Date : 2024-12-18 DOI:10.1016/j.bulsci.2024.103569

Liang Huang , Hanli Tang , Caifeng Zhang

引用次数: 0

Abstract

In this paper we prove the reverse endpoint estimate of fractional integral operators

\frac{n}{α} v_{n}^{1 - \frac{α}{n}} {‖ f ‖}_{L^{1}} \leq {‖ I_{α} f ‖}_{L^{\frac{n}{n - α}, \infty}}

when

α > n

for nonnegative function

f \in L^{1} (R^{n})

. And we also show that

{‖ I_{α} f ‖}_{L^{\frac{n}{n - α}, \infty}} \leq v_{n}^{1 - \frac{α}{n}} {‖ f ‖}_{L^{1}},

where

α > n

0 \leq f \in L_{n - α}^{1} (R^{n})

and the constant

v_{n}^{1 - \frac{α}{n}}

is sharp. Moreover we establish the limiting weak type behaviors for fractional integral operators when

α > n

. Specifically, there holds

\lim_{λ \to 0} λ | {x \in R^{n} : I_{α} (f) (x) < λ} |^{\frac{n - α}{n}} = \infty for any 0 < f \in L^{1} (R^{n}), \lim_{λ \to + \infty} λ | {x \in R^{n} : I_{α} (f) (x) < λ} |^{\frac{n - α}{n}} = v_{n}^{\frac{n - α}{n}} {‖ f ‖}_{L^{1}} for any 0 \leq f \in L_{n - α}^{1} (R^{n}) .

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α > n

证明了非负函数f∈L1（Rn）在α>；n时分数阶积分算子snα αvn1−αn‖f‖L1≤‖Iαf‖Lnn−α，∞的反端点估计。并证明了‖Iαf‖Lnn−α，∞≤vn1−αn‖f‖L1，其中α>；n, 0≤f∈Ln−α1(Rn)，且常数vn1−αn是尖锐的。此外，我们建立了分数阶积分算子在α>；n。而言,还有holdslimλ→0⁡λ| {x∈Rn:α(f) (x) & lt;λ}| n−α为任何0 & lt; n =∞limλ∈f L1 (Rn)→+∞⁡λ| {x∈Rn:α(f) (x) & lt;λ}| n−αn = vnn−αn为L1为任何0≤f∈Ln−α1 (Rn)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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