若干反向Hardy-Littlewood-Sobolev型不等式

中文摘要 AI 辅助

我们在\\(\mathbb{R}^n\\)和\\(\mathbb{R}_+^n\\)上建立了若干尖锐的反向Hardy-Littlewood-Sobolev（HLS）型不等式。利用算子表示，我们克服了半空间情形下对称二重积分结构不可用的困难。在\\(\mathbb{R}^n\\)上，当\\(1 \le n < \alpha\\)、\\(\frac{n}{\alpha} < t < 1\\)且\\(0 < q < 1\\)时，对于非负函数\\(f\\)，存在满足\\(\mathscr{C}(n,\alpha,q,t)>0\\)的常数，使得\\(\\|E_\alpha f \\|_{L^{t^\prime}(\mathbb{R}^n)} \ge \mathscr{C}(n,\alpha,q,t) \\|f \\|_{L^1(\mathbb{R}^n)}^{\gamma} \\|f \\|_{L^q(\mathbb{R}^n)}^{1-\gamma}\\)成立，其中\\(\gamma:= \frac{n - q\alpha - \frac{n}{t^\prime}q}{n(1-q)}\\)，当且仅当\\(q>\frac{n}{\alpha}\\)时该式成立；此处\\(E_\alpha\\)是带有Riesz核的延拓算子，\\(t^\prime\\)是\\(t\\)的共轭指数，尖锐常数在\\(\frac{n t^\prime}{n + \alpha t^\prime} \le q < 1\\)时取得。在\\(\mathbb{R}_+^n\\)上，当\\(2 \le n < \alpha\\)、\\(\frac{n}{\alpha} < t < 1\\)且\\(0 < q < 1\\)时，对于非负函数\\(f\\)，存在满足\\(\widetilde{\mathscr{C}}(n,\alpha,q,t)>0\\)的常数，使得\\(\\|\widetilde{E}_\alpha f \\|_{L^{t^\prime}(\mathbb{R}_+^n)} \ge\widetilde{\mathscr{C}}(n,\alpha,q,t) \\|f\\|_{L^1(\partial \mathbb{R}_+^n)}^{\widetilde{\gamma}} \\|f\\|_{L^q(\partial \mathbb{R}_+^n)}^{1-\widetilde{\gamma}}\\)成立，其中\\(\widetilde{\gamma}:= \frac{(n-1) - q(\alpha-1) - \frac{n}{t^\prime}q}{(n-1)(1-q)}\\)，当且仅当\\(q > \frac{n-1}{\alpha-1}\\)时该式成立；此处\\(\widetilde{E}_\alpha\\)是带有泊松型核的延拓算子，尖锐常数在\\(\frac{t^\prime(n-1)}{n + t^\prime(\alpha-1)} \le q < 1\\)时取得。我们还将结果推广到\\(q\ge1\\)的情形，证明过程用到重排不等式、尖锐Carlson-Levin不等式以及Riesz和泊松型位势的精细逐点下界，所得结果统一并推广了经典的反向HLS不等式，尤其是在\\(\mathbb{R}_+^n\\)上的结果。

英文摘要

We establish some sharp reverse Hardy-Littlewood-Sobolev (HLS) type inequalities on \(\mathbb{R}^n\) and \(\mathbb{R}_+^n\). Using an operator representation, we overcome the difficulty that the symmetric double-integral structure is unavailable in the half-space setting. On \(\mathbb{R}^n\), for \(1 \le n < α\), \(\frac{n}α < t < 1\), and \(0 < q < 1\), there holds for nonnegative \(f \) that \[ \|E_αf \|_{L^{t^\prime}(\mathbb{R}^n)} \ge \mathscr{C}(n,α,q,t) \|f \|_{L^1(\mathbb{R}^n)}^γ \|f \|_{L^q(\mathbb{R}^n)}^{1-γ}, \quad γ:= \frac{n - qα- \frac{n}{t^\prime}q}{n(1-q)} \] for some $\mathscr{C}(n,α,q,t)>0$ iff \(q>\frac{n}α\), where \(E_α\) is the extension operator with Riesz kernel and \(t^\prime\) is the conjugate of \(t\). The sharp constant is achieved when \(\frac{n t^\prime}{n + αt^\prime} \le q < 1\). On \(\mathbb{R}_+^n\), with \(2 \le n < α\), \(\frac{n}α < t < 1\), and \(0 < q < 1\), we show for nonnegative \(f \) that \[ \|\widetilde{E}_αf \|_{L^{t^\prime}(\mathbb{R}_+^n)} \ge\widetilde{\mathscr{C}}(n,α,q,t) \|f\|_{L^1(\partial \mathbb{R}_+^n)}^{\widetildeγ} \|f\|_{L^q(\partial \mathbb{R}_+^n)}^{1-\widetildeγ}, \quad \widetildeγ := \frac{(n-1) - q(α-1) - \frac{n}{t^\prime}q}{(n-1)(1-q)}, \] for some $\widetilde{\mathscr{C}}(n,α,q,t)>0$ iff \(q > \frac{n-1}{α-1}\), where \(\widetilde{E}_α\) is the extension operator with Poisson-type kernel. The sharp constant is achieved when \(\frac{t^\prime(n-1)}{n + t^\prime(α-1)} \le q < 1\). We further extend results to \(q\ge1\). The proofs use rearrangement inequalities, the sharp Carlson--Levin inequality, and refined pointwise lower bounds for the Riesz and Poisson-type potentials. Our results unify and extend the classical reverse HLS inequalities, especially on \(\mathbb{R}_+^n\).

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32 pages