arXiv 1706.086572017-06-27math.CA

$0<q< 1 \le p<\infty$情形下正并元算子的双权$L^p\to L^q$界

Two-weight $L^p\to L^q$ bounds for positive dyadic operators in the case $0<q< 1 \le p<\infty$

Timo S. Hänninen, Igor E. Verbitsky

备注：: 25 pages

摘要：

令$σ$，$ω$是$\mathbb{R}^d$上的测度，令$\{λ_Q\}_{Q\in\mathcal{D}}$是由$\mathbb{R}^d$中并元立方体的集合$\mathcal{D}$索引的非负实数族。在可积指数的困难区间$0<q<1 \le p<\infty$内，刻画了正并元算子\begin{equation*} T_λ(fσ):= \sum_{Q\in \mathcal{D}} λ_Q \, \Big(\frac{1}{σ(Q)} \int_Q f\mathrm{d}σ\Big) \, 1_Q \end{equation*}的双权范数不等式\begin{equation*} \lVert T_λ(fσ)\rVert_{L^q(ω)}\le C \, \lVert f \rVert_{L^p(σ)}\quad \text{for every $f\in L^p(σ)$,} \end{equation*}.这一系列的指数$p, q$最近出现在非线性偏微分方程的应用中，这是我们研究的动机之一。此外，我们引入了一个规模的离散沃尔夫势条件，单调依赖于一个可积性参数，并证明了这样的条件是必要的（但不是充分的）小参数，充分的（但不是必要的）大参数。我们的表征适用于Riesz势$I_α(f σ) = (-Δ)^{-\fracα{2}} (fσ) $（$0<α<d$），因为它是已知的，它们可以由模型并矢算子控制。在$p, q$的这个范围内的Riesz势的加权范数不等式以前只在$σ$是Lebesgue测度的特殊情况下才被刻画出来。

英文摘要：

Let $σ$, $ω$ be measures on $\mathbb{R}^d$, and let $\{λ_Q\}_{Q\in\mathcal{D}}$ be a family of non-negative reals indexed by the collection $\mathcal{D}$ of dyadic cubes in $\mathbb{R}^d$. We characterize the two-weight norm inequality, \begin{equation*} \lVert T_λ(fσ)\rVert_{L^q(ω)}\le C \, \lVert f \rVert_{L^p(σ)}\quad \text{for every $f\in L^p(σ)$,} \end{equation*} for the positive dyadic operator \begin{equation*} T_λ(fσ):= \sum_{Q\in \mathcal{D}} λ_Q \, \Big(\frac{1}{σ(Q)} \int_Q f\mathrm{d}σ\Big) \, 1_Q \end{equation*} in the difficult range $0<q<1 \le p<\infty$ of integrability exponents. This range of the exponents $p, q$ appeared recently in applications to nonlinear PDE, which was one of the motivations for our study. Furthermore, we introduce a scale of discrete Wolff potential conditions that depends monotonically on an integrability parameter, and prove that such conditions are necessary (but not sufficient) for small parameters, and sufficient (but not necessary) for large parameters. Our characterization applies to Riesz potentials $I_α(f σ) = (-Δ)^{-\fracα{2}} (fσ) $ ($0<α<d$), since it is known that they can be controlled by model dyadic operators. The weighted norm inequality for Riesz potentials in this range of $p, q$ has been characterized previously only in the special case where $σ$ is Lebesgue measure.

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摘要：

英文摘要：

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