arXiv 1410.34022014-10-13math.CA

强极大函数的加权Solyanik估计

Weighted Solyanik estimates for the strong maximal function

Paul A. Hagelstein, Ioannis Parissis

备注：: 19 pages, submitted for publication
发表信息：: Publ. Mat. 62 (2018), no. 1, 133--159
DOI：: 10.5565/PUBLMAT6211807

摘要：

设$\mathsf M_{\mathsf S}$表示$\mathbb R^n$上的强极大算子，$w$是非负的局部可积函数.对于$α\in(0,1)$，我们定义了加权尖Tauberian常数$\mathsf C_{\mathsf S}$与$\mathsf M_{\mathsf S}$由$$ \mathsf C_{\mathsf S} (α):= \sup_{\substack {E\subset \mathbb R^n \\ 0<w(E)<+\infty}}\frac{1}{w(E)}w(\{x\in\mathbb R^n:\, \mathsf M_{\mathsf S}(\mathbf{1}_E)(x)>α\}). $$。我们证明了，$\lim_{α\to 1^-} \mathsf C_{\mathsf S} (α)=1$当且仅当$w\in A_\infty ^*$，即当且仅当$w$是一个强Muckenhoupt重量。这通过估计值$\mathsf C_{\mathsf S}(α)-1\lesssim_{n} (1-α)^{(cn [w]_{A_\infty ^*})^{-1}}$量化为$α\to 1^-$，其中$c>0$是数值常数;该估计值在指数$1/(cn[w]_{A_\infty ^*})$不能根据$[w]_{A_\infty ^*}$改进的意义上是尖锐的。作为推论，我们得到了$\mathbb R^n$中强Muckenhoupt权的一个尖锐的逆Hölder不等式，以及$A_\infty^*$到$A_{p}^*$中的一个定量嵌入.我们还考虑了$\mathbb R^n$上与权$w$相关联的强极大算子，记为$\mathsf M_{\mathsf S} ^w$。在这种情况下，相应的尖锐的陶伯常数$\mathsf C_{\mathsf S} ^w$被定义为$$ \mathsf C_{\mathsf S} ^w α) := \sup_{\substack {E\subset \mathbb R^n \\ 0<w(E)<+\infty}}\frac{1}{w(E)}w(\{x\in\mathbb R^n:\, \mathsf M_{\mathsf S} ^w (\mathbf{1}_E)(x)>α\}).$$我们表明，存在一些常数$c_{w,n}>0$只依赖于$w$和尺寸$n$，使$\mathsf C_{\mathsf S} ^w (α)-1 \lesssim_{w,n} (1-α)^{c_{w,n}}$作为$α\to 1^-$时，$w\in A_\infty ^*$是一个强Muckenhoupt重量。

英文摘要：

Let $\mathsf M_{\mathsf S}$ denote the strong maximal operator on $\mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $α\in(0,1)$ we define the weighted sharp Tauberian constant $\mathsf C_{\mathsf S}$ associated with $\mathsf M_{\mathsf S}$ by $$ \mathsf C_{\mathsf S} (α):= \sup_{\substack {E\subset \mathbb R^n \\ 0<w(E)<+\infty}}\frac{1}{w(E)}w(\{x\in\mathbb R^n:\, \mathsf M_{\mathsf S}(\mathbf{1}_E)(x)>α\}). $$ We show that $\lim_{α\to 1^-} \mathsf C_{\mathsf S} (α)=1$ if and only if $w\in A_\infty ^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $\mathsf C_{\mathsf S}(α)-1\lesssim_{n} (1-α)^{(cn [w]_{A_\infty ^*})^{-1}}$ as $α\to 1^-$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_\infty ^*})$ can not be improved in terms of $[w]_{A_\infty ^*}$. As corollaries, we obtain a sharp reverse Hölder inequality for strong Muckenhoupt weights in $\mathbb R^n$ as well as a quantitative imbedding of $A_\infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $\mathbb R^n$ associated with the weight $w$ and denoted by $\mathsf M_{\mathsf S} ^w$. In this case the corresponding sharp Tauberian constant $\mathsf C_{\mathsf S} ^w$ is defined by $$ \mathsf C_{\mathsf S} ^w α) := \sup_{\substack {E\subset \mathbb R^n \\ 0<w(E)<+\infty}}\frac{1}{w(E)}w(\{x\in\mathbb R^n:\, \mathsf M_{\mathsf S} ^w (\mathbf{1}_E)(x)>α\}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $\mathsf C_{\mathsf S} ^w (α)-1 \lesssim_{w,n} (1-α)^{c_{w,n}}$ as $α\to 1^-$ whenever $w\in A_\infty ^*$ is a strong Muckenhoupt weight.

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