与算子相关的局部Hardy空间及加倍度量测度空间上的球拟Banach函数空间及其应用
Local Hardy Spaces Associated with Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications
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中文总结 AI 辅助
本文在加倍度量测度空间框架下，研究与球拟Banach函数空间及满足高斯上界热核的非负自伴算子相关的局部Hardy空间，建立其多种刻画并应用于几类函数空间。
中文摘要 AI 辅助
设$(\boldsymbol{\textit{X}},d,\boldsymbol{\textit{u}})$为加倍度量测度空间，$X$为$\boldsymbol{\textit{X}}$上的球拟Banach函数空间，$L$为$L^2(\boldsymbol{\textit{X}})$上的非负自伴算子，其热核满足高斯上界。本文研究同时与$X$和$L$相关的局部Hardy空间$h_{X,L}(\boldsymbol{\textit{X}})$，首先建立该空间的原子与分子刻画；作为这些刻画的应用，得到$h_{X,L}(\boldsymbol{\textit{X}})$与全局Hardy空间$H_{X,L}(\boldsymbol{\textit{X}})$、$H_{X,L+mI}(\boldsymbol{\textit{X}})$之间的关系，还建立该空间的径向与非切向极大函数刻画。利用这些极大函数刻画，在热核满足守恒性质及Hölder正则性估计的额外假设下，进一步证明在相应指标范围内，$h_{X,L}(\boldsymbol{\textit{X}})$与局部原子Hardy空间$h_{X,\text{at}}^p(\boldsymbol{\textit{X}})$等价（拟范数等价）；最后将上述结果应用于Lebesgue空间、Orlicz空间、加权Lebesgue空间及变Lebesgue空间。
英文摘要
Let $(\mathcal{X},d,μ)$ be a doubling metric measure space, $X$ a ball quasi-Banach function space on $\mathcal{X}$, and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernel satisfies a Gaussian upper bound. In this article, we study the local Hardy space $h_{X,L}(\mathcal{X})$ associated with both $X$ and $L$. We first establish the atomic and molecular characterizations of $h_{X,L}(\mathcal{X})$. As applications of these characterizations, we obtain the relations between $h_{X,L}(\mathcal{X})$ and the global Hardy spaces $H_{X,L}(\mathcal{X})$ and $H_{X,L+mI}(\mathcal{X})$. We also establish the radial and non-tangential maximal function characterizations of $h_{X,L}(\mathcal{X})$. Using these maximal function characterizations, under the additional assumptions that the heat kernel satisfies the conservation property and a Hölder regularity estimate, we further show that $h_{X,L}(\mathcal{X})$ coincides with the local atomic Hardy space $h_{X,\mathrm{at}}^p(\mathcal{X})$ with equivalent quasi-norms in the corresponding range of indices. Finally, we apply the above results to Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces.
