arXiv math/06100782006-10-02math.APmath.FA

基于热核界的新旧Morry空间

Old and New Morry Spaces via Heat Kernel Bounds

X. Duong, L. Yan, J. Xiao

摘要：

给定$p\in [1,\infty)$和$λ\in (0,n)$，研究了${\mathbb R}^n$上所有局部可积复值函数$f$的Morrey空间${\rm L}^{p,λ}({\mathbb R}^n)$，使得对于每个半径为$r_B$的开欧氏球$B\subset{\mathbb R}^n$，（取决于$f$）和$c=c(f,B)$（依赖于$f$和$B$）满足$$ r_B^{-λ}\int_{B}|f(x)-c|^pdx\le C $$并导出旧的和新的，选择$c=f_B=|B|^{-1}\int_{B}f(y)dy$或用$P_{t_B}(x)=\int_{t_B}p_{t_B}(x,y)f(y)dy$代替$c$所引起的两种本质上不同的情况--其中$t_B$被缩放为$r_B$，$p_t(\cdot,\cdot)$是上的解析半群$\{e^{-tL}\}_{t\ge0}$的无穷小生成元$L$的核${\rm L}^2({\mathbb R}^n)$。因此，我们导致同时表征旧的和新的Morrey空间，但也表明，对于一个合适的操作$L$，新的Morrey空间是等价的旧的。

英文摘要：

Given $p\in [1,\infty)$ and $λ\in (0,n)$, we study Morrey space ${\rm L}^{p,λ}({\mathbb R}^n)$ of all locally integrable complex-valued functions $f$ on ${\mathbb R}^n$ such that for every open Euclidean ball $B\subset{\mathbb R}^n$ with radius $r_B$ there are numbers $C=C(f)$ (depending on $f$) and $c=c(f,B)$ (relying upon $f$ and $B$) satisfying $$ r_B^{-λ}\int_{B}|f(x)-c|^pdx\le C $$ and derive old and new, two essentially different cases arising from either choosing $c=f_B=|B|^{-1}\int_{B}f(y)dy$ or replacing $c$ by $P_{t_B}(x)=\int_{t_B}p_{t_B}(x,y)f(y)dy$ -- where $t_B$ is scaled to $r_B$ and $p_t(\cdot,\cdot)$ is the kernel of the infinitesimal generator $L$ of an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ on ${\rm L}^2({\mathbb R}^n)$. Consequently, we are led to simultaneously characterize the old and new Morrey spaces, but also to show that for a suitable operator $L$, the new Morrey space is equivalent to the old one.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文