论文标题
准希尔伯特式Sobolev空间的张力
Tensorization of quasi-Hilbertian Sobolev spaces
论文作者
论文摘要
Sobolev空间的张力问题询问了如何从其因素中确定产品度量量$ x \ times y $的SOBOLOLEV空间。我们表明，文献中的Sobolev空间的两个自然描述是重合的，$ W^{1,2}（x \ times y）= J^{1,2}（x，y）$，从而解决了Sobolev空间的张力问题，在$ p = 2 $的情况下，$ x $和$ x $ y $ y $ y ne $ y ne $ y ne $ y ne $ y ne $ y nesimian quasie quasie quasiimian quasiimian quasiimian quasiimian quasiemian quasiimian quasiimian quasiemiani。 $ w^{1,2} $允许通过dirichlet表单重新置换。该课程特别包括公制度量空间$ x，有限的Hausdorff尺寸以及无限的Hilbertian空间。 More generally for $p\in (1,\infty)$ we obtain the norm-one inclusion $\|f\|_{J^{1,p}(X,Y)}\le \|f\|_{W^{1,p}(X\times Y)}$ and show that the norms agree on the algebraic tensor product $ w^{1，p}（x）\ otimes w^{1，p}（y）\ subset w^{1，p}（x \ times y）$。当$ p = 2 $和$ x $和$ y $是无限的准希尔伯特式时代时，标准的dirichlet形式理论产生了$ w^{1,2}（x）（x）\ otimes w^{1,2}（y）$ in $ j^{1,2}（x，y）$的密度。我们的方法提出了$ w^{1，p}（x）（x）\ otimes w^{1，p}（y）$的密度的问题。
The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $X\times Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, $W^{1,2}(X\times Y)=J^{1,2}(X,Y)$, thus settling the tensorization problem for Sobolev spaces in the case $p=2$, when $X$ and $Y$ are infinitesimally quasi-Hilbertian, i.e. the Sobolev space $W^{1,2}$ admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces $X,Y$ of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally for $p\in (1,\infty)$ we obtain the norm-one inclusion $\|f\|_{J^{1,p}(X,Y)}\le \|f\|_{W^{1,p}(X\times Y)}$ and show that the norms agree on the algebraic tensor product $W^{1,p}(X)\otimes W^{1,p}(Y)\subset W^{1,p}(X\times Y)$. When $p=2$ and $X$ and $Y$ are infinitesimally quasi-Hilbertian, standard Dirichlet form theory yields the density of $W^{1,2}(X)\otimes W^{1,2}(Y)$ in $J^{1,2}(X,Y)$ thus implying the equality of the spaces. Our approach raises the question of the density of $W^{1,p}(X)\otimes W^{1,p}(Y)$ in $J^{1,p}(X,Y)$ in the general case.
