论文标题
高阶加权迪里奇型积分的本地道格拉斯公式
A local Douglas formula for higher order weighted Dirichlet-type integrals
论文作者
论文摘要
我们证明了一个局部的道格拉斯公式，用于高阶加权dirichlet型积分。借助此公式，我们研究了相关的高阶加权dirichlet-type空间的乘数代数$ \ MATHCAL H _ {\pMBμ}，$由$ m $ -m $ -tuple $ \pmbμ=（μ__1，\ ldots，\ ldots，μ__{m}）$ nontial-nortial-nortial-nonite bon-noniite bon-nonite b inte-nontial-boniite conto。尤其是表明，任何加权的dirichlet型订单$ m，$ $ m \ geqslant 3，$ $ $ $ m \ geqslant 3，$在点式产品下形成代数。我们还证明，每个非零关闭的$ M_z $ -Invariant子空间为$ \ MATHCAL H _ {\PMBμ}，如果$ M \ geqslant 3 $ 3 $或$μ_2$，则有一个codimension $ 1 $属性。作为本文获得的本地道格拉斯公式的另一种应用，可以表明，对于任何$ m \ geqslant 2，$加权的dirichlet型订单$ m $的空间都不与任何de branges-rovnyak space $ \ Mathcal $ \ Mathcal H（b）$相吻合。
We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces $\mathcal H_{\pmbμ},$ induced by an $m$-tuple $\pmb μ=(μ_1,\ldots,μ_{m})$ of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order $m,$ for $m\geqslant 3,$ forms an algebra under pointwise product. We also prove that every non-zero closed $M_z$-invariant subspace of $\mathcal H_{\pmbμ},$ has codimension $1$ property if $m\geqslant 3$ or $μ_2$ is finitely supported. As another application of local Douglas formula obtained in this article, it is shown that for any $m\geqslant 2,$ weighted Dirichlet-type space of order $m$ does not coincide with any de Branges-Rovnyak space $\mathcal H(b)$ with equivalence of norms.
