Vector calculus for tamed Dirichlet spaces (Q6661169)

Inspired by [\textit{N. Gigli}, Nonsmooth differential geometry -- an approach tailored for spaces with Ricci curvature bounded from below. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1404.53056)], this monograph aims to construct a functional first and second order calculus over Dirichlet spaces that are tamed by signed extended Kato class measures. In the author's general approach, the following quantities are most important:\N\begin{itemize}\N\item the Hessian operator on appropriate functions, along with provint that sufficiently many of these do exist, and\N\item a measure-valued Ricci curvature.\N\end{itemize}\N\NIn addition, the tamed analogue of the finite-dimensional \(\mathrm{BE}_{2}(K,N)\) condition [Zbl 1486.35008], \(K\in\boldsymbol{R}\) and \(N\in\lbrack1,\infty\}\), following the \(\mathrm{RCD}^{\ast}(K,N)\)-treatise [Zbl 1395.53046] which is not essentially different from [\textit{N. Gigli}, Nonsmooth differential geometry -- an approach tailored for spaces with Ricci curvature bounded from below. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1404.53056)].\N\NThe synopsis of the monograph goes as follows.\N\N\begin{itemize}\N\item[Chapter 1] deals with first order differential calculus. The author first recapitulates basic notions about Dirichlet forms and -modules in \S 1.1. The appropriate analogs of differential 1-forms and vector fields are introduced in \S 1.2 and \S 1.3. The following theorem is established.\N\NTheorem. \(L^{2}(T^{\ast}M)\) and \(L^{2}(TM)\) are \(L^{2}\)-normed \(L^{\infty}\)-modules with pointwise norms both denoted by \(\left\vert \cdot\right\vert \). They come with a linear differential\N\[\Nd:\mathcal{F}_{e}\rightarrow L^{2}(T^{\ast}M)\N\]\Nand a linear gradient\N\[\N\nabla:\mathcal{F}_{e}\rightarrow L^{2}(TM)\N\]\Nsuch that for every , we have\N\[\N\left\vert \mathrm{d}f\right\vert =\left\vert \nabla f\right\vert =\Gamma(f)^{1/2}\qquad\mathfrak{m}-a.e.\N\]\N\N\item[Chapter 2] investigates elements of a second order calculus on tamed spaces, whose properties are recorded in \S 2.1.1. The author goes on with giving a meaning to the Hessian in \S 2.2, to the covariant derivative in \S 2.3, and exterior differential and Hodge theory in \S 2.4. \S 2.5 contains the appropriate existence of a Ricci curvature and a second fundamental form. The following theorem is established.\N\NTheorem. The map\N\[\N\boldsymbol{Ric}^{\kappa}(X,X)=\Delta^{2\kappa}\frac{\left\vert X\right\vert }{2}^{2}+\overrightarrow{\Delta}X^{\flat}(X)\mathfrak{m}-\left\vert \nabla X\right\vert _{\mathrm{HS}}^{2}\mathfrak{m},\quad X^{\flat}\in\mathrm{Reg}(TM)\N\]\Nextends continuously to the closure \(H_{\#}^{1,2}(TM)\) of \(\mathrm{Reg}(TM)\) with respect to an appropriate \(H^{1,2}\)-norm with values in the space of Borel measures on \(M\) with finite total variation changing no \(\mathcal{E}\)-polar sets.\N\end{itemize}