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In the framework of quasi-regular strongly local Dirichlet form $(\mathscr{E},D(\mathscr{E}))$ on $L^2(X;\mathfrak{m})$ admitting minimal $\mathscr{E}$-dominant measure $\mu$, we construct a natural $p$-energy functional $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$ on $L^p(X;\mathfrak{m})$ and $(1,p)$-Sobolev space $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ for $p\in]1,+\infty[$. In this paper, we establish the Clarkson type inequality for $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$. As a consequence, $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, we prove that (generalized) normal contraction operates on $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$, which has been shown in the case of various concrete settings, but has not been proved for such general framework. Moreover, we prove that $(1,p)$-capacity ${\rm Cap}_{1,p}(A)<\infty$ for open set $A$ admits an equilibrium potential $e_A\in D(\mathscr{E}^{\,p})$ with $0\leq e_A\leq 1$ $\mathfrak{m}$-a.e. and $e_A=1$ $\mathfrak{m})$-a.e.~on $A$.

The notion of tamed Dirichlet space was proposed by Erbar, Rigoni, Sturm and Tamanini as a Dirichlet space having a weak form of Bakry-\'Emery curvature lower bounds in distribution sense. After their work, Braun established a vector calculus for it, in particular, the space of $L^2$-normed $L^{\infty}$-module describing vector fields, $1$-forms, Hessian in $L^2$-sense. In this framework, we establish the Hess-Schrader-Uhlenbrock inequality for $1$-forms as an element of $L^2$-cotangent module (an $L^2$-normed $L^{\infty}$-module), which extends the Hess-Schrader-Uhlenbrock inequality by Braun under an additional condition., Comment: 42 pages. arXiv admin note: substantial text overlap with arXiv:2308.12728. substantial text overlap with arXiv:2108.12374 by other authors

Let $V$ be a $C^1$-vector field on an $n$-dimensional complete Riemannian manifold $(M, g)$. We prove a Liouville theorem for $V$-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative $(m, V)$-Ricci curvature for $m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ into Cartan-Hadam\-ard manifolds, which extends Cheng's Liouville theorem proved S.~Y.~Cheng for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan-Hadamard manifolds. We also prove a Liouville theorem for $V$-harmonic maps from complete Riemannian manifolds with non-negative $(m, V)$-Ricci curvature for $m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt-Jost-Wideman and Choi. Our probabilistic proof of Liouville theorem for several growth $V$-harmonic maps into Hadamard manifolds enhances an incomplete argument by Stafford. Our results extend the results due to Chen-Jost-Qiu\cite{ChenJostQiu} and Qiu\cite{Qiu} in the case of $m=+\infty$ on the Liouville theorem for bounded $V$-harmonic maps from complete Riemannian manifolds with non-negative $(\infty, V)$-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of $V$-harmonic maps and the recurrence property of $\Delta_V$-diffusion processes on manifolds. Our results are new even in the case $V=\nabla f$ for $f\in C^2(M)$.

The notion of tamed Dirichlet space was proposed by Erbar, Rigoni, Sturm and Tamanini as a Dirichlet space having a weak form of Bakry-\'Emery curvature lower bounds in distribution sense. After their work, Braun established a vector calculus for it, in particular, the space of $L^2$-normed $L^{\infty}$-module describing vector fields, $1$-forms, Hessian in $L^2$-sense. In this framework, we establish the Littlewood-Paley-Stein inequality for $1$-forms as an element of $L^p$-cotangent bundles and boundedness of Riesz transforms, which partially solves the problem raised by Kawabi-Miyokawa., Comment: 68 pages. arXiv admin note: text overlap with arXiv:2307.12514; text overlap with arXiv:2108.12374 by other authors

The notion of tamed Dirichlet space by distributional lower Ricci curvature bounds was proposed by Erbar, Rigoni, Sturm and Tamanini as the Dirichlet space having a weak form of Bakry-\'Emery curvature lower bounds in distribution sense. In this framework, we establish the Littlewood-Paley-Stein inequality for $L^p$-functions and $L^p$-boundedness of $q$-Riesz transforms with $q>1$, which partially generalizes the result by Kawabi-Miyokawa.

We consider the Kato and the Dynkin class and their local counterparts on a smooth Riemannian manifold as Fr\'{e}chet spaces. Based on recent results by Carron, Mondello and Tewodrose we show that for a Riemannian manifold $(X,g)$ of dimension $m\geq 2$ with spectral negative part $\sigma^-_g$ of the Ricci curvature in $L^q_{\phi_g}(X,g)+L^\infty(X,g)$ for some $q>m/2$, the function $\sigma^-_g$ is in the Kato class of $(X,g)$ if and only if $(X,g)$ satisfies a Gaussian upper heat kernel bound for small times and is locally volume doubling. Here $L^q_{\phi_g}(X,g)$ is the $L^q$-space which is weighted with the inverse volume function. By establishing a localization result for the Dynkin norm, we prove that the local Kato class and the local Dynkin class do not depend on the chosen Riemannian metric and thus can be defined as Fr\'{e}chet spaces on arbitrary smooth manifolds. Moreover, we prove that smooth compactly supported functions are dense in the local Kato class and we use this result to prove that Schr\"odinger semigroups with Kato decomposable potentials are space-time continuous., Comment: Correction in statement of Theorem 3.3 in version 1

In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations, we give a necessary and sufficient condition on the stability of global two-sided (upper) estimates for fundamental solution of Feynman-Kac semigroup of X. As a corollary, under the same assumptions, a weak type global two-sided (upper) estimates holds for the fundamental solution of Feynman-Kac semigroup with (extended) Kato class conditions for measures. This generalizes all known results on the stability of global integral kernel estimates by symmetric Feynman-Kac perturbations with Kato class conditions in the framework of symmetric Markov processes.

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-\'Emery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-\'Emery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Bakry-\'Emery Ricci curvature under $m\geq n$ if the vector field is a gradient type. When $m<1$, our results are new in the literature., Comment: 27 pages. arXiv admin note: text overlap with arXiv:2001.00444

In this paper, we give sufficient conditions for the equivalence between semigroup strong Feller property and resolvent strong Feller property., Comment: 27 pages (to appear in Springer Proceedings in Mathematics and Statistics)

We study comparison geometry of manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N\in ]-\infty,1]\cup [n,+\infty]$ with $\varepsilon$-range introduced by Lu-Minguzzi-Ohta. We will conclude splitting theorems, and also comparison geometric results for inscribed radius, volume around the boundary, and smallest Dirichlet eigenvalue of the weighted $p$-Laplacian. Our results interpolate those for $N\in [n,+\infty[$ and $\varepsilon=1$, and for $N\in ]-\infty,1]$ and $\varepsilon=0$ by the second named author., Comment: 24 pages

In this paper, we introduce the notion of $L^p$-Green-tight measures of $L^p$-Kato class in the framework of symmetric Markov processes. The class of $L^p$-Green-tight measures of $L^p$-Kato class is defined by the $p$-th power of resolvent kernels. We first prove that under the $L^p$-Green tightness of the measure $\mu$, the embedding of extended Dirichlet space into $L^{2p}(E;\mu)$ is compact under the absolute continuity condition for transient Markov processes, which is an extension of recent seminal work by Takeda. Secondly, we prove the coincidence between two classes of $L^p$-Green-tightness, one is originally introduced by Zhao, and another one is invented by Chen. Finally, we prove that our class of $L^p$-Green-tight measures of $L^p$-Kato class coincides with the class of $L^p$-Green tight measures of Kato class in terms of Green kernel under the global heat kernel estimates. We apply our results to $d$-dimensional Brownian motion androtationally symmetric relativistic $\alpha$-stable processes on $\mathbb{R}^d$., Comment: 28 pages

In the present article, we provide a characterization of a lower $N$-weighted Ricci curvature bound for $N \in ]-\infty,1]\cup[n,+\infty]$ with $\varepsilon$-range introduced by Lu-Minguzzi-Ohta in terms of a convexity of entropies over Wasserstein space. We further derive various interpolation inequalities and functional inequalities., Comment: 25 pages

Lu-Minguzzi-Ohta have introduced the notion of a lower $N$-weighted Ricci curvature bound with $\varepsilon$-range, and derived several comparison geometric estimates from a Laplacian comparison theorem for weighted Laplacian. The aim of this paper is to investigate various rigidity phenomena for the equality case of their comparison geometric results. We will obtain rigidity results concerning the Laplacian comparison theorem, diameter comparisons, and volume comparisons. We also generalize their works for non-symmetric Laplacian induced from vector field potential., Comment: 20 pages

In this paper, we establish the coincidence of two classes of $L^p$-Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of $L^p$-Kato class measures is defined by the $p$-th power of positive order resolvent kernel, another is defined in terms of the $p$-th power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that $q$-th integrable functions on balls with radius $1$ having uniformity of its norm with respect to centers are of $L^p$-Kato class if $q$ is greater than a constant related to $p$ and the constants appeared in the upper and lower estimates of the heat kernel. These are complete extensions of some results by Aizenman-Simon and the recent results by the second named author in the framework of Brownian motions on Euclidean space. We further give necessary and sufficient conditions for a Radon measure with Ahlfors regularity to belong to $L^p$-Kato class. Our results can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on $d$-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on., Comment: 25 pages, minor corrections in the proof of Theorem 4.1

Let $L=\Delta-\nabla\phi\cdot \nabla$ be a symmetric diffusion operator with an invariant measure $\mu({\rm} d x)=e^{-\phi(x)}{\mathfrak m}({\rm d} x)$ on a complete non-compact smooth Riemannian manifold $(M,g)$ with its volume element ${\mathfrak m}={\rm vol}_g$, and $\phi\in C^2(M)$ a potential function. In this paper, we prove a Laplacian comparison theorem on weighted complete Riemannian manifolds with ${\rm CD}(K, m)$-condition for $m\leq 1$ and a continuous function $K$. As consequences, we give the optimal conditions on $m$-Bakry-\'Emery Ricci tensor for $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, and the Cheeger-Gromoll type splitting theorem, stochastic completeness and Feller property of $L$-diffusion processes hold on weighted complete Riemannian manifolds. Some of these results were well-studied for $m$-Bakry-\'Emery Ricci curvature for $m\geq n$ (\!\!\cite{Lot,Qian,XDLi05, WeiWylie}) or $m=1$ (\!\!\cite{Wylie:WarpedSplitting, WylieYeroshkin}). When $m<1$, our results are new in the literature.

This erratum corrects the article arXiv:0806.2044 published in Ann. Probab. 36 (2008) 931--970, Comment: Published in at http://dx.doi.org/10.1214/11-AOP684 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao's divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized It\^{o} formula. (With Errata.), Comment: Published in at http://dx.doi.org/10.1214/09-AOP516 and Errata at http://dx.doi.org/10.1214/11-AOP700 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We prove the infinitesimal version of Bishop-Gromov volume comparison condition for Alexandrov spaces.

We establish the equivalence of the analytic and probabilistic notions of subharmonicity in the framework of general symmetric Hunt processes on locally compact separable metric spaces, extending an earlier work of the first named author on the equivalence of the analytic and probabilistic notions of harmonicity. As a corollary, we prove a strong maximum principle for locally bounded finely continuous subharmonic functions in the space of functions locally in the domain of the Dirichlet form under some natural conditions.

Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery) Ricci curvature., Comment: 20 pages to appear in Tohoku Mathematical Journal

We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem., Comment: 30 pages, 1 figure

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the $L^p$-convergence, we establish a theory of variational convergences. We prove that the Poincar\'e inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are $\CAT(0)$-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature.