Existence of parabolic minimizers to the total variation flow on metric measure spaces

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

Sobolev-to-Lipschitz property on $${\mathsf {QCD}}$$-spaces and applications

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These results apply in particular to large classes of (ideal) sub-Riemannian manifolds.

Strongly Recurrent Transformation Groups

We define the notions of strong and strict recurrency for actions of countable ordered groups on $\sigma$-finite non atomic measure spaces with quasi-invariant measures. We show that strong recurrency is equivalent to non existence of weakly wandering sets of positive measure. We also show that for certain p.m.p ergodic actions the system is not strictly recurrent, which shows that strong and strict recurrency are not equivalent.

Reverse integral Hardy inequality on metric measure spaces

In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. In addition, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.

Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces

We study commutators of weighted fractional Hardy-type operators within the frameworks of local generalized Morrey spaces over quasi-metric measure spaces for a certain class of “radial” weights. Quasi-metric measure spaces may include, in particular, sets of fractional dimentsions. We prove theorems on the boundedness of commutators with CMO coefficients of these operators. Given a domain Morrey space 𝓛 p,φ (X) for the fractional Hardy operators or their commutators, we pay a special attention to the study of the range of the exponent q of the target space 𝓛 q,ψ (X). In particular, in the case of classical Morrey spaces, we provide the upper bound of this range which is greater than the known Adams exponent.