Trace theorems are an indispensable tool for the analysis of kinetic equations. They have been established in wide generality by Cessenat and co-workers for radiative transfer and related applications. A variety of trace estimates have been established for the kinetic Fokker-Planck and Kolmogorov equation, typically requiring smoothness of the underlying domain; see the recent survey by Niebel \&Valentini. In this work, we prove a new trace estimate for kinetic Sobolev spaces over the sphere for $\rho$-convex domains which, in general, may have a non-smooth boundary. Similar to the work of Cessenat, we use characteristics to obtain trace estimates in weighted trace spaces with explicit constants. For completeness, we also present a density result for the corresponding function spaces on Lipschitz domains.
We study the spatially homogeneous Boltzmann equation without angular cutoff for very soft potentials satisfying the inverse power law relation $\gamma+4s=1$. Our main results establish the existence, uniqueness, stability and regularization estimates of solutions in the critical Sobolev space $ H^{-(\gamma + 2s + \fra...
This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kernel proof on Heisenberg groups equipped with the Carnot--Carath\'eodory distance. For source densities bounded above and away from zero on bou...
We establish stability principles under compact gluing for the reverse square-root inequality and the Riesz transform. Below a common Sobolev dimension, the reverse estimate passes from finitely many model manifolds to their connected sum. A boundary-compatible Poisson parametrix also proves the Neumann reverse inequal...
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et...
T. Hytönen, Da-Chun Yang, Wen Yuan et al.· 2 citations
We establish the global well-posedness of weak solutions to the Cauchy problem for a broad class of multi-dimensional $n\times n$ Keyfitz-Kranzer type systems with homogeneous flux. Our proof relies on a novel viscous approximation that yields the necessary compactness estimates. By exploiting the specific structure of...
In this paper, we prove a boundedness result for parabolic Riesz transforms in the range $p \le 2$ associated with degenerate parabolic operators whose elliptic part is in divergence form with complex coefficients depending measurably on space and time. The degeneracy is determined by a spatial weight in the Muckenhoup...
Khalid Baadi· 1 citation
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