Bounds for subcritical best Sobolev constants in W1, p

<p style='text-indent:20px;'>This paper aims at establishing fine bounds for subcritical best Sobolev constants of the embeddings</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega),\quad 1\leq q&lt; \begin{cases} \frac{Np}{N-p},&amp; 1\leq p&lt;N\\ \infty,&amp; p = N \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ N\geq p\geq1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded smooth domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula> or the whole space. The Sobolev limiting case <inline-formula><tex-math id="M5">\begin{document}$ p = N $\end{document}</tex-math></inline-formula> is also covered by means of a limiting procedure.</p>

Poincaré and Log–Sobolev Inequalities for Mixtures

This work studies mixtures of probability measures on R n and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the χ 2 -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.

Sobolev inequalities for fractional Neumann Laplacians on half spaces

We consider different fractional Neumann Laplacians of order {s\in(0,1)} on domains {\Omega\subset\mathbb{R}^{n}} , namely, the restricted Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{R}}}} , the semirestricted Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sr}}}} and the spectral Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sp}}}} . In particular, we are interested in the attainability of Sobolev constants for these operators when Ω is a half-space.

Sandwiched Rényi Convergence for Quantum Evolutions

We study the speed of convergence of a primitive quantum time evolution towards its fixed point in the distance of sandwiched Rényi divergences. For each of these distance measures the convergence is typically exponentially fast and the best exponent is given by a constant (similar to a logarithmic Sobolev constant) depending only on the generator of the time evolution. We establish relations between these constants and the logarithmic Sobolev constants as well as the spectral gap. An important consequence of these relations is the derivation of mixing time bounds for time evolutions directly from logarithmic Sobolev inequalities without relying on notions like lp-regularity. We also derive strong converse bounds for the classical capacity of a quantum time evolution and apply these to obtain bounds on the classical capacity of some examples, including stabilizer Hamiltonians under thermal noise.