Related papers: Logarithmic Sobolev inequality for the inhomogeneo…

We provide a new characterization of the logarithmic Sobolev inequality.

We apply inverse spectral theory to study Sobolev norms of solutions to the nonlinear Schrodinger equation. For initial datum $q_0\in L^2(\mathbb{R})$ and $s\in [-1,0]$, we prove that there exists a conserved quantity that is equivalent to…

For $\alpha\geq 1$, let $g:\mathbb N\to\mathbb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k)=(k/k-1)^\alpha$, $k\geq 2$. Consider the symmetric nearest neighbour zero range process on the discrete torus $\mathbb T_L$ in which a particle jumps…

We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic ``phases"…

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost…

Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the…

We study coupled systems of nonlinear lowest Landau level equations, for which we prove global existence results with polynomial bounds on the possible growth of Sobolev norms of the solutions. We also exhibit explicit unbounded…

We are interested in the $q$ Logarithmic Sobolev inequality for probability measures on the infinite product of Heisenberg groups. We assume that the one site boundary free measure satisfies either a $q$ Log-Sobolev inequality or a U-Bound…

We consider the Sobolev norms of the pointwise product of two functions, and estimate from above and below the constants appearing in two related inequalities.

The Sobolev space $H^{\varsigma}(\mathbb{R}^{d})$, where $\varsigma > d/2$, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measurement instrument, it is desirable in…

We study the infinite-dimensional log-Sobolev inequality for spin systems on $\mathbb{Z}^d$ with interactions of power higher than quadratic. We assume that the one site measure without a boundary $e^{-\phi(x)}dx/Z$ satisfies a log-Sobolev…

In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension $d\ge3$. The main…

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on $\mathbb{R}^n$ with an explicit expression for the constant. Namely, we show that for all $0<s<\frac{n}{2}$ and $a>0$ we have the…

We study boundary representations of hyperbolic groups $\Gamma$ on the (compactly embedded) function space $W^{\log,2}(\partial\Gamma)\subset L^2(\partial\Gamma)$, the domain of the logarithmic Laplacian on $\partial\Gamma$. We show that…

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the…

We consider the most general class of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of an arbitrary order whose solutions and right-hand sides belong to appropriate Sobolev spaces. For…

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincar\'{e} inequalities, general Beckner inequalities...). We also discuss the…

In this note we will generalize the results deduced in arXiv:1905.08203 and arXiv:2103.15360 to fractional Sobolev spaces. In particular we will show that for $s\in (0,1)$, $n>2s$ and $\nu\in \mathbb{N}$ there exists constants $\delta =…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\Delta_g^{k/2}…

This work studies the instability of stochastic scalar reaction diffusion equations, driven by a multiplicative noise that is white in time and smooth in space, near to zero, which is assumed to be a fixed point for the equation. We prove…