1. Interpolation inequalities. A classical problem in analysis is to understand how “smoothness” controls norms that measure the “size” of functions. Maz’ya recognized in his classic text on Sobolev spaces the intrinsic importance of inequalities that would refine both Hardy’s inequality and Sobolev embedding. Dilation invariance and group symmetry play an essential role in determining sharp co...

We investigate the connection between the validity of certain logarithmic Sobolev inequality and the validity of suitable generalizations of Gagliardo–Nirenberg inequalities. A similar connection holds between reverse logarithmic Sobolev inequalities and a new class of reverse Gagliardo–Nirenberg inequalities, valid for a suitable class of functions. 0. Introduction The main concern of this pap...

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler–Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev or...

Abstract. We are interested in the Logarithmic Sobolev Inequality for the infinite volume Gibbs measure with no quadratic interactions. We consider unbounded spin systems on the one dimensional Lattice with interactions that go beyond the usual strict convexity and without uniform bound on the second derivative. We assume that the one dimensional single-site measure with boundaries satisfies th...

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find low...

Logarithmic Sobolev inequalities are a well-studied technique for estimating rates of convergence of Markov chains to their stationary distributions. In contrast to continuous state spaces, discrete settings admit several distinct log Sobolev inequalities, one of which is the subject of this paper. Here we derive modi!ed log Sobolev inequalities for some models of random walk, including the ran...

‎the global fom and gmres algorithms are among the effective‎ ‎methods to solve sylvester matrix equations‎. ‎in this paper‎, ‎we‎ ‎study these algorithms in the case that the coefficient matrices‎ ‎are real symmetric (real symmetric positive definite) and extract‎ ‎two cg-type algorithms for solving generalized sylvester matrix‎ ‎equations‎. ‎the proposed methods are iterative projection metho...

We are concerned with the short- and large-time behavior of $L^2$-propagator norm Fokker-Planck equations linear drift, i.e. $\partial_t f=\mathrm{div}_{x}{(D \nabla_x f+Cxf)}$. With a coordinate transformation these can be normalized such that diffusion drift matrices linked as $D=C_S$, symmetric part $C$. The main result this paper is connection between their drift-ODE $\dot x=-Cx$: Their nor...