This paper is concerned with the relationships between L differentiability and Sobolev functions. It is shown that if / is a Sobolev function with weak derivatives up to order k in L , and 0 s / s k, then / has an L derivative of order / everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an 2V derivative P everywhere excep...

μ being a nonnegative measure satisfying some Log-Sobolev inequality, we give conditions on F for the Boltzmann measure ν = eμ to also satisfy some Log-Sobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied. Résumé. μ étant une mesure positive satisfaisant une ...

We consider a varying discrete Sobolev inner product involving the Laguerre weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and of their zeros. We are interested in Mehler–Heine type formulas because they describe the asymptotic differences between these Sobolev orthogonal polynomials and the classical Laguerre polynomials. Moreover, they give u...

A new class of alternative dual frames is introduced in the setting of finite frames for Rd. These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (Σ∆) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approxim...

Several inviscid models in hydrodynamics and geophysics such as the incompressible Euler vorticity equations, the surface quasi-geostrophic equation and the Boussinesq equations are not known to have even local well-posedness in the corresponding borderline Sobolev spaces. Here H is referred to as a borderline Sobolev space if the L∞-norm of the gradient of the velocity is not bounded by the H-...

In 1994, Lions, Perthame and Tadmor conjectured the maximal smoothing effect for multidimensional scalar conservation laws in Sobolev spaces. For strictly smooth convex flux and the one-dimensional case we detail the proof of this conjecture in the framework of Sobolev fractional spaces W s,1, and in fractional BV spaces: BV s. The BV s smoothing effect is more precise and optimal. It implies t...

There are several ways to generalize the notion of the Sobolev space to the setting of metric spaces equipped with a Borel measure. We describe next two definitions of the Sobolev space on a metric space (S, d) equipped with a Borel masure μ that is finite on every ball. Following [11], for 1 ≤ p < ∞, we define the Sobolev space M(S, d, μ) as the set of all u ∈ L(S) for which there exists 0 ≤ g...