We introduce weak slice conditions and investigate imbeddings of Sobolev spaces in various Lipschitz-type spaces.

We give a uniied approach to error estimates for periodic interpolation on full and sparse grids in certain Sobolev spaces. We imposèperiodic' Strang{Fix conditions on the underlying functions in order to obtain error bounds with explicit constants. x1. Introduction The approximation and interpolation of bivariate periodic functions have been studied for some time. While periodic interpolation ...

Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces and discuss a few applications. First we show q-variation regularity of Cameron-Martin paths associated to fractional Brownian motion and other Volterra processes. This is useful, for instance, to establis...

for A; = 0 ,1 , . . . and with suitable assumptions on V which are valid for the p(x)Laplacian operator. Following some ideas from [11] we construct a dual variational method which applies to more general type of nonlinearities than those that are subject to a Palais-Smale type condition. We relate critical values and critical points to the action functional for which (1.1) is the Euler-Lagrang...

The goal is legitimization of term-wise differentiation of L spectral expansions, so that computations producing a classical outcome are correct. We are fond of L expansions because they are what Plancherel gives. Typically, L expansions are not continuous, much less differentiable, so the issue cannot be proving classical differentiability, which does not hold. To say that L spectral expansion...

for some function φ : Ω → R. The objective is to find a lifting φ “as regular as u permits.” For example, if u is continuous one may choose φ to be continuous and if u ∈ C one may also choose φ to be C. A more delicate result asserts that if u ∈ VMO (= vanishing means oscillation), then one may choose φ to be also VMO (see R. Coifman and Y. Meyer [1] and H. Brezis and L. Nirenberg [1]). In this...

On domains with conical points, weighted Sobolev spaces with powers of the distance to the conical points as weights form a classical framework for describing the regularity of solutions of elliptic boundary value problems, cf. papers by Kondrat’ev and Maz’ya-Plamenevskii. Two classes of weighted norms are usually considered: Homogeneous norms, where the weight exponent varies with the order of...

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 ≥ 3. The main consequence is an improvement of Sobolev’s inequality when d ≥ 5, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension...

In this paper, we consider the commutators of fractional integrals ∫ Rd b(x)− b(y) |x− y| f(y)dy on the Sobolev spaces . H s p where b is a locally integrable function and r ∈ (0, d). We establish the equivalence between the boundedness of the commutators and the paraproducts of J.M. Bony. Then, we establish necessary and sufficient conditions for the boundedness of the paraproduit operator fro...

All affinely covariant convex-body-valued valuations on the Sobolev space W (R) are completely classified. It is shown that there is a unique such valuation for Blaschke addition. This valuation turns out to be the operator which associates with each function f ∈W (R) the unit ball of its optimal Sobolev norm. 2000 AMS subject classification: 46B20 (46E35, 52A21,52B45) Let ‖ ·‖ denote a norm on...