The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to N. Korevaar and R. Schoen on the one hand, and J. Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to L. Ambrosio (for BV maps into metric spaces), Y.G. Reshetnyak and finally to the notion of Newtonian-Sobol...

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...

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 this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames for L2(R). In particular, we show that a Sobolev space of an arbitrary fixed order of smoothness can be characterized in terms of the weighted `2-norm of the analysis wavelet coefficient sequences using a fixed compactly supported nonstationary tight wavelet frame in L...

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...