We give a characterization of Sobolev spaces of bivariate periodic functions with dominating smoothness properties in terms of Sobolev spaces of univariate functions. The mixed Sobolev norm is proved to be a uniform crossnorm. This property can be used as a powerful tool in approximation theory. x1. Introduction Beside the approximation of functions from the usual isotropic periodic Sobo-lev sp...

Action of p ? on distributions is examined in the context of Sobolev spaces, weighted L 2 spaces and weighted Sobolev spaces, respectively. The results obtained are as follows: Let k be a real number. (1) If f is in H

Various issues with regard to chaos and recurrence in infinite dimensions are discussed. The doctrine we are trying to derive is that Sobolev spaces over bounded spatial domains do host chaos and recurrence, while Sobolev spaces over unbounded spatial domains are lack of chaos and recurrence. Local Sobolev spaces over unbounded spatial domains can host chaos and are natural phase spaces e.g. fo...

This paper generalizes the mixed extension principle in L2(R) of [50] to a pair of dual Sobolev spaces H(R) and H−s(Rd). In terms of masks for φ, ψ, . . . , ψ ∈ H(R) and φ̃, ψ̃, . . . , ψ̃ ∈ H−s(Rd), simple sufficient conditions are given to ensure that (X(φ;ψ, . . . , ψ), X−s(φ̃; ψ̃, . . . , ψ̃)) forms a pair of dual wavelet frames in (Hs(Rd),H−s(Rd)), where X(φ;ψ, . . . , ψ) := {φ(· − k) : k ∈ Zd} ...

We study the pathwise regularity of the map φ 7→ I(φ) = ∫ T 0 〈φ(Xt), dXt〉 where φ is a vector function on R belonging to some Banach space V , X is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A stochastic current is a continuous version of this map, seen as a random element of the topological dual of V . We give sufficient conditio...

Abstract The initialand boundary-value problem for the Kawahara equation, a fifthorder KdV type equation, is studied in weighted Sobolev spaces. This functional framework is based on the dual-Petrov–Galerkin algorithm, a numerical method proposed by Shen (2003 SIAM J. Numer. Anal. 41 1595–619) to solve third and higher odd-order partial differential equations. The theory presented here includes...

We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables s...

This paper is concerned with the construction of biorthogonal wavelet bases on n-dimensional cubes which provide Riesz bases for Sobolev and Besov spaces with homogeneous Dirichlet boundary conditions on part of the boundary. The essential point is that the primal and dual wavelets satisfy certain corresponding complementary boundary conditions. These results form the key ingredients of the con...

In the fundamental and inspiring book of Y. Meyer [Mey01] patterns are characterized as oscillating functions, which in turn are considered elements of dual Sobolev spaces. The concept of oscillating patterns can be used for pattern recognition and edge enhancing techniques with regularization methods exploiting the Bregman distance, a concept which has been established by Burger et al. Recentl...

We prove some general results on the existence of partitions of unity in Sobolev type spaces on various innnite dimensional manifolds. As special cases we obtain in particular, (continuous) partitions of unity a) in the Malliavin test functions on an abstract Wiener space; b) in the rst order Sobolev space on pinned and free loop spaces; c) in the rst order Sobolev spaces associated with revers...