We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight #j moderates the behavior of functions with respect to the jth variable. For #j #1, we obtain the classical Sobolev spaces whereas for decreasing #j 's the weighted Sobolev spaces consist of functions...

In a previous paper, the authors showed that the information complexity of the Fredholm problem of the second kind is essentially the same as that of the approximation problems over the spaces of kernels and right-hand sides. This allowed us to give necessary and sufficient conditions for the Fredholm problem to exhibit a particular level of tractability (for information complexity) over weight...

We prove dimension-invariant imbedding theorems for Sobolev spaces using extrapolation means.

Abstract. We propose a mathematical framework to effectively study lattice materials with periodic and non-periodic structures over entire spaces in one, two, and three dimensions. The existence and uniqueness of solutions for periodic lattice problems with absolute terms are proved in discrete Sobolev spaces. By Fourier transform discrete lattice problems are converted to semi-discrete problem...

A multilinear version of Schur’s test is obtained for products of L spaces and is used to derive boundedness for multilinear multiplier operators acting on Sobolev and Besov spaces.

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a non-linear and non-local version of the ground state representation.

We present isocapacitary characterizations of Sobolev inequalities in very general metric measure spaces.

The I-method in its first version as developed by Colliander et al. in [2] is applied to prove that the Cauchy-problem for the generalised Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space H(R → R), provided s > − 1 42 .

1 Some Basic Analysis 3 1.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Fourier inversion on S and L . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Sobolev spa...

The function spaces Dk(R) are introduced and studied. The definition of these spaces is based on a regularity property for the critical Sobolev spaces Ws,p(Rn), where sp = n, obtained by J. Bourgain, H. Brezis, New estimates for the Laplacian, the div–curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (7) (2004) 539–543 (see also J. Van Schaftingen, Estimates for L1-vector fields...