The time local and global well-posedness for the Maxwell-Schrödinger equations is considered in Sobolev spaces in three spatial dimensions. The Strichartz estimates of Koch and Tzvetkov type are used for obtaining the solutions in the Sobolev spaces of low regularities. One of the main results is that the solutions exist time globally for large data. §

Besov as well as Sobolev spaces of dominating mixed smoothness are shown to be tensor products of Besov and Sobolev spaces defined on R. Based on this we derive several useful characterizations from the the one-dimensional case to the ddimensional situation. Finally, consequences for hyperbolic cross approximations, in particular for tensor product splines, are discussed.

The Smolyak algorithm represents one possible approach to the approximation of functions of many variables. The natural domains of de nition are given by tensor products of function spaces de ned on R or on some interval I ⊂ R. Here Besov as well as Sobolev spaces of dominating mixed smoothness come into play. They are tensor products of Besov and Sobolev spaces de ned on R.

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hubert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer o...

In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete. We also prove the density of the polynomials in these spaces for non-closed compact curves and, finally, we find conditions under which the multiplication operator is bounded on the completion of polynomials. These results have applications to the study of ...

We characterize the restrictions of first order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator. Let (X, d, µ) be a metric space (X, d) equipped with a Borel measure µ, which is non-negative and outer regular, and is finite on every bounded subset. In this paper we describe the restrictions of first order ...

We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of Lx estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate L 4 n−1 t Lx for n = 2, 3 even for data in inhomogeneous Sobolev spaces.

We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of Lx estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate L 4 n−1 t Lx for n = 2, 3 even for data in inhomogeneous Sobolev spaces.

A new class of weak weighted derivatives and its associated Sobolev spaces is introduced and studied. The proposed notion uses a variational formulation in its definition which generalizes the usual weighting of the classical weak derivative. Such a construction naturally leads to Sobolev spaces containing classes of discontinuous functions. Weak closedness with respect to both varying function...