We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in t...

In this paper we construct a class of locally supported wavelet bases for C Lagrange nite element spaces on possibly non uniform meshes on n dimensional domains or manifolds The wavelet bases are stable in the Sobolev spaces Hs for jsj jsj on Lipschitz manifolds and the wavelets can in principal be arranged to have any desired order of vanishing moments As a consequence these bases can be used ...

We describe a recent, one-parameter family of characterizations Sobolev and BV functions on $\mathbb{R}^n$ n, using sizes superlevel sets suitable difference quotients. This provides an alternative point view to the BBM formula by Bourgain, Brezis, Mironescu, complements in case some results Cohen, Dahmen, Daubechies, DeVore about wavelet coefficients such functions. An application towards Gagl...

We prove a regularity result on polyhedral domains P ⊂ R using the weighted Sobolev spaces Ka (P). In particular, we show that there is no loss of Ka –regularity for solutions of strongly elliptic systems with smooth coefficients. In the proof, we identify Ka (P) with the Sobolev spaces on P associated to the metric r P gE , where gE is the Euclidean metric and rP(x) is a smoothing of the Eucli...

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for...

Moduli spaces M of self-dual SU(2) connections (“instantons”) over a compact Riemannian 4.manifold (hl, g) carry a natural L2 metric g, which is generally incomplete. For instantons of Pontryagin index 1 over a compact, simply connected, oriented, positive-definite base manifold, the completion M is Donaldson’s compactification; in fact the boundary of the completion is an isometric copy of (M,...

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...

We study the Sobolev spaces of exponential type associated with the Dunkl-Bessel Laplace operator. Some properties including completeness and the imbedding theorem are proved. We next introduce a class of symbols of exponential type and the associated pseudodifferential-difference operators, which naturally act on the generalized Dunkl-Sobolev spaces of exponential type. Finally, using the theo...

In this paper, metric complexities of certain classes of continuous-time systems are studied, using the time-domain sampling approach and the concepts of Kolmogorov, Gel’fand and sampling n-widths for certain classes of Sobolev space. A sampling theorem is obtained which extends Shannon’s sampling theorem to systems with possibly non-band-limited spectra. The theorem demonstrates that continuou...

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. §