In this paper we prove an optimal local well-posedness result for the 1+2 dimensional system of nonlinear wave equations (NLW) with quadratic null-form derivative nonlinearities Qμν . The Cauchy problem for these equations is known to be ill-posed for data in the Sobolev space H with s ≤ 5/4 for all the basic null-forms, except Q0, thus leaving a gap to the critical regularity of sc = 1. Follow...

This paper proposes a combination of contour deformation modelling in Sobolev spaces and the Condensation filter to track an object over a sequence of images. As Sobolev spaces are smoothness spaces this allows to control the smoothness of the contour deformation extending previous wavelet representations. We also introduce a probabilistic model for the wavelet deformation of the contour that i...

In this paper we analyze a class of projection operators with values in a subspace of polynomials. These projection operators are related to the Hubert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthogonality relat...

In the case ω = 1, this space is denoted W (Ω). Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is “disturbed” in the sense that some degeneration or singularity appears. This “bad” behaviour can be caused by the coefficient...

We study a Γ -convergence problem related to a new characterization of Sobolev spaces W1,p(RN) (p > 1) established in H.-M. Nguyen [H.-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689–720] and J. Bourgain and H.-M. Nguyen [J. Bourgain, H.-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 75–80]. We can also hand...

If A is a unital C∗-algebra and if H is a complex Hilbert space, then the set SH(A) of all unital completely positive linear maps from A to the algebra B(H) of continuous linear operators on H is an operator-valued, or generalised, state space of A. The usual state space of A occurs with the one-dimensional Hilbert space C. The structure of the extreme points of generalised state spaces was det...

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz’ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also gi...