We investigate properties of measures in infinite dimensional spaces in terms of Poincaré inequalities. A Poincaré inequality states that the L2 variance of an admissible function is controlled by the homogeneous H1 norm. In the case of Loop spaces, it was observed by L. Gross [17] that the homogeneous H1 norm alone may not control the L2 norm and a potential term involving the end value of the...

The aim of the paper is to characterize the trace space of vector-valued Sobolev spaces W p (R , E) , where E is an arbitrary Banach space. In particular, we do not assume that the underlying Banach space E has the UMD property. Vector-valued Sobolev and Besov spaces are widely used in abstract evolution equations, cf. e.g. Amann [1, 2, 4], Veraar and Weis [57] or Denk, Hieber, Prüss, Saal, and...

We consider error estimates for the interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated \native" Hilbert spaces are shown to be norm-equivalent to Sobolev spaces. Thus we can derive approximation orders for fu...

Let Ω R ω be an open cone in Rn, n ¥ 3, where ω € Sn 1 is a smooth subdomain of the unit sphere. Denote by K and S the double and single layer potential operators associated to Ω and the Laplace operator ∆. Let r be the distance to the origin. We consider a natural class of operators on BΩ, called Mellin convolution operators with good mapping properties between weighted Sobolev spaces and show...

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalizations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results t...

For various spaces of potentials we prove that the set of finite gap potentials of the ZakharovShabat system is dense. In particular our result holds for Sobolev spaces and for spaces of analytic potentials of a given type. Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-21875 Originally published at: Grébert, B; Kappeler, T (2003). D...

We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the HardyLittlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing L∞x (R ) by BMOx(R) and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with...

Abstract In this paper, we study the resonance problem of a class of singular quasilinear parabolic equations with respect to its higher near-eigenvalues. Under a generalized Landesman-Lazer condition, it is proved that the resonance problem admits at least one nontrivial solution in weighted Sobolev spaces. The proof is based upon applying the Galerkin-type technique, the Brouwer’s fixedpoint ...

In this paper we extend the technique of computer{assisted proofs to xed point problems in Sobolev spaces. Up to now the method was limited to the case of spaces of analytic functions. The possibility to work with Sobolev spaces is an important progress and opens up many new domains of applications. Our discussion is centered around a concrete problem that arises in the theory of critical pheno...

In our recent paper [12] we developed a new principle of “symmetrization by truncation” to obtain symmetrization inequalities of Sobolev type via truncation. In this note we consider the corresponding results for Sobolev spaces on domains, without assuming that the Sobolev functions vanish at the boundary. The explicit connection between Sobolev-Poincaré inequalities and isoperimetric inequalit...