Abstract. The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x 1 = 0 and x n = 0 are applied to functions belonging to quasi-homogeneous, mixed-norm Lizorkin-Triebel spaces F s, a p,q ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x 1 = 0. For x 1 the trace spaces are proved to be mixed-norm Lizorkin-Triebel spaces with a specific sum exponent; for x n they are similarly defined Besov spaces. The treatment includes continuous right-inverses and higher order traces. The results rely on a sequence version of Nikol ′ skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria.
This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax-Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.
It is shown that para-multiplication applies to a certain product n(u, v) defined for appropriate u and v in Y(IR"). Boundedness of n(., .) is investigated for the anisotropic Besov and Triebel-Lizorkin spaces -i.e., for Bf$S and FE*,S with s E R and p and q in 10, a] (though p < 03 in
The Boutet de Monvel calculus of pseudo-differential boundary operators is generalised to the scales of Besov and Triebel-Lizorkin spaces, B s p,q and F s p,q , with s ∈ R and p and q ∈ ]0, ∞] (though with p < ∞ for the F s p,q spaces). The continuity and Fredholm properties proved here extend those in [Fra86a] and [Gru90], and the results on range complements of surjectively elliptic Green operators improve the earlier known, even for the classical spaces with 1 < p < ∞.The symbol classes treated are the x-uniformly estimated ones. On R n + a trace operator T and a singular Green operator G, both of class 0, are defined in general to be (Γ) M to the space D ′ (Ω) N ′ × D ′ (Γ) M ′ , then both T and P Ω + G have class ≤ r − 1.
A study is made of a recent integral identity of B. Helffer and J. Sjöstrand, which for a not yet fully determined class of probability measures yields a formula for the covariance of two functions (of a stochastic variable); in comparison with the Brascamp-Lieb inequality, this formula is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using a fine version of the Closed Range Theorem, the identity's validity is shown to be equivalent to some explicitly given spectral properties of Witten-Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp-Lieb's inequality is obtained. For a certain class of measures occuring in statistical mechanics, explicit criteria for the Witten-Laplacians are found from the Persson-Agmon formula, from compactness of embeddings and from the Weyl calculus, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.In the proofs of Helffer and Sjöstrand, Φ has had a rather specific nature, e.g. with Φ ′′ jk bounded and3 below. Formula (1.2) has also been used by A. Naddaf and T. Spencer [NS97], V. Bach, T. Jecko and J. Sjöstrand [BJS98], B. Helffer [Hel97b] and others. Indirectly A 0 , A 1 appeared earlier in [Sjö93, HS94, Hel95].Concerning formula (1.2), it should be noted that g j only enters in the covariance through Pg j := g j − g j dµ , which is the orthogonal projection onto L 2 (µ) ⊖ C, i.e. onto the complement of the constant functions. Because the gradient provides another means to remove the part of g j lying in C, it is natural to have ∇g 1 and ∇g 2 on the right hand side of (1.2) and to have an inverse of a second order differential operator, like A −1 1 , to counteract the gradients. Viewed thus, (1.2) may seem plausible, and this article presents a study of it, resulting first of all in more general sufficient conditions for the formula; secondly, conditions that are equivalent to (1.2) are given at an abstract level (although these two improvements are formally substantial, the consequences for statistical mechanics are to be investigated). Thirdly a more systematic and streamlined approach to (1.2) is presented.Remark 1.1. For the general importance of formula (1.2), recall that for a stochastic variable X : Ω → R n with distribution µ on R n , the left hand side of (1.2) equals cov(g 1 (X ), g 2 (X )). So when µ is of the type treated here, then (1.2) provides a formula also for such covariances. However, this is clear, and hence this consequence shall not be treated below.1.2. Summary. In this paper various -abstract and explicit -conditions are deduced for (1.2). To give an application of these, (1.1) is derived in the general strictly convex case from (1.2) and moreover extended to the case f ∈ H 1 loc (R n ) ∩ L 2 (µ); this supplements an explanation of B. Helffer of how (1.2) implies the validity of (1.1) for certain f...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.