The present paper discusses the relations between regularity, Dirichlet, and Neumann problems. Among other results, we prove that the solvability of the regularity problem does not imply the solvability of the dual Dirichlet problem for general elliptic operators with complex bounded measurable coefficients. This is strikingly different from the case of real operators, for which such an implica...

We solve the Kato problem for divergence form elliptic operators whose heat kernels satisfy a pointwise Gaussian upper bound. More precisely, given the Gaussian hypothesis, we establish that the domain of the square root of a complex uniformly elliptic operator L = −div(A∇) with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate ‖ √ Lf‖2 ∼ ‖∇f‖2...

This paper studies the relationship between the singular trace of a weak trace class operator and the asymptotic behaviour of its ζ-function at its leading singularity. For Dixmier measurable and universally measurable operators we describe their measurability in terms of the behaviour of the ζ-function. We use recent advances in singular trace theory and a new approach based on Tauberian theor...

We discuss the division method for subspectra which appears to be one of the key approaches in the study of spectral properties of self-adjoint differential vector-operators, that is operators generated as a direct sum of self-adjoint extensions on an Everitt-Markus-Zettl multi-interval system. In the current work we show how the division method may be applied to obtain the ordered spectral rep...

We study non-autonomous observation systems , where is a strongly measurable family of closed operators on Banach space and bounded from to . Based an abstract uncertainty principle dissipation estimate, we prove that the system satisfies final-state observability estimate in for subsets present applications above result families uniformly elliptic differential as well Ornstein–Uhlenbeck with I...

We prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables. The result is sharp, even for smooth amplitudes. Nevertheless, in the case when the amplitude contains the oscillatory factor ξ 7→ ei|ξ| , the result can be substantially improved. We extend the Lpboundedness of pseudo-pseudodifferential operators to certain w...

Theoretical predictions of several measurable quantities, which are relevant in K-, Dand Bmeson phenomenology, depend on the matrix elements of some ∆F = 2 four-fermion operators. Examples are given by FCNC effects in SUSY extensions of the Standard Model [1]-[2], by the B s– B̄ s width difference [3] and by the O(1/m 3 b) corrections in inclusive b-hadron decay rates (which actually depend on t...

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in [4] and [3] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are establishe...