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In this paper, bilinear pseudo‐differential operators with symbols in the bilinear Hörmander class BS0,0m$BS^m_{0,0}$ are considered. In particular, the boundedness of these operators on Sobolev spaces is established. Our main result is proved by using symbolic calculus and the boundedness of those operators with certain S0, 0‐type symbols on Lebesgue spaces. [ABSTRACT FROM AUTHOR]

We introduce a new class of smooth pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity with a finite loss of derivatives of certain systems of pseudodifferential operators is stable under perturbations by lower order systems of pseudodifferential operators whose order depends on the loss of derivatives. We also present some applications. [ABSTRACT FROM AUTHOR]

This paper presents an investigation on the global hypoellipticity problem for a class of systems of pseudo‐differential operators on the torus. The approach consists in establishing conditions on the matrix symbol of the system such that it can be transformed into a suitable triangular form involving a nilpotent upper triangular matrix. Hence, the global hypoellipticity is studied by analyzing the behavior of the eigenvalues and their averages. [ABSTRACT FROM AUTHOR]

We establish the notion of a "projective analytic vector", whose defining requirements are weaker than the usual ones of an analytic vector, and use it to prove generation theorems for one‐parameter groups on locally convex spaces. More specifically, we give a characterization of the generators of strongly continuous one‐parameter groups which arise as the result of a projective limit procedure, in which the existence of a dense set of projective analytic vectors plays a central role. An application to strongly continuous Lie group representations on Banach spaces is given, with a focused analysis on concrete algebras of functions and of pseudodifferential operators. [ABSTRACT FROM AUTHOR]

In this paper, a bisingular pseudodifferential calculus, along the lines of the one introduced by L. Rodino in his paper of 1975, is developed in the global setting of a product of compact Lie groups. The approach follows that introduced by M. Ruzhansky and V. Turunen in their book of 2010 (see also V. Fischer's paper of 2015), in that it exploits the harmonic analysis of the groups involved. [ABSTRACT FROM AUTHOR]

We study continuity and Hölder continuity of t↦Ptf$t\mapsto P_tf$, where Pt$P_t$ is a generalized Mehler semigroup in Cb(X)$C_b(X)$, the space of the continuous and bounded functions from a Banach space X to R$\mathbb {R}$, and f∈Cb(X)$f\in C_b(X)$. The generators L of such semigroups are realizations of a class of differential and pseudo‐differential operators, both in finite and in infinite dimension. Examples of operators L to which this theory is applicable include Ornstein–Uhlenbeck operators with fractional diffusion in finite dimension, and Ornstein–Uhlenbeck operators with associated strong‐Feller semigroups, in infinite dimension. [ABSTRACT FROM AUTHOR]

In a recent paper O. Gannot and M. Wrochna considered the Klein–Gordon equation on an asymptotically anti‐de Sitter spacetime subject to Robin boundary conditions, proving in particular a propagation of singularities theorem. In this work we generalize their result considering a more general class of boundary conditions implemented on the conformal boundary via pseudodifferential operators of suitable order. Using techniques proper of b‐calculus and of twisted Sobolev spaces, we prove also for the case in hand a propagation of singularity theorem along generalized broken bicharacteristics, highlighting the potential presence of a contribution due to the pseudodifferential operator encoding the boundary condition. [ABSTRACT FROM AUTHOR]

We use Operator Ideals Theory and Gershgorin theory to obtain explicit information in terms of the symbol concerning the spectrum of pseudo‐differential operators, on a smooth manifold Ω with boundary ∂Ω, in the context of the non‐harmonic analysis of boundary value problems introduced in [29] in terms of a model operator L. For symbols in the Hörmander class S1,00(Ω¯×I), we provide a non‐harmonic version of Gohberg's lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo‐differential operator is a compact operator in L2(Ω), or a Riesz operator in Lp(Ω) in the case of Riemannian manifolds with smooth boundary. We extend to the context of the non‐harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations. [ABSTRACT FROM AUTHOR]

In this paper we investigate the nuclear trace of vector‐valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo‐differential operators. First we characterise the nuclearity of pseudo‐differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector‐valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector‐valued Fourier multiplier in terms of its operator‐valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic Hörmander classes. [ABSTRACT FROM AUTHOR]

In this paper we prove sufficient conditions for the Fredholm property of a non‐smooth pseudodifferential operator P which symbol is in a Hölder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol‐smoothing. [ABSTRACT FROM AUTHOR]

Let H be a self‐adjoint isotropic elliptic pseudodifferential operator of order 2. Denote by u(t) the solution of the Schrödinger equation (i∂t−H)u=0 with initial data u(0)=u0. If u0 is compactly supported the solution u(t) is smooth for small t>0, but not for all t. We determine the wavefront set of u(t) in terms of the wavefront set of u0 and the principal and subprincipal symbol of H. [ABSTRACT FROM AUTHOR]

Abstract: We study new weighted estimates for the 2‐fold product of Hardy–Littlewood maximal operators defined by M ⊗ ( f , g ) : = M f M g. This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón–Zygmund operators, the bilinear Hardy–Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures ∥ f g ∥ L p , ∞ w 1 p / p 1 w 2 p / p 2 ≤ C ∥ f ∥ L p 1 , ∞ ( w 1 ) ∥ g ∥ L p 2 , ∞ ( w 2 ) .Unfortunately, we shall prove that this inequality does not hold, in general, and we shall have to consider a weaker version of it. [ABSTRACT FROM AUTHOR]

Abstract: We study minimal energy problems for strongly singular Riesz kernels | x − y | α − n, where n ≥ 2 and α ∈ ( − 1 , 1 ), considered for compact ( n − 1 )‐dimensional C ∞‐manifolds Γ immersed into R n. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such minimization problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator of order β = 1 − α on Γ. The measures with finite energy are shown to be elements from the Sobolev space H β / 2 ( Γ ), 0 < β < 2, and the corresponding minimal energy problem admits a unique solution. We relate our continuous approach also to the discrete one, which has been worked out earlier by D. P. Hardin and E. B. Saff. [ABSTRACT FROM AUTHOR]

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces. Moreover, we deduce composition and certain invariance properties of these classes. [ABSTRACT FROM AUTHOR]

We consider a boundary problem over an exterior subregion of [ABSTRACT FROM AUTHOR]

In this work we consider the η-invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the η-function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the K-theory of the algebra of 0-order (global) bisingular operators. With these preparations we establish the regularity properties of the η-function at the origin for global bisingular operators which are self-adjoint, elliptic and of positive orders. [ABSTRACT FROM AUTHOR]

We prove boundedness of pseudodifferential operators on anisotropic mixed-norm Besov and Triebel-Lizorkin spaces. Our proof relies only on general maximal function estimates and provides a new perspective even in the case of spaces without mixed norms. Moreover, we cover the case of Fourier multipliers on the above mentioned spaces. As application we establish boundedness of pseudodifferential operators and Fourier multipliers on anisotropic mixed-norm Sobolev spaces. [ABSTRACT FROM AUTHOR]

A table of contents for the 2016 issue of the periodical "Mathematische Nachrichten" is presented.

In the first part of the paper we study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fréchet space [ABSTRACT FROM AUTHOR]

In this work we show that if [ABSTRACT FROM AUTHOR]

We compute the K-groups of the [ABSTRACT FROM AUTHOR]

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG-pseudodifferential operators on [ABSTRACT FROM AUTHOR]

In [ABSTRACT FROM AUTHOR]