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The aim of this paper is to study $L^p$-boundedness property of the pseudo differential operator associated with a symbol, on rank one Riemannian symmetric spaces of noncompact type, where the symbol satisfies H\"ormander-type conditions…
An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with…
We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $n\in \mathbb N$…
The main goal of this paper is to give potential theoretical approach to study the Dunkl Laplacian $\Delta_k$ which is a standard example of differential-difference operators. By introducing the Green kernel relative to $\Delta_k$, we prove…
Based on the tile discretization elaborated by the author in "The Polynomial Carleson Operator", we develop a Calderon-Zygmund type decomposition of the Carleson operator. As a consequence, through a unitary method that makes no use of…
In this paper we discuss compactness of the canonical solution operator to d-bar on weigthed L^2- spaces on C^n. For this purpose we apply ideas which were used for the Witten Laplacian in the real case and various methods of spectral…
We deal with a real valued integral operator L of Laplace transformation type acting between Lebesgue spaces on the semi-axis. Sufficient conditions for belonging L to Schatten type classes are obtained. Some upper asymptotic estimates for…
Operator systems are the unital self-adjoint subspaces of the bounded operators on a Hilbert space. Complex operator systems are an important category containing the C*-algebras and von Neumann algebras, which is increasingly of interest in…
We consider the Laplacian in $\mathbb{R}^n$ perturbed by a finite number of distant perturbations those are abstract localized operators. We study the asymptotic behaviour of the discrete spectrum as the distances between perturbations tend…
Let $D\subset \mathbb{R}^d$ be a bounded Lipschitz domain, $\omega$ be a high order modulus of continuity and let $T$ be a convolution Calder\'{o}n-Zygmund operator. We characterize the bounded restricted operators $T_D$ on the Zygmund…
We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with…
We study elliptic and parabolic problems governed by singular elliptic operators \begin{equation*} \mathcal L =\sum_{i,j=1}^{N+1}q_{ij}D_{ij}+\frac c y D_y \end{equation*} in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N,…
We continue our study of topological partial *algebras, focusing our attention to the interplay between the various partial multiplications. The special case of partial *-algebras of operators is examined first, in particular the link…
Let $(V,\omega)$ be an orthosympectic $\mathbb Z_2$-graded vector space and let $\mathfrak g:=\mathfrak{gosp}(V,\omega)$ denote the Lie superalgebra of similitudes of $(V,\omega)$. When the space $\mathscr P(V)$ of superpolynomials on $V$…
We consider differential-algebraic equations in infinite dimensional state spaces and study, under which conditions we can associate a $C_{0}$-semigroup with such equations. We determine the right space of initial values and characterise…
We study the operator $L=-\Delta+q$ on a bounded domain $\Omega\subset\mathbb R^n$, where $q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well--defined with Dirichlet and…
In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of…
We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary…
We characterize bounded multiplication operators in weighted Dirichlet spaces that are power bounded, Ces\`{a}ro bounded and uniformly Kreiss. Moreover, we show the equivalence in such spaces between mean ergodicity and Ces\`{a}ro…
In the first part, we give an explicit description of the cotangent complex of differential graded (dg) operads, modeled as an operadic infinitesimal bimodule. This leads to a uniform formula for the Quillen cohomology of their associated…