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We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of $2s$-stable processes and exterior data, inhomogeneity in weighted $L^2$-spaces. This class of operators includes the fractional Laplacian.…

This is the continuation of previous article. For subspaces $M^n(t)$ and $M^{n-m}(t)$ which are invariant manifolds of the differential equation under consideration we build a change of variables which splits this equation into a system of…

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local…

For $1\le t < \infty$, a compact subset $K\subset\mathbb C$, and a finite positive measure $\mu$ supported on $K$, $R^t(K, \mu)$ denotes the closure in $L^t(\mu)$ of rational functions with poles off $K$. Let $\text{abpe}(R^t(K, \mu))$…

We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach does not require prior knowledge about the eigenfunctions,…

We establish the existence of a regular functional $M$-position, in the sense of Pisier, for geometric log-concave functions. This provides a functional analogue of Pisier's regular $M$-positions for convex bodies and yields uniform control…

Let f:X-->R be a function defined on a connected nonsingular real algebraic set X in R^n. We prove that regularity of f can be detected on either algebraic curves or surfaces in X. If dimX>1 and k is a positive integer, then f is a regular…

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk $\DD$ and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue…

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds…

Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}_{\mathcal{R}}(\vec{f})(x)=\sup_{\substack{R \ni x…

We find an explicit integral formula for the eigenfunctions of a fourth order differential operator against the kernel involving two Bessel functions. Our formula establishes the relation between K-types in two different realizations of the…

This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the…

It is proved that given $-1/2<s<1/2$, for any $f\in L^2(\mathbb{R})$, there is a unique $u\in \widehat{H}^{|s|}(\mathbb{R})$ such that $$ f=\boldsymbol{D}^{-s}u+\boldsymbol{D}^{s*}u\,, $$ where $\boldsymbol{D}^{-s}, \boldsymbol{D}^{s*}$ are…

In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…

We discuss the requirement of single valuedness and periodicity of eigenfunction of the third component of the operator of angular momentum. This condition, imposed on a non observable, is often used to derive that the eigenvalues of…

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is…

Consider the Dirichlet-Laplacian in $\Omega:= (0,L)\times (0,H)$ and choose another open set $\omega\subset \Omega$. The estimate $0<C_{\omega}\leq R_{\omega}(u):=\frac{\Vert u\Vert^{2}_{L^{2}(\omega)}}{\Vert u\Vert^{2}_{L^{2}(\Omega)}}\leq…

We consider elliptic differential operators on either the entire Euclidean space $\mathbb{R}^d$ or on subsets consisting of a cube $\Lambda_L$ of integer length $L$. For eigenfunctions of the operator, and more general solutions of elliptic…

We study linear boundary-value problems for systems of first-order ordinary differential equations with the most general boundary conditions in the normed spaces of continuously differentiable functions on a finite closed interval. The…

Given an irrational rotation $T$ on $\M T$ we settle necessary and sufficient conditions on a step function $\phi$ and $t\in \M T$ for the existence of measurable solutions to the cohomogical equation $$\exp{(2i\pi\phi)}=\e{2i\pi t}f/f\rond…