Related papers: On a Weighted Singular Integral Operator with Shif…

For $\alpha\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d} [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}}$$ and for $\alpha\in (0,1)$ we consider the same operator but where the $\nabla f$…

Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\mathbb C}$ and state space $H$. The scattering (or impulse response) functions $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator…

We investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{\alpha }u(t)+Au(t)=f(u(t)), & 1<\alpha <2, \\ u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}% \end{equation*}% where…

We prove necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions…

Let $\mathrm{Lip}(X)$, $\mathrm{Lip}^b(X)$, $\mathrm{Lip}^{\mathrm{loc}}(X)$ and $\mathrm{Lip}^\mathrm{pt}(X)$ be the vector spaces of Lipschitz, bounded Lipschitz, locally Lipschitz and pointwise Lipschitz (real-valued) functions defined…

We study the following Cauchy problems for semi-linear structurally damped $\sigma$-evolution models: \begin{equation*} u_{tt}+ (-\Delta)^\sigma u+ \mu (-\Delta)^\delta u_t = f(u,u_t),\, u(0,x)= u_0(x),\, u_t(0,x)=u_1(x) \end{equation*}…

We consider a singular Schr\"odinger operator in $L^2(\mathbb{R}^2)$ written formally as $-\Delta - \beta\delta(x-\gamma)$ where $\gamma$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the…

We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ of slowly oscillating functions of limited smoothness introduced in \cite{K09}. We show…

We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are…

In this note, we present a characterization of semistable unitary operators on $L^2(\mathbb{R})$, under the assumption that the operator is (i) translation-invariant, (ii) symmetric, and (iii) locally uniformly continuous (LUC) under…

The first goal of this paper is to establish the existence of a positive solution for the singular boundary value problem (1.1), where $\mathcal{B}$ is a general boundary operator of Dirichlet, Neumann or Robin type, either classical or…

Let $U:[0,\infty)^2 \to [0,\infty)$ be a~measurable kernel satisfying: (i) $U(x,y)$ is nonincreasing in $x$ and nondecreasing in $y$; (ii) there exists a~constant $\theta>0$ such that $U(x,z) \le \theta\left( U(x,y)+U(y,z) \right)$ for all…

This article consists of two connected parts. In the first part, we study the shift invariant subspaces in certain $\mathcal{P}^2(\mu)$-spaces, which are the closures of analytic polynomials in the Lebesgue spaces $\mathcal{L}^2(\mu)$…

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent)…

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is…

Let $\,I\subseteq\mathbb{R}\,$ be an interval and $\,\beta:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a unique fixed point $\,s_0\in I\,$ which satisfies $\,(s_0-t)(\beta(t)-t)\geq 0\,$ for all $\,t\in I$, where…

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \…

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the…

We prove weighted strong $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators. For the first family of operators the weights used can be either Sawyer's one-sided $A^+_p$ weights or Muckenhoupt's…

We solve a long-standing open problem in theory of weighted inequalities concerning iterated Copson operators. We use a constructive approximation method based on a new discretization principle that is developed here. In result, we…