Related papers: Fundamental solutions of nonlocal H\"ormander's op…

We show global and interior higher-order log-H\"older regularity estimates for solutions of Dirichlet integral equations where the operator has a nonintegrable kernel with a singularity at the origin that is weaker than that of any…

In this paper, we study the continuous dependence of the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation \[iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb…

Let $L= -\Delta+ V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative potential, $V\ne 0$, and belongs to the reverse H\"older class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a…

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob}…

We establish the existence and uniqueness, in bounded as well as unbounded Lipschitz type cylinders of the forms $U_X\times V_{Y,t}$ and $\Omega\times \mathbb R^{m}\times \mathbb R$, of weak solutions to Cauchy-Dirichlet problems for the…

We consider fractional operators of the form $$\mathcal{H}^s=(\partial_t -\mathrm{div}_{x} ( A(x,t)\nabla_{x}))^s,\ (x,t)\in\mathbb R^n\times\mathbb R,$$ where $s\in (0,1)$ and $A=A(x,t)=\{A_{i,j}(x,t)\}_{i,j=1}^{n}$ is an accretive,…

We consider the linear integro-differential operator $L$ defined by \[ Lu(x) =\int_\Rn (u(x+y) - u(x) - 1_{[1,2]}(\alpha) 1_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x)) k(x,y) \sd y . \] Here the kernel $k(x,y)$ behaves like $|y|^{-d-\alpha}$,…

In this paper we obtain the weak type (1,1) boundedness of Calderon-Zygmund operators acting over operator-valued functions. Our main tools for its solution are a noncommutative form of Calderon-Zygmund decomposition in conjunction with a…

We study the existence of a large solution to a semilinear problem in a bounded open $C^{1,1}$ set for a class of nonlocal operators obtained by an appropriate subordination of the Laplacian. These operators are classical generalisations of…

Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at the underlying Bessel-type operators with more…

In this paper, we systematically study the Fefferman-Stein inequality and Coifman-Fefferman inequality for the general commutators of singular integral operators that satisfy H\"{o}rmander conditions of Young type. Specifically, we first…

We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators…

We investigate a general parabolic initial-boundary value problem with zero Cauchy data in some anisotropic H\"ormander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate…

This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type \begin{equation*} \left\{\; \begin{aligned} -\Delta_{\Phi_{1}} u&=F_u(x,u,v)+\lambda R_u(x,u,v)\;\text{ in } \Omega& \\…

In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =f(x,u)+g(x) , \ \mbox{in} \ \Omega, \] where $\Omega \subset \mathbb{R}^{N}$, $N\geq 1$, is a bounded connected open, $g \in…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(\Omega)$ and to…

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which…

Let $\mathbb{B}^d$ be the unit ball on the complex space $\mathbb{C}^d$ with normalized Lebesgue measure $dv.$ For $\alpha\in\mathbb{R},$ denote $k_\alpha(z,w)=\frac{1}{(1-\langle z,w\rangle)^\alpha},$ the Bergman-type integral operator…

We investigate expansive Hilbert space operators $T$ that are finite rank perturbations of isometric operators. If the spectrum of $T$ is contained in the closed unit disc $\overline{\mathbb{D}}$, then such operators are of the form $T=…