Related papers: Local $L^2$ results for $\bar\partial$: The isolat…

We study the Cauchy problem for the non-linear Schr\"odinger equation with singular potentials. For point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L^{p} spaces. Specific…

We discuss criteria for a self-adjoint operator on L^2(X) to have empty essential spectrum. We state a general result for the case of a locally compact abelian group X and give examples for X=R^n.

In this note I give a description of Lyubeznik's local cohomology invariants for a certain natural class of local rings, namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity. This…

A bounded linear operator between Banach spaces is called {\it completely continuous} if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous…

We study an abstract equation in a reflexive Banach space, depending on a real parameter $\lambda$. The equation is composed by homogeneous potential operators. By analyzing the Nehari sets, we prove a bifurcation result. In some particular…

In this article, we consider metrically thin singularities A of the tangential Cauchy-Riemann operator on smoothly embedded Cauchy-Riemann manifolds M. The main result states removability within the space of locally integrable functions on…

A set of bounded linear operators from a Banach space to a Banach lattice is collectively L-weakly compact whenever union of images of the unit ball is L-weakly compact. We extend the Meyer-Nieberg duality theorem to collectively L-weakly…

In this work we consider the Cauchy problem for the cubic Schr\"odinger equation posed on cylinder $\mathbb{R}\times\mathbb{T}$ with fractional derivatives $(-\partial_y^2)^{\alpha},\, \alpha >0$, in the periodic direction. The spatial…

We study Dirichlet forms defined by nonintegrable L\'evy kernels whose singularity at the origin can be weaker than that of any fractional Laplacian. We show some properties of the associated Sobolev type spaces in a bounded domain, such as…

This paper is motivated by a claim in the classical textbook of Muskhelishvili concerning the Cauchy singular integral operator $S$ on H\"older functions with parameters. To the contrary of the claim, a counter example was constructed by…

Let X be an irreducible n-dimensional projective variety in CP^N with arbitrary singular locus. We prove that the L2-(p,1)-d-bar cohomology groups (with respect to the Fubini-Study metric) of the regular part of X are finite dimensional.

We introduce a trick of dealing with $L^2$ estimates of $\bar{\partial}$ with singular weights on complete K\"ahler domains.

Let $\varphi: \mathbb{P}^{n}_{F} \to \mathbb{P}^{n}_{F}$ where $F$ is a complete valued field. If $x$ is a fixed point, such that the action of $\varphi$ on $T_{x}$ has eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$, with $\lambda_{1},…

Consider an operator equation $F(u)=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. If $F$ is monotone $C^2_{loc}(H)$ operator, then we construct…

We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on L^2(R) which satisfy them. These operators arise from the creation operator via a functional calculus involving…

Let $E \subset \mathbb{C}$ be a Borel set such that $0<\mathcal{H}^1(E)<\infty$. David and L\'eger proved that the Cauchy kernel $1/z$ (and even its coordinate parts $\textrm{Re}\, z/|z|^2$ and $\textrm{Im}\, z/|z|^2$, $z\in…

We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schr\"{o}dinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the…

We consider the $L^2$-boundedness of the solution itself of the Cauchy problem for wave equations with time-dependent wave speeds. We treat it in the one-dimensional Euclidean space. To study these, we adopt a simple multiplier method by…

This is a survey article about $L^2$ estimates for the $\bar \partial$ operator. After a review of the basic approach that has come to be called the "Bochner-Kodaira Technique", the focus is on twisted techniques and their applications to…

Let D be a bounded domain in n-dimensional Eucledian space with a smooth boundary. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex {A_i} of first…