Related papers: Weighted $L^2$ theory for the Euclidean Dirac oper…

In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-\Delta_p u= f(x) F(u) $ in $\mathbb{R}^n$ where $p\ge 2$, $0\le f\in C(\mathbb{R}^n)$ and $F\in…

A characterization is obtained for those pairs of weights $v$ and $w$ on $\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\mathbb{R}^2_+)$ to…

We derive bounds on the location of non-embedded eigenvalues of Dirac operators on the half-line with non-Hermitian $L^1$-potentials. The results are sharp in the non-relativistic or weak-coupling limit. In the massless case, the absence of…

The properties of highest-weight representations of the N=2 superconformal algebra in two dimensions can be considerably simplified when re-expressed in terms of relaxed ^sl(2) representations. This applies to the appearance of submodules…

We extend the work of Hellerman (arxiv:0902.2790) to derive an upper bound on the conformal dimension $\Delta_2$ of the next-to-lowest nontrival primary operator in unitary two-dimensional conformal field theories without chiral primary…

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem \begin{equation}\label{ab eqn} \begin{cases} \partial_t u(t,x)=\psi(t,-i\nabla)u(t,x)+f(t,x),\quad &(t,x)\in(0,T)\times\mathbb{R}^d,\\…

Let $L:=-\Delta+V$ be the Schr\"{o}dinger operator on $\mathbb{R}^n$ with $n\geq 3$, where $V$ is a non-negative potential which belongs to certain reverse H\"{o}lder class $RH_q(\mathbb{R}^n)$ with $q\in (n/2,\,\infty)$. In this article,…

In this paper we offer alternate upper bound for the operator $\Pi_b^*\Pi_d$ to the ones present in literature, thus extending the known upper bounds from the $L^2(\mathbb{R})$ setting to $L^p(w)$, for $1<p<\infty,$ and a Muckenhoupt weight…

We study higher regularity for weak solutions of the $p$-Laplace equation $-\Delta_p u = f$ in a domain $\Omega \subset \mathbb{R}^n$ for $p$ sufficiently close to 2. For $m \ge 3$, assuming that $f$ satisfies suitable Sobolev and H\"older…

In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class $M_{\rho, \Lambda}^m(\mathbb{ T}\times \mathbb{Z})$ (associated to a suitable weight function $\Lambda$ on…

We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in…

Potential algebras can be used effectively in the analysis of the quantum systems. In the article, we focus on the systems described by a separable, 2x2 matrix Hamiltonian of the first order in derivatives. We find integrals of motion of…

We show that the Lusin area integral or the square function on the unit ball of $\C^n$, regarded as an operator in weighted space $L^2(w)$ has a linear bound in terms of the invariant $A_2$ characteristic of the weight. We show a…

In this article, we consider a higher-order elliptic equation with nonsmooth coefficients with respect to Orlicz spaces on the domain $\Omega\subset\mathbb{R}^{n}$. The separable subspace of this space is distinguished in which infinitely…

The main goal of this paper is to prove a two-weight criteria for multidimensio-nal Hardy type operator from weighted Lebesgue spaces into $p$-convex weighted Banach function spaces. Analogously problem for the dual operator is considered.…

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If ${\mathcal L}_0=\mbox{div}…

The p-Laplace operator in the entire N-dimensional Euclidean space, subject to external electromagnetic potentials, is investigated. In the general case 1<p<N, the existence of at least one solution of mountain pass type to a weighted…

We give a min-max characterization of the weighted Dirac eigenvalues, and show that the weighted eigenvalues and eigenspaces of Dirac operators are continuous with respect to weak $L^p$ convergence of the inverse weight, for any $p>n$.…

This work is concerned with extending the results of Calder\' on and Vaillancourt proving the boundedness of Weyl pseudo differential operators Op_h^{weyl} (F) in L^2(\R^n). We state conditions under which the norm of such operators has an…

By H\"ormander's $L^2$-m\'ethode, we study some operators in the Hilbert space of weight $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$. We prove in each case of operator the existence of its inverse which is also a bounded operator.