Related papers: Explicit fundamental solution for the operator $L+…

We examine unitary and nonunitary representations of the Heisenberg-Weyl Lie algebra $\mathfrak{hw}_n$, with particular emphasis on tensor products of unitary representations and on indecomposable nonunitary representations. In the unitary…

Let M be a compact manifold without boundary. Associated to a metric g on M there are various Laplace operators, for example the de Rham Laplacian on forms and the conformal Laplacian on functions. For a general Laplace type operator we…

The periodic unfolding method is one of the latest tool after multi-scale convergence to study multi-scale problems like homogenization problems. It provides a good understanding of various micro scales involved in the problem which can be…

Let $B(H)$ be the algebra of all bounded operators on a Hilbert space $H$. Let $T=V|T|$ be the polar decomposition of an operator $T\in B(H)$. The mean transform of $T$ is defined by $M(T)=\frac{T+|T|V}{2}$. In this paper, we discuss…

We introduce a generalization of the Heisenberg algebra which is written in terms of a functional of one generator of the algebra, $f(J_0)$, that can be any analytical function. When $f$ is linear with slope $\theta$, we show that the…

Let $U$ be an algebraic subgroup of the group of $n\times n$ upper-triangular matrices with units on the diagonal over a finite field of large enough characteristic, and $\mathfrak{n}$ be the Lie algebra of $U$. The main tool in…

Given a singular filtration on a manifold, e.g. a subriemannian setting, one can understand the elliptic regularity problems through a special kind of calculus. The principal symbol in this calculus involves the unitary representations of a…

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \,…

In this paper, we introduce $m$-subharmonic functions in quaternionic space $\mathbb{H}^{n}$, we define the quaternionic Hessian operator and solve the homogeneous Dirichlet problem for the quaternionic Hessian equation on the unit ball…

Given a bounded linear operator $T$ on separable Hilbert space, we develop an approach allowing one to construct a matrix representation for $T$ having certain specified algebraic or asymptotic structure. We obtain matrix representations…

Let H be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let u be a semi-concave function defined on M. If E (u) is the set of all the super-differentials of u and (\phi t) the Hamiltonian flow…

A quadratic Lie conformal algebra corresponds to a Hamiltonian pair in \cite{GD}, which plays fundamental roles in completely integrable systems. Moreover, it also corresponds to certain compatible pairs of a Lie algebra and a Novikov…

For the Lie algebras $g_n= \mathfrak{o}_{2n+1},\mathfrak{sp}_{2n},\mathfrak{o}_{2n}$ a simple construction of a base in an irreducible representation is given. The construction of this base uses the method of $Z$-invariants of Zhelobenko…

Let $\lambda$ be a complex number in the closed unit disc $\overline{\Bbb D}$, and $\cal H$ be a separable Hilbert space with the orthonormal basis, say, ${\cal E}=\{e_n:n=0,1,2,\cdots\}$. A bounded operator $T$ on $\cal H$ is called a {\em…

We prove the existence of a global fundamental solution $\Gamma(x;y)$ (with pole $x$) for any H\"ormander operator $\mathcal{L}=\sum_{i=1}^m X_i^2$ on $\mathbb{R}^n$ which is $\delta$-homogeneous of degree $2$. By means of a global Lifting…

We consider the possibility of adding a Grassmann-odd function \nu to the odd Laplacian. Requiring the total \Delta operator to be nilpotent leads to a differential condition for \nu, which is integrable. It turns out that the odd function…

In our paper~\cite{KR} we began a systematic study of representations of the universal central extension $\widehat{\Cal D}\/$ of the Lie algebra of differential operators on the circle. This study was continued in the paper~\cite{FKRW} in…

In this paper we continue the study of the superconformal index of four-dimensional $\mathcal{N}=2$ theories of class $\mathcal{S}$ in the presence of surface defects. Our main result is the construction of an algebra of difference…

We prove an entanglement principle for fractional Laplace operators on $\mathbb R^n$ for $n\geq 2$ as follows; if different fractional powers of the Laplace operator acting on several distinct functions on $\mathbb R^n$, which vanish on…

We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-\alpha q},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+\alpha}}…