Related papers: Sobolev and Schatten Estimates for the Complex Gre…
The purpose of this paper is to describe certain CR-covariant differential operators on a strictly pseudoconvex CR manifold $M$ as residues of the scattering operator for the Laplacian on an ambient complex K\"{a}hler manifold $X$ having…
We establish a sharp Sobolev trace inequality on the Siegel domain $\Omega_{n+1}$ involving the weighted norm-$W^{2,2}(\Omega_{n+1}, \rho^{1-2[\gamma]})$. The inequality is closely related the realization of fractional powers of the…
In this paper we consider conformal spectral estimates of the Dirichlet-Laplace operator in conformal regular domains $\Omega \subset \mathbb R^2$. This study is based on the geometric theory of composition operators on Sobolev spaces that…
The Hamiltonian of a Coulomb plus polynomial potential on the Coulomb-Sturmian basis has an infinite symmetric band-matrix structure. A band matrix can always be considered as a block-tridiagonal matrix. So, the corresponding Green's…
We discuss kernels on complact classical groups $G=\U(n)$, $\OO(2n)$, $\Sp(n)$ defined by the formula $K(z,u)=|\det(1-zu^*)|^s$. We obtain the explicit Plancherel formula for these kernels and the interval of positive-definiteness. We also…
In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/\lambda_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower…
We provide Green's function estimates for parabolic operators on polyhedrons and polyhedral cones in $\mathbb{R}^3$. These estimates incorporate mixed weights, which include appropriate powers of the distances to the vertices, the edges,…
A method for the computation of scattering data and of the Green function for the one-dimensional Schr\"{o}dinger operator $H:=-\frac{d^2}{dx^2}+q(x)$ with a decaying potential is presented. It is based on representations for the Jost…
In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace-Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such…
In this paper, we characterize all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral bi-Carleman operator in $L_2(R)$ with bounded and arbitrarily smooth kernel on $R^2$. In addition, we give…
Assume that $M$ is a CR compact manifold without boundary and CR Yamabe invariant $\mathcal{Y}(M)$ is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows \begin{equation*} \Bigl| \int_M\int_M…
We find that correlation functions at one dimension are crucially affected by the curvature of the bare single particle spectrum. Parabolic curvature leads to two closely related phenomena: the Green's function exhibits oscillation (as a…
We derive the spectrum of the Laplace-Beltrami operator on the quotient orbifold of the non hyperbolic triangle groups.
The main objective of the work is to provide sharp two-sided estimates of $\lambda$-Green function of hyperbolic Brownian motion of a half-space. We strongly rely on recent results obtained by K. Bogus and J. Malecki [3], regarding precise…
This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this…
In this article, we define the spherical $\pi$-operator over domains in the $(n-1)$-D unit sphere $S^n$ of $R^n$ and develop new and analogous results on the operator it self and its mapping properties. We introduce the spherical Dirac…
Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions associated with the lowest eigenvalue, whose inverse is the orthogonal Green operator. In this study, we show that elliptic Schr\"{o}dinger…
In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are…
In this paper we will extend to non-abelian groups inverse spectral results, proved by us in an earlier paper, for compact abelian groups, i.e. tori. More precisely, Let $\mathsf G$ be a compact Lie group acting isometrically on a compact…
Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the…