Let Lα = − 12 ∑n j=1 ( ZjZj + ZjZj ) + iαT be the sub-Laplacian on the nonisotropic Heisenberg group Hn where Zj , Z̄j for j = 1, 2, · · · , n and T are a basis of the Lie algebra hn. We apply the Laguerre calculus to obtain the fundamental solution of the heat kernel exp{−sLα}, the Schrödinger operator exp{−isLα} and the operator ∆λ,α = − 12 ∑n j=1 λj(ZjZ̄j + Z̄jZj) + iαT. We also discuss some ba...

Abstract. We consider the fractional Laplacian −(−1)α/2 on an open subset in Rd with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian inC1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric α-stable process killed upon leaving a C1,1 open set. Our results are the first sharp twoside...

We use boundary triples to find a parametrization of all self-adjoint extensions the magnetic Schrodinger operator, in quasi-convex domain~$\Omega$ with compact boundary, and potentials components $\textrm{W}^{1}_{\infty}(\overline{\Omega})$. This gives also new characterization Laplacian nonregular domains. Then we discuss gauge transformations for such generalize equivalence Dirichlet operato...

We provide a detailed description of the relationships between fractional Laplacian order 2s∈(0,n) on Rn and s-polyharmonic extension operator.

In this paper, we discuss the existence and non-existence of weak solutions to non-linear problem with a fractional p-Laplacian introduced by ?-Hilfer operator, combining technique Nehari manifolds fibering maps. Also, obtain some results on operator manifold via Euler functional.