On invariant functions for positive operators

Translation-invariant bilinear operators with positive kernels

We study L (or Lr,∞) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on R. We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on R, while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some suffic...

Differential Operators Commuting with Invariant Functions

On linear operators with an invariant subspace of functions

Let us denote V, the finite dimensional vector spaces of functions of the form ψ(x) = pn(x)+ f(x)pm(x) where pn(x) and pm(x) are arbitrary polynomials of degree at most n andm in the variable x while f(x) represents a fixed function of x. Conditions onm,n and f(x) are found such that families of linear differential operators exist which preserve V. A special emphasis is accorded to the cases wh...

Invariant Differential Operators On

For two positive integers m and n, we let Pn be the open convex cone in R consisting of positive definite n × n real symmetric matrices and let R be the set of all m×n real matrices. In this article, we investigate differential operators on the non-reductive manifold Pn × R (m,n) that are invariant under the natural action of the semidirect product group GL(n,R)⋉R on the MinkowskiEuclid space P...

Invariant Operators on Manifolds

This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. conformal Riemannian and almost quaternionic geometries. Exploiting some finite dimensional representation theory of simple Lie algebras, we give explicit formulae for distinguished invariant curve...