A symmetrized conjugacy scheme for orthogonal expansions

TL;DR

The paper introduces a symmetrization method for orthogonal expansions linked to a second order differential operator, enabling a classical conjugacy scheme that facilitates analysis of Riesz transforms and conjugate Poisson integrals.

Contribution

It develops a symmetrization procedure that extends conjugacy schemes to general orthogonal expansions, connecting differential operators with differential-difference Laplacians.

Findings

Partial derivatives are skew-symmetric in the extended scheme.

The scheme allows commuting Riesz transforms and conjugate Poisson integrals.

Provides new insights into higher order Riesz transforms for orthogonal expansions.

Abstract

We establish a symmetrization procedure in a context of general orthogonal expansions associated with a second order differential operator $L$, a `Laplacian'. Combined with a unified conjugacy scheme furnished in our earlier article it allows, via a suitable embedding, to associate a differential-difference `Laplacian' $\mathbb{L}$ with the initially given orthogonal system of eigenfunctions of $L$, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related `partial derivatives' decomposing $\mathbb{L}$ are skew-symmetric in an appropriate $L^2$ space and they commute with Riesz transforms and conjugate Poisson integrals. The results shed also some new light on the question of defining higher order Riesz transforms for general orthogonal expansions.