How to define the Riesz transform for Fourier-Neumann expansions?

Autor: Roncal Gómez, LuzCiaurri Ramírez, OscarVarona Malumbres, Juan Luis; Torrea Hernández, José Luis; 

Tipo de documento: Capítulo de libro

Libro: CONTRIBUCIONES CIENTIFICAS EN HONOR DE MIRIAN ANDRES.

Año: 2010  Páginas: 203-215

 Texto completo open access 

Resumen: A self-adjoint differential operator L with positive eigenvalues is considered. The authors provide a general view to the method followed by definitions, in a spectral way, of the heat semigroup e−tL, the Poisson semigroup e−tL√ and the Riesz potentials L−a, and arriving at the decomposition L=δ∗δ (with δ and δ∗ two adjoint first-order differential operators). Then they define the Riesz transform R=δL−1/2. In particular, for the differential operator Lα corresponding to the Fourier-Neumann expansions of order α>−1, the authors find, with the help of a computer algebra system, a cluster of strange decompositions Lα=δδ∗ (or similar). None of them seems to be useful to define the Riesz transform, because the operators δ and δ∗ are too weird: the expressions of δ and δ∗ contain functions with an infinite quantity of poles, so integrable functions cannot be obtained by applying δ.