Imaginary powers of a Laguerre differential operator

Author: Wróbel B.  

Publisher: Springer Publishing Company

ISSN: 0236-5294

Source: Acta Mathematica Hungarica, Vol.124, Iss.4, 2009-09, pp. : 333-351

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Abstract

Imaginary powers associated to the Laguerre differential operator $$ L_alpha = - Delta + |x|^2 + sum _{i = 1}^d frac{1} {{x_i^2 }}(alpha _i^2 - 1/4) $$ are investigated. It is proved that for every multi-index α = (α1,...α d ) such that α i ≧ −1/2, α i ∉ (−1/2, 1/2), the imaginary powers $$ mathcal{L}_alpha ^{ - igamma } ,gamma in mathbb{R} $$ , of a self-adjoint extension of L α, are Calderón-Zygmund operators. Consequently, mapping properties of $$ mathcal{L}_alpha ^{ - igamma } $$ follow by the general theory.