Our main goal in this article is to study mixed norm estimates for the
Ces\`{a}ro means associated with Dunkl--Hermite expansions on Rd.
These expansions arise when one consider the Dunkl--Hermite operator (or Dunkl
harmonic oscillator) Hκ​:=−Δκ​+∣x∣2, where
Δκ​ stands for the Dunkl--Laplacian. It is shown that the desired
mixed norm estimates are equivalent to vector-valued inequalities for a
sequence of Ces\`{a}ro means for Laguerre expansions with shifted parameter. In
order to obtain the latter, we develop an argument to extend these operators
for complex values of the parameters involved and apply a version of three
lines lemma.Comment: 24 pages. Revised version following referee's comments. To appear in
Transactions of the American Mathematical Societ