  We develop a technique of proving standard estimates in the setting of
Laguerre function expansions of convolution type, which works for all
admissible type multi-indices $\alpha$ in this context. This generalizes a
simpler method existing in the literature, but being valid for a restricted
range of $\alpha$. As an application, we prove that several fundamental
operators in harmonic analysis of the Laguerre expansions, including maximal
operators related to the heat and Poisson semigroups, Riesz transforms,
Littlewood-Paley-Stein type square functions and multipliers of Laplace and
Laplace-Stieltjes transforms type, are (vector-valued) Calder\'on-Zygmund
operators in the sense of the associated space of homogeneous type.
