  We study negative powers of Laguerre differential operators in $\R$, $d\ge1$.
For these operators we prove two-weight $L^p-L^q$ estimates, with ranges of $q$
depending on $p$. The case of the harmonic oscillator (Hermite operator) has
recently been treated by Bongioanni and Torrea by using a straightforward
approach of kernel estimates. Here these results are applied in certain
Laguerre settings. The procedure is fairly direct for Laguerre function
expansions of Hermite type, due to some monotonicity properties of the kernels
involved. The case of Laguerre function expansions of convolution type is less
straightforward. For half-integer type indices $\alpha$ we transfer the desired
results from the Hermite setting and then apply an interpolation argument based
on a device we call the {\sl convexity principle} to cover the continuous range
of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the
Dunkl harmonic oscillator in the context of a finite reflection group acting on
$\R$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we
obtain in this setting are essentially consequences of those for Laguerre
function expansions of convolution type.
