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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) hasrecently been treated by Bongioanni and Torrea by using a straightforwardapproach of kernel estimates. Here these results are applied in certainLaguerre settings. The procedure is fairly direct for Laguerre functionexpansions of Hermite type, due to some monotonicity properties of the kernelsinvolved. The case of Laguerre function expansions of convolution type is lessstraightforward. For half-integer type indices $\alpha$ we transfer the desiredresults from the Hermite setting and then apply an interpolation argument basedon a device we call the {\sl convexity principle} to cover the continuous rangeof $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of theDunkl 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 weobtain in this setting are essentially consequences of those for Laguerrefunction expansions of convolution type.

Negative powers of Laguerre operators