Hermite and Laguerre Sobolev spaces with weights

B. BONGIOANNI, J.L. TORREA

Miraflores de la Sierra, Madrid, España

Congreso; HAOS-2007; 2007

Universida Autónoma de Madrid

Weighted Sobolev spaces associated to the Hermite operator $H=-\Delta + |x|^2$ are described for weights $w$ in the $A_p$ classes, $1<p<\infty$, using composition of annihilation operators $A_j = \partial_j + x_j$ as derivatives. In this sense, a function $f$ has $k$ derivatives in $L^p(w)$ if and only if $H^{-k/2}f$ belongs to $L^p(w)$. Also we present power weighted Sobolev spaces associated to Laguerre systems of type $\alpha>-1$. In this context the derivatives used are composition of $\delta^{\alpha+k} = \sqrt{x}\frac{d}{dx} + \frac{1}{2} \left(\sqrt{x} - \frac{\alpha+k}{\sqrt{x}}\right)$ for integers $k\ge 0$, and we deal with higher order Riesz transforms type operators related to them.