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 $delta^{alpha+k} = sqrt{x}rac{d}{dx} + rac{1}{2} left(sqrt{x} - rac{alpha+k}{sqrt{x}} ight)$ for integers $kge 0$, and we deal with higher order Riesz transforms type operators related to them.