Fractional powers of the parabolic Hermite operator. Regularity properties

Let $\mathcal{L}={\mathrm{\partial }}_t-{\mathrm{\Delta }}_x+\mathrm{\mid }x{\mathrm{\mid }}^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $\alpha >0$ define the Parabolic Hermite-Zygmund spaces $${\mathrm{\Lambda }}_{\mathcal{L}}^{\alpha }=\{f:f\in L^{\mathrm{\infty }}({\mathbb{R}}^{n+1})\mathrm{a}\mathrm{n}\mathrm{d}{\parallel {\mathrm{\partial }}_y^k{e}^{-y\sqrt{\mathcal{L}}}f\parallel }_{L^{\mathrm{\infty }}({\mathbb{R}}^{n+1})}\le C_k{y}^{-k+\alpha },\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}k=[\alpha ]+1,y>0.\},$$ with the obvious norm. It is shown that these spaces have a pointwise description of H"older type. The fractional powers ${\mathcal{L}}^{\pm \beta }$ are well defined in these spaces and the following regularity properties are proved: \begin{eqnarray*} \alpha, \beta >0, \quad |$\mathcal{L}$^{-\beta} f|{ \Lambda^{\alpha+2\beta}{$\mathcal{L}$}}\le C |f|{ \Lambda^\alpha{$\mathcal{L}$}}. \end{eqnarray*} \begin{eqnarray*} 0< 2\beta < \alpha, \quad |$\mathcal{L}$^\beta f|{\Lambda{$\mathcal{L}$}^{\alpha-2\beta}}\le C |f|{\Lambda^\alpha{$\mathcal{L}$}}. \end{eqnarray*} Parallel results are obtained for the Hermite operator $-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{.}$ The proofs use in a fundamental way the semigroup definition of the operators ${\mathcal{L}}^{\pm \beta }$ and $(-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{\pm \beta }$. The non-convolution structure of the operators produce an extra difficulty of the arguments.