Let $\mathcal{ L}= \partial_t- \Delta_x+|x|^2$. Consider its Poisson semigroup $e^{ -y\sqrt{ \mathcal{ L}}}$. For $\alpha >0$ define the Parabolic Hermite-Zygmund spaces $$ \Lambda^\alpha_{ \mathcal{ L}}=\left\{ f: \:f\in L^\infty(\mathbb{ R}^{ n+1})\:\; { \rm and} \:\; \left\|\partial_y^k e^{ -y\sqrt{ \mathcal{ L}}} f \right\|_{ L^\infty(\mathbb{ R}^{ n+1})}\leq C_k y^{ -k+\alpha},\;\: { \rm with }\, k=[\alpha]+1, y>0. \right\}, $$ 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 $- \Delta +|x|^2.$ The proofs use in a fundamental way the semigroup definition of the operators $\mathcal{ L}^{ \pm \beta}$ and $(-\Delta+|x|^2)^{ \pm \beta}$. The non-convolution structure of the operators produce an extra difficulty of the arguments.
Categories: math.FA