Fractional powers of the parabolic Hermite operator. Regularity properties

Abstract

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 $${\Lambda }_{\mathcal{L}}^{\alpha }=\left\{f:f\in L^{\infty }({\mathbb{R}}^{n+1})\mathrm{and}{\Vert {\partial }_y^k{e}^{-y\sqrt{\mathcal{L}}}f\Vert }_{L^{\infty }({\mathbb{R}}^{n+1})}\le C_k{y}^{-k+\alpha },\mathrm{with}k=[\alpha ]+1,y>0.\right\},$$ with the obvious norm. It is shown that these spaces have a pointwise description of Hölder type. The fractional powers ${\mathcal{L}}^{\pm \beta }$ are well defined in these spaces and the following regularity properties are proved: \begin{eqnarray*} α, β>0, \quad \|\mathcal{L}^{-β} f\|_{ Λ^{α+2β}_{\mathcal{L}}}\le C \|f\|_{ Λ^α_{\mathcal{L}}}. \end{eqnarray*} \begin{eqnarray*} 0< 2β< α, \quad \|\mathcal{L}^βf\|_{Λ_{\mathcal{L}}^{α-2β}}\le C \|f\|_{Λ^α_{\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.

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