Parabolic Hermite Lipschitz Spaces: Regularity of Fractional Operators

Abstract

We introduce a pointwise definition of Lipschitz (also called Hölder) spaces adapted to the parabolic Hermite operator $\mathbb {H}= \partial _t- \Delta _x+|x|^2$ on $\mathbb {{R}}^{n+1}$. Also for every $\alpha >0$, we define the following spaces by means of the Poisson semigroup of $\mathbb {H}$, $\mathcal {P}_y^{\mathbb {H}}=e^{-y\sqrt{\mathbb {H}}}$:

$$\begin{aligned} \Lambda _\alpha ^{\mathcal {P}^\mathbb {H}}= & {} \left\{ f: \;f\in L^\infty (\mathbb {R}^{n+1})\, \mathrm{and} \, \left\| \partial _y^k e^{-y\sqrt{\mathbb {H}}} f \right\| _{L^\infty (\mathbb {R}^{n+1})}\right. \\&\left. \le C_k y^{-k+\alpha },\, \mathrm {for}\, k=[\alpha ]+1,\;y>0 \right\} , \end{aligned}$$

with the obvious norm. We prove that both spaces do coincide and their norms are equivalent. For the harmonic oscillator, $\mathcal {{H}}=-\Delta _x+|x|^2$, Stinga and Torrea introduced in 2011 adapted Hölder classes. Parallel to the parabolic case, we characterize these pointwise Hölder spaces via the $L^\infty$ norm of the derivatives of the Poisson and heat semigroups, $e^{-y\sqrt{\mathcal {{H}}}}$ and $e^{-\tau \mathcal {{H}}}$, respectively. As important applications of these semigroups characterizations, we get regularity results regarding the boundedness in these adapted Lipschitz spaces of operators related to $\mathbb {H}$ and $\mathcal {{H}}$ as fractional (positive and negative) powers, Bessel potentials, Hermite Riesz transforms, and Laplace transform multipliers, in a more direct way. The proofs use in a fundamental way the semigroup definition of the operators considered along the paper. The non-convolution structure of the operators produces an extra difficulty on the arguments.