##### Sesión Analisis (II)

Diciembre 13, 11:40 ~ 12:00

## Fractional powers of the parabolic Hermite operator. Regularity properties

### de León Contreras, Marta

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 $$\Lambda^\alpha_{\mathcal{L}}=\left\{f: \:f\in L^\infty(\mathbb{R}^{n+1})\:\; {\rm and} \:\; \left\|\partial_y^kte^{-y\sqrt{\mathcal{L}}} f \right\|_{L^\infty(\mathbb{R}^{n+1})}\leq C_k y^{-k+\alpha},\;\: {\rm with }\, k=[\alpha]+1, y>0. \right\},$$ with the obvious norm. In this talk we will characterize these spaces and show that they have a pointwise description of H\"older type. Also, we will study regularity properties of some fractional operators such as H\"older and Schauder estimates for the fractional parabolic harmonic oscillator, $\mathcal{L}^{\pm \beta}$. Parallel results will be given for the Hermite operator $- \Delta +|x|^2.$ The proof 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.

Autores: de León Contreras, Marta / Torrea, José Luis.