# Lipschitz spaces adapted to Schr\"odinger operators and regularity properties

## TL;DR

This paper introduces Lipschitz spaces adapted to Schr"odinger operators and establishes their equivalence with heat semigroup-based spaces, leading to new regularity results for fractional powers, Riesz transforms, and multipliers associated with these operators.

## Contribution

It defines new Lipschitz spaces linked to Schr"odinger operators and proves their equivalence with semigroup-based spaces, extending regularity theory for related operators.

## Key findings

- Equivalence of Lipschitz spaces and heat semigroup spaces for certain alpha.
- Regularity properties for fractional powers of Schr"odinger operators.
- Results on Schr"odinger Riesz transforms and multipliers.

## Abstract

Consider the Schr\"odinger operator $\mathcal{L}=-\Delta+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse H\"older condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B V(y)^qdy\right)^{1/q}\le \frac{C}{|B|}\int_B V(y)dy, \, \text{{ for some }}q>n/2. \end{equation*} We define $\Lambda^\alpha_{\mathcal{L}},\, 0<\alpha <2,$ the class of measurable functions such that $$ \|\rho(\cdot)^{-\alpha}f(\cdot)\|_\infty<\infty \quad \, \, \text{and}\:\: \quad \sup_{|z|>0}\frac{\|f(\cdot+z)+f(\cdot-z)-2f(\cdot)\|_\infty}{|z|^\alpha}<\infty, $$ where $\rho$ is the critical radius function associated to $\mathcal{L}$. Let $W_y f = e^{-y\mathcal{L}}f$ be the heat semigroup of $\mathcal{L}$. Given $\alpha >0,$ we denote by $\Lambda_{\alpha/2}^{{W}}$ the set of functions $f$ which satisfy \begin{equation*} \|\rho(\cdot)^{-\alpha}f(\cdot)\|_\infty<\infty \hbox{ and } \Big\|\partial_y^k{W}_y f \Big\|_{L^\infty(\mathbb{R}^{n})}\leq C_\alpha y^{-k+\alpha/2},\;\: \, {\rm with }\, k=[\alpha/2]+1, y>0. \end{equation*} We prove that for $0<\alpha \le 2-n/q$, $\Lambda^\alpha_{\mathcal{L}} = \Lambda_{\alpha/2}^{{W}}.$ As application, we obtain regularity properties of fractional powers (positive and negative) of the operator $\mathcal{L}$, Schr\"odinger Riesz transforms, Bessel potentials and multipliers of Laplace transforms type. The proofs of these results need in an essential way the language of semigroups. Parallel results are obtained for the classes defined through the Poisson semigroup, $P_yf= e^{-y\sqrt{\mathcal{L}}}f.$