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

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, $P_yf= e^{-y\sqrt{\mathcal{L}}}f.$