Lipschitz spaces adapted to Schrödinger operators and regularity properties

Abstract

Consider the Schrödinger operator $\mathcal {{L}}=-\Delta +V$ in $\mathbb {{R}}^n, n\ge 3,$ where V is a nonnegative potential satisfying a reverse Hölder condition of the type

$$\begin{aligned} \left( \frac{1}{|B|}\int _B V(y)^qdy\right) ^{1/q}\le \frac{C}{|B|}\int _B V(y)dy, \, \text {{ for some }}q>n/2. \end{aligned}$$

We define $\Lambda ^\alpha _\mathcal {{L}},\, 0<\alpha <2,$ the class of measurable functions such that

$$\begin{aligned} \Vert \rho (\cdot )^{-\alpha }f(\cdot )\Vert _\infty<\infty \quad \, \, \text {and}\,\, \quad \sup _{|z|>0}\frac{\Vert f(\cdot +z)+f(\cdot -z)-2f(\cdot )\Vert _\infty }{|z|^\alpha }<\infty , \end{aligned}$$

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

$$\begin{aligned} \Vert \rho (\cdot )^{-\alpha }f(\cdot )\Vert _\infty <\infty \hbox { and } \Big \Vert \partial _y^k{W}_y f \Big \Vert _{L^\infty (\mathbb {R}^{n})}\le C_\alpha y^{-k+\alpha /2},\;\, \, \mathrm{with }\, k=[\alpha /2]+1, y>0. \end{aligned}$$

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