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{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 ${\Lambda }_{\mathcal{L}}^{\alpha },0<\alpha <2,$ the class of measurable functions such that $$\Vert \rho (\cdot {)}^{-\alpha }f(\cdot ){\Vert }_{\infty }<\infty \text{and}\underset{|z|>0}{\sup }\frac{\Vert f(\cdot +z)+f(\cdot -z)-2f(\cdot ){\Vert }_{\infty }}{|z{|}^{\alpha }}<\infty ,$$ 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*} \|ρ(\cdot)^{-α}f(\cdot)\|_\infty<\infty \hbox{ and } \Big\|\partial_y^k{W}_y f \Big\|_{L^\infty(\mathbb{R}^{n})}\leq C_αy^{-k+α/2},\;\: \, {\rm with }\, k=[α/2]+1, y>0. \end{equation*} We prove that for $0<\alpha \le 2-n/q$, ${\Lambda }_{\mathcal{L}}^{\alpha }={\Lambda }_{\alpha /2}^W.$ 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, $P_{y}f=e^{-y\sqrt{\mathcal{L}}}f.$

26 pages

References in corpus (1)

• Fractional powers of the parabolic Hermite operator. Regularity properties

Cited by in corpus (2)

• Besov spaces associated with non-negative operators on Banach spaces
• Regularity property of some operators on Lipschitz spaces and homogeneous Lipschitz spaces related to biharmonic operator