Lipschitz spaces adapted to Schrödinger operators and regularity properties
Consider the Schrödinger operator L= - Δ + V in Rn, n≥ 3 , where V is a nonnegative potential satisfying a reverse Hölder condition of the type (1|B|∫BV(y)qdy)1/q ≤C|B|∫BV(y)dy, for some q > n/2. We define ΛαL, 0 < α < 2, the class of measurable functions such that ‖ρ(·)-αf(·)‖∞ < ∞ and...
| Autores: | , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2020 |
| País: | España |
| Recursos: | Universidad Autónoma de Madrid |
| Repositório: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglês |
| OAI Identifier: | oai:repositorio.uam.es:10486/700462 |
| Acesso em linha: | http://hdl.handle.net/10486/700462 https://dx.doi.org/10.1007/s13163-020-00357-9 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Fractional Laplacian Hölder estimates Lipschitz Hölder Zygmund spaces Semigroups Matemáticas |
| Resumo: | Consider the Schrödinger operator L= - Δ + V in Rn, n≥ 3 , where V is a nonnegative potential satisfying a reverse Hölder condition of the type (1|B|∫BV(y)qdy)1/q ≤C|B|∫BV(y)dy, for some q > n/2. We define ΛαL, 0 < α < 2, the class of measurable functions such that ‖ρ(·)-αf(·)‖∞ < ∞ and sup|z>0‖f(·+z) + f(·-z)-2f(·)‖∞|z|α < ∞,where ρ is the critical radius function associated to L. Let Wyf = e-yLf be the heat semigroup of L. Given α > 0 , we denote by Λ Wα/2 the set of functions f which satisfy ‖ρ(·)-αf(·)‖∞ <∞ and‖∂kyWyf‖L∞(Rn)≤Cαy-k+α/2, withk=[α/2]+1,y >0. We prove that for 0 < α ≤ 2 - n/q, ΛαL = Λ Wα/2. As application, we obtain regularity properties of fractional powers (positive and negative) of the operator 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, Py f = e-y √Lf |
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