On L^p estimates for positivity-preserving Riesz transforms related to Schrödinger operators

2022/03/29 by Maciej Kucharski, Kucharski, Maciej, Błażej Wróbel +1
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2203.15530

Abstract

We study the L^p, 1\leqslant p\leqslant ∞, boundedness for Riesz transforms of the form V^a(-(1)/(2)Δ+V)^-a, where a>0 and V is a non-negative potential. We prove that V^a(-(1)/(2)Δ+V)^-a is bounded on L^p(ℝ^d) with 1< p\leqslant 2 whenever a\leqslant 1/p. We demonstrate that the L^∞(ℝ^d) boundedness holds if V satisfies an a-dependent integral condition that is resistant to small perturbations. Similar results with stronger assumptions on V are also obtained on L¹(ℝ^d). In particular our L^∞ and L¹ results apply to non-negative potentials V which globally have a power growth or an exponential growth. We also discuss a counterexample showing that the L^∞(ℝ^d) boundedness may fail.

Related

• Dimension-free estimates for positivity-preserving Riesz transforms related to Schrödinger operators with certain potentials
• Riesz transforms associated to Schrödinger operators with negative potentials
• Behaviour of Schrödinger Riesz transforms over smoothness spaces
• Maximal inequalities and Riesz transform estimates on L^p spaces for Schrödinger operators with nonnegative potentials
• Riesz transforms in one dimension