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Remarks on L p -limiting absorption principle of Schrödinger operators and applications to spectral m ...

Remarks on L p -limiting absorption principle of Schrödinger operators and applications to... AbstractThis paper comprises two parts. We first investigate an Lp{L^{p}}-type of limiting absorption principle for Schrödinger operators H=-Δ+V{H=-\Delta+V}on ℝn{\mathbb{R}^{n}}(n≥3{n\geq 3}), i.e., we prove the ϵ-uniform L2⁢(n+1)/(n+3){L^{{2(n+1)}/({n+3})}}–L2⁢(n+1)/(n-1){L^{{2(n+1)}/({n-1})}}-estimates of the resolvent (H-λ±i⁢ϵ)-1{(H-\lambda\pm i\epsilon)^{-1}}for all λ>0{\lambda>0}under the assumptions that the potential V belongs to some integrable spaces and a spectral condition of H at zero is satisfied. As applications, we establish a sharp Hörmander-type spectral multiplier theorem associated with Schrödinger operators H and deduce Lp{L^{p}}-bounds of the corresponding Bochner–Riesz operators. Next, we consider the fractional Schrödinger operator H=(-Δ)α+V{H=(-\Delta)^{\alpha}+V}(0<2⁢α<n{0<2\alpha<n}) and prove a uniform Hardy–Littlewood–Sobolev inequality for (-Δ)α{(-\Delta)^{\alpha}}, which generalizes the corresponding result of Kenig–Ruiz–Sogge [20]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Remarks on L p -limiting absorption principle of Schrödinger operators and applications to spectral m ...

Forum Mathematicum , Volume 30 (1): 13 – Jan 1, 2018

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References (46)

Publisher
de Gruyter
Copyright
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2016-0162
Publisher site
See Article on Publisher Site

Abstract

AbstractThis paper comprises two parts. We first investigate an Lp{L^{p}}-type of limiting absorption principle for Schrödinger operators H=-Δ+V{H=-\Delta+V}on ℝn{\mathbb{R}^{n}}(n≥3{n\geq 3}), i.e., we prove the ϵ-uniform L2⁢(n+1)/(n+3){L^{{2(n+1)}/({n+3})}}–L2⁢(n+1)/(n-1){L^{{2(n+1)}/({n-1})}}-estimates of the resolvent (H-λ±i⁢ϵ)-1{(H-\lambda\pm i\epsilon)^{-1}}for all λ>0{\lambda>0}under the assumptions that the potential V belongs to some integrable spaces and a spectral condition of H at zero is satisfied. As applications, we establish a sharp Hörmander-type spectral multiplier theorem associated with Schrödinger operators H and deduce Lp{L^{p}}-bounds of the corresponding Bochner–Riesz operators. Next, we consider the fractional Schrödinger operator H=(-Δ)α+V{H=(-\Delta)^{\alpha}+V}(0<2⁢α<n{0<2\alpha<n}) and prove a uniform Hardy–Littlewood–Sobolev inequality for (-Δ)α{(-\Delta)^{\alpha}}, which generalizes the corresponding result of Kenig–Ruiz–Sogge [20].

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 2018

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