\(L^p\) estimates of Schrödinger operators on nilpotent groups
type: Item
instance of: scholarly article
title: \(L^p\) estimates of Schrödinger operators on nilpotent groups
zbMATH Open document ID: 0929.22005
author: Hong-Quan Li
published in: Journal of Functional Analysis
publication date: 2000-01-23T00:00:00Z
review text: Let \(G\) be a simply connected nilpotent Lie group with Lie algebra \({\mathcal G}\) and dimension \(D\) at infinity. Suppose \(\dim {\mathcal G} \geq 3\). Let \(X_1,\ldots,X_m \in {\mathcal G}\) be a Hörmander system. Next let \(\Delta = \sum_{i=1}^m X_i^2\) be the associated sub-Laplacian and \(\rho\) the Carnot--Carathéodory distance on \(G\). Fix \(q_0 > D/2\) and a measurable function \(W\) on \(G\). Suppose throughout that \(W \geq 0\), \(W \neq 0\) and \(W \in B_{q_0}\), i.e., \(W \in L^{q_0}_{\text{loc}}(G)\) and there exists a \(C > 0\) such that \[ \Biggl( | B(x,r)| ^{-1} \int_{B(x,r)} W^{q_0}(y)\,dy \Biggr)^{1/q_0} \leq C | B(x,r)| ^{-1} \int_{B(x,r)} W(y)\,dy \] for all \(x \in G\) and \(r > 0\), where \(| B(x,r)| \) is the Haar measure of the ball \(B(x,r) = \{ y \in G ; \rho(x,y) < r \} \) with centre \(x\) and radius \(r\). The main theorems of this paper are as follows. Theorem A. If \(m \geq 3\) then for all \(1 < p \leq q_0\) there exists a \(C_p > 0\) such that \(\| \nabla^2 (- \Delta + W)^{-1} f\| _p \leq C_p \| f\| _p\) for all \(f \in L^p(G)\). Theorem B. Set \(p_0^{-1} = 3/(2 q_0) - D^{-1}\) if \(q_0 < D\) and \(p_0^{-1} = 1/(2q_0)\) if \(q_0 \geq D\). Then for all \(1 \leq p < p_0\) there exists a \(C_p > 0\) such that \(\| W^{1/2} \nabla (- \Delta + W)^{-1} f\| _p \leq C_p \| f\| _p\) for all \(f \in L^p(G)\). Theorem C. Suppose in addition that \(q_0 < D\) and set \(p_0^{-1} = q_0^{-1} - D^{-1}\). Then for all \(1 < p < p_0\) there exists a \(C_p > 0\) such that \(\| \nabla (- \Delta + W)^{-1/2} f\| _p \leq C_p \| f\| _p\) for all \(f \in L^p(G)\). Theorem D. If \(1 \leq p < 2 q_0\) then there exists a \(C_p > 0\) such that \(\| W^{1/2} (- \Delta + W)^{-1/2} f\| _p \leq C_p \| f\| _p\) for all \(f \in L^p(G)\). Theorem E. If \(1 \leq p \leq 2 q_0\) then there exists a \(C_p > 0\) such that \(\| W^{1/2} (- \Delta + W)^{-1} \nabla f\| _p \leq C_p \| f\| _p\) for all \(f \in L^p(G)\) if either \(q_0 > D\) or if \(q_0 < D\) and in addition \(p…
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 1271586
zbMATH Keywords: Riesz transforms
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: reverse Hölder class
MaRDI profile type: Publication
full work available at URL: jfan.1998.3347
OpenAlex ID: W2082608503
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DOI: 10.1006/JFAN.1998.3347
description: scientific article; zbMATH DE number 1271586