\(L^p\) estimates of Schrödinger operators on nilpotent groups
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 \(p \geq p_0'\), where \(1/p_0' = 1 - (q_0^{-1} - D^{-1})\). These theorems were proved before by \textit{Z.-W. Shen} [Ann. Inst. Fourier 45 No. 2, 513--546 (1995; Zbl 0818.35021)] in case \(G = {\mathbb R}^m\) with \(m \geq 3\).
scientific article; zbMATH DE number 1271586
scientific article