Riesz transforms associated to Schrödinger operators with negative potentials. (English) Zbl 1247.42012

Summary: The goal of this paper is to study the Riesz transforms \(\nabla A^{-1/2}\) where \(A\) is the Schrödinger operator \(-\nabla-V\) , \(V \geq 0\), under different conditions on the potential \(V\). We prove that if \(V\) is strongly subcritical, \(\nabla A^{-1/2}\) is bounded on \(L^p(\mathbb R ^N)\), \(N \geq 3\), for all \(p \in (p_0'; 2]\) where \(p_0'\) is the dual exponent of \(p_0\) where \(2 <\frac{2N}{N- 2}< p_0 < \infty\); and we give a counterexample to the boundedness on \(L^p(\mathbb R ^N)\) for \(p \in (1; p_0') \cup (p_{0*};\infty)\) where \(p_{0*} :=\frac{p_0 N}{N+p_0}\) is the reverse Sobolev exponent of \(p_0\). If the potential is strongly subcritical in the Kato subclass \(K^\infty_N\), then \(\nabla A^{-1/2}\) is bounded on \(L^p(\mathbb R ^N)\) for all \(p \in (1; 2]\), moreover if it is in \(L^{N/2}_\omega(\mathbb R ^N)\) then \(L^p(\mathbb R ^N)\) is bounded on \(L^p(\mathbb R ^N)\) for all \(p \in (1; N)\). We also prove the boundedness of \(V^{1/2}A^{-1/2}\) with the same conditions on the same spaces. Finally we study these operators on manifolds. We prove that our results hold on a class of Riemannian manifolds.


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
35J10 Schrödinger operator, Schrödinger equation
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