The Hodge–de Rham Laplacian and [formula omitted]-boundedness of Riesz transforms on non-compact manifolds

Abstract

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform dΔ−12 on both Hardy spaces Hp and Lebesgue spaces Lp under two different conditions on the negative part R− of the Ricci curvature. First we prove that if R− is α-subcritical for some α∈[0,1), then the Riesz transform d∗Δ⃗−12 on differential 1-forms is bounded from the associated Hardy space HΔ⃗p(Λ1T∗M) to Lp(M) for all p∈[1,2]. As a consequence, dΔ−12 is bounded on Lp for all p∈(1,p0) where p0>2 depends on α and the constant appearing in the doubling property. Second, we prove that if∫01‖|R−|12v(⋅,t)1p1‖p1dtt+∫1∞‖|R−|12v(⋅,t)1p2‖p2dtt<∞, for some p1>2 and p2>3, then the Riesz transform dΔ−12 is bounded on Lp for all 1<p<p2. Furthermore, we study the boundedness of the Riesz transform of Schrödinger operators A=Δ+V on Lp for p>2 under conditions on R− and the potential V. We prove both positive and negative results on the boundedness of dA−12 on Lp.