The Hodge–de Rham Laplacian and LpLp-boundedness of Riesz transforms on non-compact manifolds

Let MM 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 HpHp and Lebesgue spaces LpLp under two different conditions on the negative part R−R− of the Ricci curvature. First we prove that if R−R− is αα-subcritical for some α∈[0,1)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)Lp(M) for all p∈[1,2]p∈[1,2]. As a consequence, dΔ−12 is bounded on LpLp for all p∈(1,p0)p∈(1,p0) where p0>2p0>2 depends on αα 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>2p1>2 and p2>3p2>3, then the Riesz transform dΔ−12 is bounded on LpLp for all 12p>2 under conditions on R−R− and the potential VV. We prove both positive and negative results on the boundedness of dA−12 on LpLp.