Riesz transforms on non-compact manifolds

الأصل · EN

Let M be a complete non-compact Riemannian manifold satisfying the doubling volume property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform dΔ⁻¹/² on both Hardy spaces Hᵖ and Lebesgue spaces Lᵖ under two different conditions on the negative part of the Ricci curvature R-. First we prove that if R- is α-subcritical for some α∈ [0,1), then the Riesz transform d*Δ⁻¹/² on differential 1-forms is bounded from the associated Hardy space HᵖΔ(Λ¹T*M) to Lᵖ(M) for all p∈ [1,2]. As a consequence, the Riesz transform (on functions) 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-|¹/²v(·,√t)¹/ᵖ¹ₚ₁dt√t+∫₁∞ |R-|¹/²v(·,√t)¹/ᵖ²ₚ₂dt√t<∞, for some p₁>2 and p₂>3, then the Riesz transform dΔ⁻¹/² is bounded on Lᵖ for all 1<p<p₂. In the particular case where v(x, r) ≥ C rᵈ for all r ≥ 1 and |R-| ∈ Lᵈ/² ⁻η ∩ Lᵈ/² ⁺ η for some η> 0, then dΔ⁻¹/² is bounded on Lᵖ for all 1<p< D. 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⁻¹/² on Lᵖ