Riesz transforms associated to Schrödinger operators with negative potentials

Abstract

The goal of this paper is to study the Riesz transforms ∇A −1/2 where A is the Schrödinger operator −∆ − V, V ≥ 0, under different conditions on the potential V . We prove that if V is strongly sub-0 is the dual exponent of p 0 where 2 < 2N N −2 < p 0 < ∞; and we give a counterexample to the boundedness onis the reverse Sobolev exponent of p 0 . If the potential is strongly subcritical in the Kato subclassWe prove also 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.

Riesz transforms through reverse Hölder and Poincaré inequalities

“…Let us focus on the case V ≥ 0, otherwise the situation is more difficult and we refer the reader to [2,3] (and references therein) for some works giving assumptions on V to guarantee the boundedness of the Riesz transform. If V is non-negative and belongs to some Muckenhoupt reverse Hölder space, then boundedness of the Riesz transform has been obtained in [40,5].…”

Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

“…where θ = N 2m (1 − 2 r ). Thus, by a standard duality argument, we have that e −tL satisfies the L p − L 2 estimate for all p ∈ (1,2].…”

“…On the other hand, even for classic Schrödinger operator L = −∆ + V with V ≥ 0, Shen's counterexample in [48] showed that extra regularity conditions on V are necessary if one wants to the L q boundedness of ∇L −1/2 for q > 2. In fact, one can see [5] and [48] for V belonging to reverse Hölder class, [55] for V satisfying Fefferman condition and [1] for signed subcritical potential V belonging to Kato class.…”

“…Remark 1.1. (i) When N ≤ 2m, it is easy to see that the lower bound of the interval (1,2] in Theorem 1.1 is optimal. However, it is still unknown to us if the upper bound 2 is sharp in the following sense: for given q > 2, there exist L ∈ A m such that the Riesz transforms ∇ m L −1/2 is not bounded on L q (R N ).…”

Sobolev spaces adapted to the Schrödinger operator with inverse-square potential

“…See also [4,5] for prior partial results. Moreover, the earlier paper [13] showed that for s = 1, the range of p stated above cannot be enlarged, even if one restricts f to be spherically symmetric.…”