Abstract
Let d∈{3,4,5,...} and a weight w∈A ∞ ρ . We consider the second-order Riesz transform T=∇ 2 L -1 associated with the Schrödinger operator L=-Δ+V, where V∈RH σ with σ>d 2. We present three main results. First T is bounded on the weighted Hardy space H w,L 1 (ℝ d ) associated with L if w enjoys a certain stable property. Secondly T is bounded on the weighted BMO space BMO w,ρ (ℝ d ) associated with L if w also belongs to an appropriate doubling class. Thirdly BMO w,ρ (ℝ d ) is the dual of H w,L 1 (ℝ d ) when w∈A 1 ρ .
Highlights
Manuscript received 11th November 2020, revised 8th March 2021 and 14th March 2021, accepted 12th April 2021. It is well-known that classical Calderón–Zygmund operators are bounded on Lp spaces for p ∈ (1, ∞)
Hardy and B MO spaces as well as their variances have found their places in this context as substitutions for L1 and L∞ spaces respectively
Various results on the boundedness of Calderón–Zygmund operators on these two spaces and their variances can be found in the vast literature
Summary
It is well-known that classical Calderón–Zygmund operators are bounded on Lp spaces for p ∈ (1, ∞). Our main contributions in this paper include: (1) Proving the boundedness of T on the weighted Hardy space Hw1 ,L(Rd ) when w ∈ Aρ∞ (see Theorem 5). This extends the unweighted versions in [14, Theorem 1.2] and [8, Theorem 1.6]. The result is new even in the special case when w is a Muckenhoupt weight This provides a very first concrete example that verifies the characterizations of boundedness on B MOw,ρ(Rd ) for the generalized Calderón–Zygmund operators in [4, Theorem 2].
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