Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights

“…Properties of Muckenhoupt weights A p and reverse Hölder classes RH s are reviewed in [AM,Section 2]. If w ∈ A ∞ (µ), one can define r w = inf{p > 1 : w ∈ A p (µ)} ∈ [1, ∞) and s w = sup{s > 1 : w ∈ RH s (µ)} ∈ (1, ∞].…”

“…It is stated in the Euclidean setting, see [AM,Section 5] for the extension to spaces of homogeneous type ‡ Here and subsequently, the subscript c means with compact support.…”

“…In [AM,Lemma 4.6], examples of weights in A p (µ) ∩ RH q (µ) are given. The computations are done in the Euclidean setting, but most of them can be carried out in spaces of homogeneous type.…”

“…The following result, used to prove Theorem 1.2, (ii), is taken from [AM,Theorem 8.8 & Remark 8.10] (see also [AM,Section 8.4] for the extension to spaces of homogeneous type)…”

“…Fix w ∈ A 1 (µ) ∩ RH (q + ) ′ (µ). By [AM,Proposition 2.1], there exists 1 < q < q + such that w ∈ A q (µ) ∩ RH (q + /q) ′ (µ). This means that q ∈ W w (1, q + ), therefore by (i), T = |∇∆ −1/2 | is bounded on L q (M, w) and so (a) holds.…”