Use of Hardy spaces and interpolation

Vertical versus conical square functions

“…Namely it suffices to prove an L 1 estimate on some abstract atoms (that is an H 1 F,ato to L 1 estimate with H 1 F,ato as in Section 3.3 of [BZ]) and then interpolate. By checking details and values (left to readers) from the clear presentation in [Be2], one exactly finds the range for L p boundedness when p < 2. This theory, compared to the [HM] theory, has the advantage of not caring much about the "right" definition of the Hardy spaces as this is not needed for the purpose of interpolation.…”

Pseudodifferential Operators Associated with a Semigroup of Operators

“…Using the assumed off-diagonal estimates, we still have off-diagonal estimates with the ψ t function replaced by any φ t function, satisfying only decay at 0 (see [21,Lemma 4.12] or [14,Corollary 3.6] in a specific case). Then applying with the particular function φ t (•) := 1 − e −t• , we can apply the extrapolation result of [12], [13,Theorem 5.11] and obtain L p -boundedness of the operator T .…”

Abstract Hardy–Sobolev spaces and interpolation

“…This allows us to apply the Whitney covering theorem to Ω 1 and consider the Calderón-Zygmund decomposition of Proposition 3.1 (in Section 3) for b with p = β. We obtain b = j h j + g 0 (11) with h j , g 0 satisfying the properties of Proposition 3.1. We have…”

“…We refer the reader to [14,11,13] for details concerning the use of "finite atomic Hardy space" instead of the whole atomic Hardy space. The use of this last one brings technical problems (we do not know how to solve them) that are not important and are twisted by the use of the atomic Hardy space.…”