The molecular characterization of Hardy spaces

On the classification of homogeneous multipliers bounded on 𝐻¹(𝑅²)

“…This classification, the Lp result indicated above (see Theorem 7) and conditions found by Taibleson and Weiss [12] are the most complete sufficiency results for R .…”

“…We also prove that these necessary conditions are sufficient for Tm to map Lp boundedly to Lp for p > 1. In [12], Taibleson and Weiss have shown that if Tm is in B(HP) for p < 1 then m is continuous on R \{0}. The replacement of L by H for this study is natural as it is well known that Tm is not bounded on L if m is not constant.…”

Bilinear decompositions and commutators of singular integral operators

“…Furthermore, is bounded from H 1 (R n ) into itself. Similarly to the proof of Theorem 3.1, we first consider f a ψ-atom related to the cube Q = Q[x 0 , r] and note that Let ε ∈ (0, 1), recall that (see [41]) g is an ε-molecule for H 1 (R n ) centered at y 0 if R n g(x)dx = 0 and g 1/2 L q g| · −y 0 | 2nε 1/2 L q =: N(g) < ∞, where q = 1/(1 − ε). It is well known that if g is an ε-molecule for H 1 (R n ) centered at y 0 , then g ∈ H 1 (R n ) and g H 1 ≤ CN(g) where C > 0 depends only on n, ε.…”

On the necessity of the Hörmander condition for multipliers on 𝐻^{𝑝}(𝑅ⁿ)

“…In [4] Taibleson and Weiss show that if m satisfies (#) for some / > n/2, then for \/p -1/2 < t/n, Tm sends (p,2, t -l)-atoms boundedly to (p,2,[n(\/p -1)], t/n -l/2)-molecules; and hence, Tm is a bounded operator on H^R"). The purpose of this work is to provide a converse to this result for homogeneous and related multipliers.…”

“…This line of study was continued by Taibleson and Weiss [4] where they provide the details for the molecular decomposition of H''-spaces associated with R" and the unit disk, and show how these decompositions can be used to obtain multiplier theorems. In this paper we will provide a converse to the main multiplier theorem of Taibleson and Weiss for a certain class of multipliers.…”