Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group

Abstract

This is surprising in that these multipliers are invariant under a two-parameter group of dilations on ‫ރ‬ n × ‫,ޒ‬ while there is no two-parameter group of automorphic dilations on ‫ވ‬ n . This lack of automorphic dilations underlies the failure of such multipliers to be in general bounded on the classical Hardy space H 1 on the Heisenberg group, and also precludes a pure product Hardy space theory.

Boundedness of singular integrals on the flag Hardy spaces on Heisenberg group

“…As a consequence of Theorem 1.4 and the duality of H 1 f lag (H n ) with BM O f lag (H n ) as given in [11,12], we obtain The main idea to show our results is to apply the discrete Calderón reproducing formula, almost orthogonal estimates associated with the flag structure and the Fefferman-Stein vector valued maximal function.…”

“…Moreover, they construct a singular integral with a flag kernel on the Heisenberg group, which is not bounded on the classical Hardy spaces H 1 (H n ). Since, as pointed out in [11,12], the flag Hardy space…”

“…However, the lack of automorphic dilations underlies the failure of such multipliers to be in general bounded on the classical Hardy space H 1 and also precludes a pure product Hardy space theory on the Heisenberg group. This was the original motivation in [11] (see also [12]) to develop a theory of flag Hardy spaces H p f lag on the Heisenberg group, 0 < p ≤ 1, that is in a sense 'intermediate' between the classical Hardy spaces H p (H n ) and the product Hardy spaces H p product (C n × R) (A. Chang and R. Fefferman ([1], [2], [6], [7], [8]). They show that singular integrals with flag kernels, which include the aforementioned Marcinkiewicz multipliers, are bounded on H p f lag , as well as from H p f lag to L p , for 0 < p ≤ 1.…”

“…See [11,12] for more details about structures of dyadic cubes and strictly vertical rectangles, test function space M M +δ f lag (H n ) and its dual M M +δ f lag (H n ) ′ . The main results in this note are the following Theorem 1.4.…”

“…Note that it was proved in [11,12] that L 2 (H n ) ∩ H p f lag (H n ) is dense in H p f lag (H n ). To show Theorem 1.4, by the Definition 1.3 of the flag Hardy space, it suffices to prove that there exists a constant C such that for every f ∈ L 2 (H n ) ∩ H p f lag (H n ),…”