On the Calderon-Zygmund decomposition lemma on the
Walsh-Paley group
##
On the Calderon-Zygmund decomposition lemma on the
Walsh-Paley group

## G. Gát

It is well-known that the Calderon-Zygmund decomposition
lemma plays an extremely prominent role in the theory of harmonic analysis on
the Walsh-Paley group. However, the proof of this lemma uses the fact that the
$2^n$th partial sums of the Walsh-Fourier series of an integrable function
converges a.e. to the function. This later proved by techniques known in the
martingale theory. In this paper we give a ``purely dyadic harmonic analysis''
proof for the Calderon-Zygmund decomposition lemma.

* Mathematics Subject Classification:
*
42C10, 43A75