**SMS scnews item created by Daniel Daners at Fri 18 Apr 2014 1106**

Type: Seminar

Distribution: World

Expiry: 28 Apr 2014

**Calendar1: 28 Apr 2014 1400-1500**

**CalLoc1: AGR Carslaw 829**

Auth: daners@d110-33-100-217.mas800.nsw.optusnet.com.au (ddan2237) in SMS-WASM

### PDE Seminar

# Littlewood-Paley and Hardy space theory on spaces of homogeneous type in the sense of Coifman and Weiss via orthonormal wavelet bases

### Li

Ji Li

Macquarie University

28 April 2014 14:00-15:00, Carslaw Room 829 (AGR)

## Abstract

Spaces of homogeneous type were introduced by Coifman and Weiss in
the early 1970s. They include many special spaces in analysis and have
many applications in the theory of singular integrals and function spaces.
For instance, Coifman and Weiss introduced the atomic Hardy space
${H}_{CW}^{p}\left(X\right)$ on
$\left(X,d,\mu \right)$ and proved
that if $T$
is a Calderón-Zygmund singular integral operator that is bounded on
${L}_{2}\left(X\right)$, then
$T$ is bounded
from ${H}_{CW}^{p}\left(X\right)$
to ${L}^{p}\left(X\right)$ for
some $p\le 1$.
However, for some applications, additional assumptions were imposed
on these general spaces of homogeneous type, because the quasi-metric
$d$ may
have no regularity and quasi-metric balls, even Borel sets, may not be
open.

Using the remarkable orthonormal wavelet basis constructed recently by
Auscher and Hytönen, we establish the theory of product Hardy spaces on spaces
$X={X}_{1}\times {X}_{2}\times \cdots \times {X}_{n}$, where each
factor ${X}_{i}$
is a space of homogeneous type in the sense of Coifman and Weiss.
The main tool we develop is the Littlewood-Paley theory on
$X$,
which in turn is a consequence of a corresponding theory on each factor
space.

We make no additional assumptions on the quasi-metric or the doubling
measure for each factor space, and thus we extend to the full generality of
product spaces of homogeneous type the aspects of both one-parameter and
multiparameter theory involving the Littlewood-Paley theory and function spaces.
Moreover, our methods would be expected to be a powerful tool for developing
function spaces and the boundedness of singular integrals on spaces of
homogeneous type.

This is joint work with Yongsheng Han and Lesley Ward.

Check also the PDE
Seminar page. Enquiries to Daniel Hauer or Daniel Daners.

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