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# Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type

## Recent Developments in Harmonic Analysis May 15, 2017 - May 19, 2017

May 16, 2017 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): Lesley Ward (University of South Australia--Mawson Lakes Campus)
Location: MSRI: Simons Auditorium
Tags/Keywords
• harmonic analysis

• product Hardy spaces

• heat kernel bounds

• Calderon-Zygmund theory

• multiplier operators

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

#### Ward

Abstract

Much effort has been devoted to generalizing the Calder\'on--Zygmund theory from Euclidean spaces to metric measure spaces, or spaces of homogeneous type. Here the underlying space $\mathbb{R}^n$ with Euclidean metric and Lebesgue measure is replaced by a set $X$ with a general metric or quasi-metric and a doubling measure. Further, one can replace the Laplacian operator that underpins the Calder\'on--Zygmund theory by more general operators~$L$ satisfying heat kernel estimates. I will present recent joint work with P.~Chen, X.T.~Duong, J.~Li and L.X.~Yan along these lines. We develop the theory of product Hardy spaces $H^p_{L_1,L_2}(X_1 \times X_2)$, for $1 \leq p < \infty$, defined on products of spaces of homogeneous type, and associated to operators $L_1$, $L_2$ satisfying Davies--Gaffney estimates. This theory includes definitions of Hardy spaces via appropriate square functions, an atomic Hardy space, a Calder\'on--Zygmund decomposition, interpolation theorems, and the boundedness of a class of Marcinkiewicz-type spectral multiplier operators.

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