**Memoirs of the American Mathematical Society**

2016;
108 pp;
Softcover

MSC: Primary 28; 42;

**Print ISBN: 978-1-4704-2260-8
Product Code: MEMO/245/1159**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-3607-0
Product Code: MEMO/245/1159.E**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

# \(L^{p}\)-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets

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*Steve Hofmann; Dorina Mitrea; Marius Mitrea; Andrew J. Morris*

The authors establish square function estimates for integral
operators on uniformly rectifiable sets by proving a local
\(T(b)\) theorem and applying it to show that such estimates
are stable under the so-called big pieces functor. More generally,
they consider integral operators associated with Ahlfors-David
regular sets of arbitrary codimension in ambient quasi-metric
spaces. The local \(T(b)\) theorem is then used to establish
an inductive scheme in which square function estimates on so-called
big pieces of an Ahlfors-David regular set are proved to be
sufficient for square function estimates to hold on the entire
set.

Extrapolation results for \(L^p\) and Hardy space versions
of these estimates are also established. Moreover, the authors prove
square function estimates for integral operators associated with
variable coefficient kernels, including the Schwartz kernels of
pseudodifferential operators acting between vector bundles on
subdomains with uniformly rectifiable boundaries on
manifolds.

#### Table of Contents

# Table of Contents

## $L^{p}$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Analysis and Geometry on Quasi-Metric Spaces 1320
- Chapter 3. ๐(1) and local ๐(๐) Theorems for Square Functions 3542
- Chapter 4. An Inductive Scheme for Square Function Estimates 5158
- Chapter 5. Square Function Estimates on Uniformly Rectifiable Sets 5764
- Chapter 6. ๐ฟ^{๐} Square Function Estimates 7178
- Chapter 7. Conclusion 101108
- References 105112
- Back Cover Back Cover1120