- 1st ed.
- 1 online resource (120 pages)
- Memoirs of the American Mathematical Society ; v.245 .
- Memoirs of the American Mathematical Society .
Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Analysis and Geometry on Quasi-Metric Spaces -- 2.1. A metrization result for general quasi-metric spaces -- 2.2. Geometrically doubling quasi-metric spaces -- 2.3. Approximations to the identity on quasi-metric spaces -- 2.4. Dyadic Carleson tents -- Chapter 3. (1) and local ( ) Theorems for Square Functions -- 3.1. An arbitrary codimension (1) theorem for square functions -- 3.2. An arbitrary codimension local ( ) theorem for square functions -- Chapter 4. An Inductive Scheme for Square Function Estimates -- Chapter 5. Square Function Estimates on Uniformly Rectifiable Sets -- 5.1. Square function estimates on Lipschitz graphs -- 5.2. Square function estimates on (BP)^LG sets -- 5.3. Square function estimates for integral operators with variable kernels -- Chapter 6. ^ Square Function Estimates -- 6.1. Mixed norm spaces -- 6.2. Estimates relating the Lusin and Carleson operators -- 6.3. Weak ^ square function estimates imply ² square function estimates -- 6.4. Extrapolating square function estimates -- Chapter 7. Conclusion -- References -- Back Cover.
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.