Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets

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January 2014, 21: 8-18. doi: 10.3934/era.2014.21.8

Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets

Steve Hofmann ^1, , Dorina Mitrea ^1, , Marius Mitrea ^1, and Andrew J. Morris ^1,

1.	University of Missouri, Columbia, MO 65211, United States, United States, United States, United States

Received November 2013 Published February 2014

Abstract
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We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation results for $L^2$ square function estimates related to the analysis of integral operators that act on Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The inductive scheme is a natural application of the local $T(b)$ theorem and it implies the stability of $L^2$ square function estimates under the so-called big pieces functor. In particular, this analysis implies $L^p$ and Hardy space square function estimates for integral operators on uniformly rectifiable subsets of the Euclidean space.

Keywords: Ahlfors-David regular, tent space, quasi-metric space, Square function, local $T(b)$ theorem for the square function, singular integral operator, space of homogeneous type, variable coefficient kernel., uniformly rectifiable set.

Mathematics Subject Classification: Primary: 28A75, 42B20; Secondary: 28A78, 42B25, 42B3.

Citation: Steve Hofmann, Dorina Mitrea, Marius Mitrea, Andrew J. Morris. Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets. Electronic Research Announcements, 2014, 21: 8-18. doi: 10.3934/era.2014.21.8

References:

[1]	P. Auscher, Lectures on the Kato square root problem,, in Surveys in Analysis and Operator Theory* (Canberra*, (2001), 1.
[2]	P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $\mathbbR^n$,, Annals of Math. (2), 156 (2002), 633. doi: 10.2307/3597201.
[3]	J. Azzam and R. Schul, Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps,, Geom. Funct. Anal., 22 (2012), 1062. doi: 10.1007/s00039-012-0189-0.
[4]	D. Brigham, D. Mitrea, I. Mitrea and M. Mitrea, Triebel-Lizorkin sequence spaces are genuine mixed-norm spaces,, Math. Nachr., 286 (2013), 503. doi: 10.1002/mana.201100184.
[5]	M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral,, Colloq. Math., 60/61 (1990), 601.
[6]	R. R. Coifman, A. McIntosh and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $L^{2}$ pour les courbes lipschitziennes,, Ann. of Math. (2), 116 (1982), 361. doi: 10.2307/2007065.
[7]	R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis,, J. Funct. Anal., 62 (1985), 304. doi: 10.1016/0022-1236(85)90007-2.
[8]	R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis,, Bull. Amer. Math. Soc., 83 (1977), 569. doi: 10.1090/S0002-9904-1977-14325-5.
[9]	G. David, Morceaux de graphes lipschitziens et intégrales singulières sur une surface,, Rev. Mat. Iberoamericana, 4 (1988), 73. doi: 10.4171/RMI/64.
[10]	G. David and S. Semmes, Singular integrals and rectifiable sets in $\mathbbR^n$: Beyond Lipschitz graphs,, Astérisque, 193 (1991).
[11]	G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets,, Mathematical Surveys and Monographs, (1993).
[12]	S. Hofmann, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials,, Duke Math. J., 90 (1997), 209. doi: 10.1215/S0012-7094-97-09008-6.
[13]	S. Hofmann, Local $Tb$ Theorems and applications in PDE,, in International Congress of Mathematicians, (2006), 1375.
[14]	S. Hofmann, M. Lacey and A. McIntosh, The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds,, Annals of Math. (2), 156 (2002), 623. doi: 10.2307/3597200.
[15]	S. Hofmann and J. L. Lewis, Square functions of Calderón type and applications,, Rev. Mat. Iberoamericana, 17 (2001), 1. doi: 10.4171/RMI/287.
[16]	S. Hofmann and A. McIntosh, The solution of the Kato problem in two dimensions,, in Proceedings of the Conference on Harmonic Analysis and PDE* (El Escorial*, (2000), 143. doi: 10.5565/PUBLMAT_Esco02_06.
[17]	S. Hofmann and A. McIntosh, Boundedness and applications of singular integrals and square functions: A survey,, Bull. Math. Sci., 1 (2011), 201. doi: 10.1007/s13373-011-0014-3.
[18]	R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type,, Adv. in Math., 33 (1979), 257. doi: 10.1016/0001-8708(79)90012-4.
[19]	R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type,, Adv. in Math., 33 (1979), 271. doi: 10.1016/0001-8708(79)90013-6.
[20]	D. Mitrea, I. Mitrea and M. Mitrea, Weighted mixed-normed spaces on spaces of homogeneous type,, preprint, (2012).
[21]	D. Mitrea, I. Mitrea, M. Mitrea and S. Monniaux, Groupoid Metrization Theory. With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis,, Applied and Numerical Harmonic Analysis, (2013). doi: 10.1007/978-0-8176-8397-9.

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References:

[1]	P. Auscher, Lectures on the Kato square root problem,, in Surveys in Analysis and Operator Theory* (Canberra*, (2001), 1.
[2]	P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $\mathbbR^n$,, Annals of Math. (2), 156 (2002), 633. doi: 10.2307/3597201.
[3]	J. Azzam and R. Schul, Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps,, Geom. Funct. Anal., 22 (2012), 1062. doi: 10.1007/s00039-012-0189-0.
[4]	D. Brigham, D. Mitrea, I. Mitrea and M. Mitrea, Triebel-Lizorkin sequence spaces are genuine mixed-norm spaces,, Math. Nachr., 286 (2013), 503. doi: 10.1002/mana.201100184.
[5]	M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral,, Colloq. Math., 60/61 (1990), 601.
[6]	R. R. Coifman, A. McIntosh and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $L^{2}$ pour les courbes lipschitziennes,, Ann. of Math. (2), 116 (1982), 361. doi: 10.2307/2007065.
[7]	R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis,, J. Funct. Anal., 62 (1985), 304. doi: 10.1016/0022-1236(85)90007-2.
[8]	R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis,, Bull. Amer. Math. Soc., 83 (1977), 569. doi: 10.1090/S0002-9904-1977-14325-5.
[9]	G. David, Morceaux de graphes lipschitziens et intégrales singulières sur une surface,, Rev. Mat. Iberoamericana, 4 (1988), 73. doi: 10.4171/RMI/64.
[10]	G. David and S. Semmes, Singular integrals and rectifiable sets in $\mathbbR^n$: Beyond Lipschitz graphs,, Astérisque, 193 (1991).
[11]	G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets,, Mathematical Surveys and Monographs, (1993).
[12]	S. Hofmann, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials,, Duke Math. J., 90 (1997), 209. doi: 10.1215/S0012-7094-97-09008-6.
[13]	S. Hofmann, Local $Tb$ Theorems and applications in PDE,, in International Congress of Mathematicians, (2006), 1375.
[14]	S. Hofmann, M. Lacey and A. McIntosh, The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds,, Annals of Math. (2), 156 (2002), 623. doi: 10.2307/3597200.
[15]	S. Hofmann and J. L. Lewis, Square functions of Calderón type and applications,, Rev. Mat. Iberoamericana, 17 (2001), 1. doi: 10.4171/RMI/287.
[16]	S. Hofmann and A. McIntosh, The solution of the Kato problem in two dimensions,, in Proceedings of the Conference on Harmonic Analysis and PDE* (El Escorial*, (2000), 143. doi: 10.5565/PUBLMAT_Esco02_06.
[17]	S. Hofmann and A. McIntosh, Boundedness and applications of singular integrals and square functions: A survey,, Bull. Math. Sci., 1 (2011), 201. doi: 10.1007/s13373-011-0014-3.
[18]	R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type,, Adv. in Math., 33 (1979), 257. doi: 10.1016/0001-8708(79)90012-4.
[19]	R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type,, Adv. in Math., 33 (1979), 271. doi: 10.1016/0001-8708(79)90013-6.
[20]	D. Mitrea, I. Mitrea and M. Mitrea, Weighted mixed-normed spaces on spaces of homogeneous type,, preprint, (2012).
[21]	D. Mitrea, I. Mitrea, M. Mitrea and S. Monniaux, Groupoid Metrization Theory. With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis,, Applied and Numerical Harmonic Analysis, (2013). doi: 10.1007/978-0-8176-8397-9.

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