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

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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), Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, 2002, 1-18.
[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 $\mathbb{R}^{N}$, Annals of Math. (2), 156 (2002), 633-654. 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), 1062-1123. 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), 503-517. 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), 601-628.
[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), 361-387. 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), 304-335. 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), 569-645. 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), 73-114. doi: 10.4171/RMI/64.
[10]	G. David and S. Semmes, Singular integrals and rectifiable sets in $\mathbb{R}^{N}$: Beyond Lipschitz graphs, Astérisque, 193 (1991), 152 pp.
[11]	G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, 38, AMS, Providence, RI, 1993.
[12]	S. Hofmann, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, Duke Math. J., 90 (1997), 209-259. doi: 10.1215/S0012-7094-97-09008-6.
[13]	S. Hofmann, Local $Tb$ Theorems and applications in PDE, in International Congress of Mathematicians, Vol. II, European Math. Soc., Zürich, 2006, 1375-1392.
[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), 623-631. doi: 10.2307/3597200.
[15]	S. Hofmann and J. L. Lewis, Square functions of Calderón type and applications, Rev. Mat. Iberoamericana, 17 (2001), 1-20. 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), Vol. Extra, Publ. Mat., 2002, 143-160. 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), 201-244. 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), 257-270. 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), 271-309. 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, Birkhäuser/Springer, New York, 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), Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, 2002, 1-18.
[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 $\mathbb{R}^{N}$, Annals of Math. (2), 156 (2002), 633-654. 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), 1062-1123. 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), 503-517. 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), 601-628.
[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), 361-387. 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), 304-335. 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), 569-645. 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), 73-114. doi: 10.4171/RMI/64.
[10]	G. David and S. Semmes, Singular integrals and rectifiable sets in $\mathbb{R}^{N}$: Beyond Lipschitz graphs, Astérisque, 193 (1991), 152 pp.
[11]	G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, 38, AMS, Providence, RI, 1993.
[12]	S. Hofmann, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, Duke Math. J., 90 (1997), 209-259. doi: 10.1215/S0012-7094-97-09008-6.
[13]	S. Hofmann, Local $Tb$ Theorems and applications in PDE, in International Congress of Mathematicians, Vol. II, European Math. Soc., Zürich, 2006, 1375-1392.
[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), 623-631. doi: 10.2307/3597200.
[15]	S. Hofmann and J. L. Lewis, Square functions of Calderón type and applications, Rev. Mat. Iberoamericana, 17 (2001), 1-20. 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), Vol. Extra, Publ. Mat., 2002, 143-160. 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), 201-244. 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), 257-270. 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), 271-309. 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, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-0-8176-8397-9.

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