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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

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.  Google Scholar

[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), 633-654. doi: 10.2307/3597201.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61 (1990), 601-628.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. David and S. Semmes, Singular integrals and rectifiable sets in $\mathbbR^n$: Beyond Lipschitz graphs, Astérisque, 193 (1991), 152 pp.  Google Scholar

[11]

G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, 38, AMS, Providence, RI, 1993.  Google Scholar

[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.  Google Scholar

[13]

S. Hofmann, Local $Tb$ Theorems and applications in PDE, in International Congress of Mathematicians, Vol. II, European Math. Soc., Zürich, 2006, 1375-1392.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Mitrea, I. Mitrea and M. Mitrea, Weighted mixed-normed spaces on spaces of homogeneous type, preprint, (2012). Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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), 633-654. doi: 10.2307/3597201.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61 (1990), 601-628.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. David and S. Semmes, Singular integrals and rectifiable sets in $\mathbbR^n$: Beyond Lipschitz graphs, Astérisque, 193 (1991), 152 pp.  Google Scholar

[11]

G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, 38, AMS, Providence, RI, 1993.  Google Scholar

[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.  Google Scholar

[13]

S. Hofmann, Local $Tb$ Theorems and applications in PDE, in International Congress of Mathematicians, Vol. II, European Math. Soc., Zürich, 2006, 1375-1392.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Mitrea, I. Mitrea and M. Mitrea, Weighted mixed-normed spaces on spaces of homogeneous type, preprint, (2012). Google Scholar

[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.  Google Scholar

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