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

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.
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.

show all references

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.

[1]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[2]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155

[3]

Ryan Alvarado, Irina Mitrea, Marius Mitrea. Whitney-type extensions in quasi-metric spaces. Communications on Pure & Applied Analysis, 2013, 12 (1) : 59-88. doi: 10.3934/cpaa.2013.12.59

[4]

Raffaele Chiappinelli. Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem. Conference Publications, 2007, 2007 (Special) : 260-268. doi: 10.3934/proc.2007.2007.260

[5]

Toshiyuki Suzuki. Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space. Conference Publications, 2015, 2015 (special) : 1019-1024. doi: 10.3934/proc.2015.1019

[6]

Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-27. doi: 10.3934/dcdsb.2018221

[7]

Onur Alp İlhan. Solvability of some volterra type integral equations in hilbert space. Conference Publications, 2007, 2007 (Special) : 28-34. doi: 10.3934/proc.2007.2007.28

[8]

Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445

[9]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819

[10]

Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191

[11]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[12]

Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅱ. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074

[13]

Shu-Cherng Fang, David Y. Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅰ. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1291-1305. doi: 10.3934/jimo.2016073

[14]

Seung Jun Chang, Jae Gil Choi. A Cameron-Storvick theorem for the analytic Feynman integral associated with Gaussian paths on a Wiener space and applications. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2225-2238. doi: 10.3934/cpaa.2018106

[15]

Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure & Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837

[16]

Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure & Applied Analysis, 2015, 14 (2) : 527-548. doi: 10.3934/cpaa.2015.14.527

[17]

Baiyu Liu, Li Ma. Blow up threshold for a parabolic type equation involving space integral and variational structure. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2169-2183. doi: 10.3934/cpaa.2015.14.2169

[18]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061

[19]

Eric L. Grinberg, Haizhong Li. The Gauss-Bonnet-Grotemeyer Theorem in space forms. Inverse Problems & Imaging, 2010, 4 (4) : 655-664. doi: 10.3934/ipi.2010.4.655

[20]

Feimin Zhong, Jinxing Xie, Jing Jiao. Solutions for bargaining games with incomplete information: General type space and action space. Journal of Industrial & Management Optimization, 2018, 14 (3) : 953-966. doi: 10.3934/jimo.2017084

2016 Impact Factor: 0.483

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (0)

[Back to Top]