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$L^p$-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets
About this Title
Steve Hofmann, University of Missouri, Columbia, Dorina Mitrea, University of Missouri, Columbia, Marius Mitrea, University of Missouri, Columbia and Andrew J. Morris, University of Missouri, Columbia
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 245, Number 1159
ISBNs: 978-1-4704-2260-8 (print); 978-1-4704-3607-0 (online)
DOI: https://doi.org/10.1090/memo/1159
Published electronically: July 25, 2016
Keywords: Square function,
quasi-metric space,
space of homogeneous type,
Ahlfors-David regularity,
singular integral operators,
area function,
Carleson operator,
$T(1)$ theorem for the square function,
local $T(b)$ theorem for the square function,
uniformly rectifiable sets,
tent spaces,
variable coefficient kernels
MSC: Primary 28A75, 42B20; Secondary 28A78, 42B25, 42B30
Table of Contents
Chapters
- 1. Introduction
- 2. Analysis and Geometry on Quasi-Metric Spaces
- 3. $T(1)$ and local $T(b)$ Theorems for Square Functions
- 4. An Inductive Scheme for Square Function Estimates
- 5. Square Function Estimates on Uniformly Rectifiable Sets
- 6. $L^p$ Square Function Estimates
- 7. Conclusion
Abstract
We 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, we 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, we 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.- Ryan Alvarado, Irina Mitrea, and Marius Mitrea, Whitney-type extensions in quasi-metric spaces, Commun. Pure Appl. Anal. 12 (2013), no. 1, 59–88. MR 2972422, DOI 10.3934/cpaa.2013.12.59
- Ryan Alvarado and Marius Mitrea, Hardy spaces on Ahlfors-regular quasi metric spaces, Lecture Notes in Mathematics, vol. 2142, Springer, Cham, 2015. A sharp theory. MR 3310009
- Pascal Auscher, Lectures on the Kato square root problem, Surveys in analysis and operator theory (Canberra, 2001) Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40, Austral. Nat. Univ., Canberra, 2002, pp. 1–18. MR 1953477
- Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on ${\Bbb R}^n$, Ann. of Math. (2) 156 (2002), no. 2, 633–654. MR 1933726, DOI 10.2307/3597201
- Pascal Auscher, Steve Hofmann, John L. Lewis, and Philippe Tchamitchian, Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators, Acta Math. 187 (2001), no. 2, 161–190. MR 1879847, DOI 10.1007/BF02392615
- P. Auscher, S. Hofmann, C. Muscalu, T. Tao, and C. Thiele, Carleson measures, trees, extrapolation, and $T(b)$ theorems, Publ. Mat. 46 (2002), no. 2, 257–325. MR 1934198, DOI 10.5565/PUBLMAT_{4}6202_{0}1
- Pascal Auscher and Eddy Routin, Local $Tb$ theorems and Hardy inequalities, J. Geom. Anal. 23 (2013), no. 1, 303–374. MR 3010282, DOI 10.1007/s12220-011-9249-1
- Pascal Auscher and Qi Xiang Yang, BCR algorithm and the $T(b)$ theorem, Publ. Mat. 53 (2009), no. 1, 179–196. MR 2474120, DOI 10.5565/PUBLMAT_{5}3109_{0}8
- Jonas Azzam and Raanan Schul, Hard Sard: quantitative implicit function and extension theorems for Lipschitz maps, Geom. Funct. Anal. 22 (2012), no. 5, 1062–1123. MR 2989430, DOI 10.1007/s00039-012-0189-0
- Dan Brigham, Dorina Mitrea, Irina Mitrea, and Marius Mitrea, Triebel-Lizorkin sequence spaces are genuine mixed-norm spaces, Math. Nachr. 286 (2013), no. 5-6, 503–517. MR 3048128, DOI 10.1002/mana.201100184
- D. L. Burkholder and R. F. Gundy, Distribution function inequalities for the area integral, Studia Math. 44 (1972), 527–544. MR 340557, DOI 10.4064/sm-44-6-527-544
- A. P. Calderón, On a theorem of Marcinkiewicz and Zygmund, Trans. Amer. Math. Soc. 68 (1950), 55–61. MR 32864, DOI 10.1090/S0002-9947-1950-0032864-0
- A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 177312, DOI 10.1073/pnas.53.5.1092
- Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1104656
- Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
- Michael Christ and Jean-Lin Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159 (1987), no. 1-2, 51–80. MR 906525, DOI 10.1007/BF02392554
- R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, DOI 10.1016/0022-1236(85)90007-2
- R. R. Coifman and Yves Meyer, Nonlinear harmonic analysis, operator theory and P.D.E, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 3–45. MR 864370
- Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948
- Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
- Björn E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math. 67 (1980), no. 3, 297–314. MR 592391, DOI 10.4064/sm-67-3-297-314
- Björn E. J. Dahlberg, David S. Jerison, and Carlos E. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat. 22 (1984), no. 1, 97–108. MR 735881, DOI 10.1007/BF02384374
- Guy David, Morceaux de graphes lipschitziens et intégrales singulières sur une surface, Rev. Mat. Iberoamericana 4 (1988), no. 1, 73–114 (French). MR 1009120, DOI 10.4171/RMI/64
- Guy David and Jean-Lin Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397. MR 763911, DOI 10.2307/2006946
- G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56 (French). MR 850408, DOI 10.4171/RMI/17
- G. David and S. Semmes, Singular integrals and rectifiable sets in $\textbf {R}^n$: Beyond Lipschitz graphs, Astérisque 193 (1991), 152 (English, with French summary). MR 1113517
- Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061
- Donggao Deng and Yongsheng Han, Harmonic analysis on spaces of homogeneous type, Lecture Notes in Mathematics, vol. 1966, Springer-Verlag, Berlin, 2009. With a preface by Yves Meyer. MR 2467074
- Eugene Fabes, Layer potential methods for boundary value problems on Lipschitz domains, Potential theory—surveys and problems (Prague, 1987) Lecture Notes in Math., vol. 1344, Springer, Berlin, 1988, pp. 55–80. MR 973881, DOI 10.1007/BFb0103344
- Charles Fefferman, Recent progress in classical Fourier analysis, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 95–118. MR 0510853
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- Tadeusz Figiel, Singular integral operators: a martingale approach, Geometry of Banach spaces (Strobl, 1989) London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 95–110. MR 1110189
- A. Grau de la Herran, Ph.D. Thesis, University of Missouri, 2012.
- Ana Grau de la Herran and Mihalis Mourgoglou, A local $Tb$ theorem for square functions in domains with Ahlfors-David regular boundaries, J. Geom. Anal. 24 (2014), no. 3, 1619–1640. MR 3223569, DOI 10.1007/s12220-013-9388-7
- Yongsheng Han, Detlef Müller, and Dachun Yang, Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), no. 13-14, 1505–1537. MR 2269253, DOI 10.1002/mana.200610435
- Y. S. Han and E. T. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc. 110 (1994), no. 530, vi+126. MR 1214968, DOI 10.1090/memo/0530
- Christopher Heil, A basis theory primer, Expanded edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2011. MR 2744776
- Steve Hofmann, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, Duke Math. J. 90 (1997), no. 2, 209–259. MR 1484857, DOI 10.1215/S0012-7094-97-09008-6
- S. Hofmann, A proof of the local $Tb$ Theorem for standard Calderón-Zygmund operators, unpublished manuscript, http://www.math.missouri.edu/$\sim$hofmann/
- Steve Hofmann, Local $Tb$ theorems and applications in PDE, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1375–1392. MR 2275650
- Steve Hofmann, A local $Tb$ theorem for square functions, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 175–185. MR 2500492, DOI 10.1090/pspum/079/2500492
- Steve Hofmann, Michael Lacey, and Alan McIntosh, The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds, Ann. of Math. (2) 156 (2002), no. 2, 623–631. MR 1933725, DOI 10.2307/3597200
- Steve Hofmann and John L. Lewis, Square functions of Calderón type and applications, Rev. Mat. Iberoamericana 17 (2001), no. 1, 1–20. MR 1846089, DOI 10.4171/RMI/287
- Steve Hofmann and José María Martell, Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 3, 577–654 (English, with English and French summaries). MR 3239100, DOI 10.24033/asens.2223
- Steve Hofmann, José María Martell, and Ignacio Uriarte-Tuero, Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^p$ imply uniform rectifiability, Duke Math. J. 163 (2014), no. 8, 1601–1654. MR 3210969, DOI 10.1215/00127094-2713809
- Steve Hofmann and Alan McIntosh, The solution of the Kato problem in two dimensions, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), 2002, pp. 143–160. MR 1964818, DOI 10.5565/PUBLMAT_{E}sco02_{0}6
- Steve Hofmann and Alan McIntosh, Boundedness and applications of singular integrals and square functions: a survey, Bull. Math. Sci. 1 (2011), no. 2, 201–244. MR 2901001, DOI 10.1007/s13373-011-0014-3
- Steve Hofmann, Marius Mitrea, and Michael Taylor, Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains, J. Geom. Anal. 17 (2007), no. 4, 593–647. MR 2365661, DOI 10.1007/BF02937431
- Guoen Hu, Dachun Yang, and Yuan Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type, Taiwanese J. Math. 13 (2009), no. 1, 91–135. MR 2489309, DOI 10.11650/twjm/1500405274
- Tuomas P. Hytönen, An operator-valued $Tb$ theorem, J. Funct. Anal. 234 (2006), no. 2, 420–463. MR 2216905, DOI 10.1016/j.jfa.2005.11.001
- Tuomas Hytönen and Henri Martikainen, Non-homogeneous $Tb$ theorem and random dyadic cubes on metric measure spaces, J. Geom. Anal. 22 (2012), no. 4, 1071–1107. MR 2965363, DOI 10.1007/s12220-011-9230-z
- Tuomas Hytönen and Henri Martikainen, On general local $Tb$ theorems, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4819–4846. MR 2922611, DOI 10.1090/S0002-9947-2012-05599-1
- Tuomas Hytönen and Lutz Weis, A $T1$ theorem for integral transformations with operator-valued kernel, J. Reine Angew. Math. 599 (2006), 155–200. MR 2279101, DOI 10.1515/CRELLE.2006.081
- David S. Jerison and Carlos E. Kenig, Hardy spaces, $A_{\infty }$, and singular integrals on chord-arc domains, Math. Scand. 50 (1982), no. 2, 221–247. MR 672926, DOI 10.7146/math.scand.a-11956
- Peter W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 24–68. MR 1013815, DOI 10.1007/BFb0086793
- Carlos E. Kenig, Weighted $H^{p}$ spaces on Lipschitz domains, Amer. J. Math. 102 (1980), no. 1, 129–163. MR 556889, DOI 10.2307/2374173
- Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257–270. MR 546295, DOI 10.1016/0001-8708(79)90012-4
- Roberto A. Macías and Carlos Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 271–309. MR 546296, DOI 10.1016/0001-8708(79)90013-6
- Alan McIntosh and Yves Meyer, Algèbres d’opérateurs définis par des intégrales singulières, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 8, 395–397 (French, with English summary). MR 808636
- D. Mitrea, I. Mitrea, and M. Mitrea, Weighted mixed-normed spaces on spaces of homogeneous type, preprint, (2012).
- Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Sylvie Monniaux, Groupoid metrization theory, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013. With applications to analysis on quasi-metric spaces and functional analysis. MR 2987059
- D. Mitrea, I. Mitrea and M. Mitrea, A Treatise on the Theory of Elliptic Boundary Value Problems, Singular Integral Operators, and Smoothness Spaces in Rough Domains, book manuscript, 2011.
- Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Elia Ziadé, Abstract capacity estimates and the completeness and separability of certain classes of non-locally convex topological vector spaces, J. Funct. Anal. 262 (2012), no. 11, 4766–4830. MR 2913687, DOI 10.1016/j.jfa.2012.03.012
- Dorina Mitrea, Marius Mitrea, and Michael Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Mem. Amer. Math. Soc. 150 (2001), no. 713, x+120. MR 1809655, DOI 10.1090/memo/0713
- Marius Mitrea, Clifford wavelets, singular integrals, and Hardy spaces, Lecture Notes in Mathematics, vol. 1575, Springer-Verlag, Berlin, 1994. MR 1295843
- Marius Mitrea, On Dahlberg’s Lusin area integral theorem, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1449–1455. MR 1239801, DOI 10.1090/S0002-9939-1995-1239801-7
- F. Nazarov, S. Treil, and A. Volberg, Accretive system $Tb$-theorems on nonhomogeneous spaces, Duke Math. J. 113 (2002), no. 2, 259–312. MR 1909219, DOI 10.1215/S0012-7094-02-11323-4
- Carlos Segovia, On the area function of Lusin, Studia Math. 33 (1969), 311–343. MR 288299, DOI 10.4064/sm-33-3-311-343
- Stephen Semmes, Square function estimates and the $T(b)$ theorem, Proc. Amer. Math. Soc. 110 (1990), no. 3, 721–726. MR 1028049, DOI 10.1090/S0002-9939-1990-1028049-2
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- E. M. Stein, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 359–376. MR 663787, DOI 10.1090/S0273-0979-1982-15040-6
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
- Hans Triebel, Fractals and spectra, Monographs in Mathematics, vol. 91, Birkhäuser Verlag, Basel, 1997. Related to Fourier analysis and function spaces. MR 1484417