Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 4 (2016), 955-1018.

Characterizing regularity of domains via the Riesz transforms on their boundaries

Dorina Mitrea, Marius Mitrea, and Joan Verdera

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Abstract

Under mild geometric measure-theoretic assumptions on an open subset Ω of n, we show that the Riesz transforms on its boundary are continuous mappings on the Hölder space Cα(Ω) if and only if Ω is a Lyapunov domain of order α (i.e., a domain of class C1+α). In the category of Lyapunov domains we also establish the boundedness on Hölder spaces of singular integral operators with kernels of the form P(x y)|x y|n1+l, where P is any odd homogeneous polynomial of degree l in n. This family of singular integral operators, which may be thought of as generalized Riesz transforms, includes the boundary layer potentials associated with basic PDEs of mathematical physics, such as the Laplacian, the Lamé system, and the Stokes system. We also consider the limiting case α = 0 (with VMO(Ω) as the natural replacement of Cα(Ω)), and discuss an extension to the scale of Besov spaces.

Article information

Source
Anal. PDE, Volume 9, Number 4 (2016), 955-1018.

Dates
Received: 24 January 2016
Revised: 10 February 2016
Accepted: 11 March 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843304

Digital Object Identifier
doi:10.2140/apde.2016.9.955

Mathematical Reviews number (MathSciNet)
MR3530198

Zentralblatt MATH identifier
06607581

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B37: Harmonic analysis and PDE [See also 35-XX]
Secondary: 35J15: Second-order elliptic equations 15A66: Clifford algebras, spinors

Keywords
singular integral Riesz transform uniform rectifiability Hölder space Lyapunov domain Clifford algebra Cauchy–Clifford operator BMO VMO Reifenberg flat SKT domain Besov space

Citation

Mitrea, Dorina; Mitrea, Marius; Verdera, Joan. Characterizing regularity of domains via the Riesz transforms on their boundaries. Anal. PDE 9 (2016), no. 4, 955--1018. doi:10.2140/apde.2016.9.955. https://projecteuclid.org/euclid.apde/1510843304


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References

  • S. Alexander, “Local and global convexity in complete Riemannian manifolds”, Pacific J. Math. 76:2 (1978), 283–289.
  • R. Alvarado and M. Mitrea, Hardy spaces on Ahlfors-regular quasi metric spaces: a sharp theory, Lecture Notes in Mathematics 2142, Springer, Cham, 2015.
  • R. Alvarado, D. Brigham, V. Maz'ya, M. Mitrea, and E. Ziadé, “On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf–Oleinik boundary point principle”, Probl. Mat. Anal. 57 (2011), 3–68. In Russian; translated in J. Math. Sci. (N. Y.) 176:3 (2011), 281–360.
  • P. Auscher and T. Hyt önen, “Orthonormal bases of regular wavelets in spaces of homogeneous type”, Appl. Comput. Harmon. Anal. 34:2 (2013), 266–296.
  • F. Brackx, R. Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics 76, Pitman, Boston, 1982.
  • M. Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics 77, American Mathematical Society, Providence, RI, 1990.
  • R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes: étude de certaines intégrales singulières, Lecture Notes in Mathematics 242, Springer, Berlin, 1971.
  • R. R. Coifman and G. Weiss, “Extensions of Hardy spaces and their use in analysis”, Bull. Amer. Math. Soc. 83:4 (1977), 569–645.
  • D. L. Colton and R. Kress, Integral equation methods in scattering theory, John Wiley & Sons, New York, 1983.
  • G. David, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Mathematics 1465, Springer, Berlin, 1991.
  • G. David and J.-L. Journé, “A boundedness criterion for generalized Calderón–Zygmund operators”, Ann. of Math. $(2)$ 120:2 (1984), 371–397.
  • G. David and S. Semmes, Singular integrals and rectifiable sets in ${\bf R}\sp n$: beyond Lipschitz graphs, Astérisque 193, 1991.
  • G. David and S. Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs 38, American Mathematical Society, Providence, RI, 1993.
  • E. M. Dyn'kin, “Smoothness of Cauchy type integrals”, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. $($LOMI$)$ 92 (1979), 115–133, 320–321. In Russian.
  • E. M. Dyn'kin, “Smoothness of Cauchy type integrals”, Dokl. Akad. Nauk SSSR 250:4 (1980), 794–797. In Russian; translated in Sov. Math., Dokl. 21 (1980), 199–202.
  • L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, FL, 1992.
  • E. Fabes, I. Mitrea, and M. Mitrea, “On the boundedness of singular integrals”, Pacific J. Math. 189:1 (1999), 21–29.
  • H. Federer, Geometric measure theory, Grundlehren der Math. Wissenschaften 153, Springer, New York, 1969.
  • F. D. Gakhov, Boundary value problems, edited by I. N. Sneddon, Pergamon Press, Oxford, 1966.
  • J. García-Cuerva and A. E. Gatto, “Lipschitz spaces and Calderón–Zygmund operators associated to non-doubling measures”, Publ. Mat. 49:2 (2005), 285–296.
  • A. E. Gatto, “Boundedness on inhomogeneous Lipschitz spaces of fractional integrals singular integrals and hypersingular integrals associated to non-doubling measures”, Collect. Math. 60:1 (2009), 101–114.
  • Y. Han and D. Yang, “Some new spaces of Besov and Triebel–Lizorkin type on homogeneous spaces”, Studia Math. 156:1 (2003), 67–97.
  • Y. Han, D. Müller, and D. Yang, “A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces”, Abstr. Appl. Anal. 2008 (2008), art. ID 893409.
  • S. Hofmann, M. Mitrea, and M. Taylor, “Geometric and transformational properties of Lipschitz domains, Semmes–Kenig–Toro domains, and other classes of finite perimeter domains”, J. Geom. Anal. 17:4 (2007), 593–647.
  • S. Hofmann, M. Mitrea, and M. Taylor, “Singular integrals and elliptic boundary problems on regular Semmes–Kenig–Toro domains”, Int. Math. Res. Not. 2010:14 (2010), 2567–2865.
  • S. Hofmann, M. Mitrea, and M. E. Taylor, “Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus”, Anal. PDE 8:1 (2015), 115–181.
  • G. C. Hsiao and W. L. Wendland, Boundary integral equations, Applied Mathematical Sciences 164, Springer, Berlin, 2008.
  • V. Iftimie, “Fonctions hypercomplexes”, Bull. Math. Soc. Sci. Math. R. S. Roumanie 9 (1965), 279–332.
  • D. S. Jerison and C. E. Kenig, “Boundary behavior of harmonic functions in nontangentially accessible domains”, Adv. in Math. 46:1 (1982), 80–147.
  • A. Jonsson and H. Wallin, Function spaces on subsets of ${\bf R}\sp n$, Math. Rep. 1, 1984.
  • C. E. Kenig and T. Toro, “Free boundary regularity for harmonic measures and Poisson kernels”, Ann. of Math. $(2)$ 150:2 (1999), 369–454.
  • C. E. Kenig and T. Toro, “Poisson kernel characterization of Reifenberg flat chord arc domains”, Ann. Sci. École Norm. Sup. $(4)$ 36:3 (2003), 323–401.
  • R. Kress, Linear integral equations, Applied Mathematical Sciences 82, Springer, Berlin, 1989.
  • O. Martio and J. Sarvas, “Injectivity theorems in plane and space”, Ann. Acad. Sci. Fenn. Ser. A I Math. 4:2 (1979), 383–401.
  • J. Mateu, J. Orobitg, and J. Verdera, “Extra cancellation of even Calderón–Zygmund operators and quasiconformal mappings”, J. Math. Pures Appl. $(9)$ 91:4 (2009), 402–431.
  • P. Mattila, M. S. Melnikov, and J. Verdera, “The Cauchy integral, analytic capacity, and uniform rectifiability”, Ann. of Math. $(2)$ 144:1 (1996), 127–136.
  • Y. Meyer, Ondelettes et opérateurs, II: Opérateurs de Calderón–Zygmund, Hermann, Paris, 1990.
  • M. Mitrea, Clifford wavelets, singular integrals, and Hardy spaces, Lecture Notes in Mathematics 1575, Springer, Berlin, 1994.
  • D. Mitrea, Distributions, partial differential equations, and harmonic analysis, Springer, New York, 2013.
  • I. Mitrea and M. Mitrea, Multi-layer potentials and boundary problems for higher-order elliptic systems in Lipschitz domains, Lecture Notes in Mathematics 2063, Springer, Heidelberg, 2013.
  • I. Mitrea, M. Mitrea, and M. Taylor, “Cauchy integrals, Calderón projectors, and Toeplitz operators on uniformly rectifiable domains”, Adv. Math. 268 (2015), 666–757.
  • I. Mitrea, M. Mitrea, and M. Taylor, “Riemann–Hilbert problems, Cauchy integrals, and Toeplitz operators on uniformly rectifiable domains”, book manuscript, 2016.
  • N. I. Muskhelishvili, Singular integral equations: boundary problems of function theory and their application to mathematical physics, Noordhoff, Groningen, 1953.
  • F. Nazarov, X. Tolsa, and A. Volberg, “On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1”, Acta Math. 213:2 (2014), 237–321.
  • J. Plemelj, “Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend”, Monatsh. Math. Phys. 19:1 (1908), 205–210.
  • I. I. Privalov, The Cauchy integral, Saratov, 1918. In Russian.
  • I. I. Privalov, Limiting properties of single-valued analytic functions, Publ. Moscow State University, 1941.
  • S. W. Semmes, “A criterion for the boundedness of singular integrals on hypersurfaces”, Trans. Amer. Math. Soc. 311:2 (1989), 501–513.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30, Princeton University Press, Princeton, N.J., 1970.
  • M. E. Taylor, Tools for PDE: pseudodifferential operators, paradifferential operators, and layer potentials, Mathematical Surveys and Monographs 81, American Mathematical Society, Providence, RI, 2000.
  • X. Tolsa, “Principal values for Riesz transforms and rectifiability”, J. Funct. Anal. 254:7 (2008), 1811–1863.
  • R. Wittmann, “Application of a theorem of M. G. Kreĭ n to singular integrals”, Trans. Amer. Math. Soc. 299:2 (1987), 581–599.
  • W. P. Ziemer, Weakly differentiable functions: Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120, Springer, New York, 1989.