Duke Mathematical Journal

Caloric measure in parabolic flat domains

Steve Hofmann, John L. Lewis, and Kaj Nyström

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we define parabolic chord arc domains and show that in a parabolic chord arc domain with vanishing constant, the logarithm of the density of caloric measure with respect to a certain projective Lebesgue measure is of vanishing mean oscillation. We also obtain a partial converse to this result. Our work generalizes to the heat equation recent work of Kenig and Toro for the Laplacian.

Article information

Duke Math. J., Volume 122, Number 2 (2004), 281-346.

First available in Project Euclid: 14 April 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20 35K05


Hofmann, Steve; Lewis, John L.; Nyström, Kaj. Caloric measure in parabolic flat domains. Duke Math. J. 122 (2004), no. 2, 281--346. doi:10.1215/S0012-7094-04-12222-5. https://projecteuclid.org/euclid.dmj/1081971769

Export citation


  • H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144.
  • I. Athanasopoulos, L. Caffarelli, and S. Salsa, Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems, Ann. of Math. (2) 143 (1996), 413–434.
  • –. –. –. –., Regularity of the free boundary in parabolic phase-transition problems, Acta Math. 176 (1996), 245–282.
  • R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.
  • B. E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 275–288.
  • G. David and S. Semmes, Singular Integrals and Rectifiable Sets in $ \mathbbR^n$: Beyond Lipschitz Graphs, Astérisque 193, Soc. Math. France, Montrouge, 1991.
  • ––––, Analysis of and on Uniformly Rectifiable Sets, Math. Surveys Monogr. 38, Amer. Math. Soc, Providence, 1993.
  • J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Grundlehren Math. Wiss. 262, Springer, New York, 1984.
  • L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, Amer. Math. Soc., Providence, 1998.
  • E. B. Fabes and M. V. Safonov, Behaviour near the boundary of positive solutions of second order parabolic equations, J. Fourier Anal. Appl. 3 (1997), special issue, 871–882.
  • E. B. Fabes, M. V. Safonov, and Y. Yuan, Behaviour near the boundary of positive solutions of second order parabolic equations, II, Trans. Amer. Math. Soc. 351 (1999), 4947–4961.
  • S. Hofmann, Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials, Duke Math. J. 90 (1996), 209–259.
  • S. Hofmann and J. L. Lewis, $L^2$ solvability and representation by caloric layer potentials in time-varying domains, Ann. of Math. (2) 144 (1997), 349–420.
  • ––––, The Dirichlet problem for parabolic operators with singular drift terms, Mem. Amer. Math. Soc. 151 (2001), no. 719.
  • ––––, The $L^p$ Neumann problem for the heat equation in noncylindrical domains, preprint, http://www.math.missouri.edu/~hofmann/
  • S. Hofmann, J. L. Lewis, and K. Nyström, Existence of big pieces of graphs for parabolic problems, Ann. Acad. Sci. Fenn. Math. 28 (2003), 355–384.
  • D. Jerison, Regularity of the Poisson kernel and free boundary problems, Colloq. Math. 60/61 (1990), 547–568.
  • D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), 80–147.
  • –. –. –. –., The logarithm of the Poisson kernel of a $ C^1 $ domain has vanishing mean oscillation, Trans. Amer. Math. Soc. 273 (1982), 781–794.
  • P. W. Jones, “Square functions, Cauchy integrals, analytic capacity, and harmonic measure” in Harmonic Analysis and Partial Differential Equations (El Escorial, Spain, 1987), ed. J. García-Cuerva, Lecture Notes in Math. 1384, Springer, Berlin, 1989, 24–68.
  • –. –. –. –., Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), 1–15.
  • R. Kaufman and J.-M. Wu, Parabolic measure on domains of class $\Lip 1/2$, Compositio Math. 65 (1988), 201–207.
  • C. E. Kenig and T. Toro, Harmonic measure on locally flat domains, Duke Math J. 87 (1997), 509–551.
  • –. –. –. –., Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math. (2) 150 (1999), 369–454.
  • –. –. –. –., Poisson kernel characterization of Reifenberg flat chord arc domains, Ann. Sci. École Norm. Sup. (4) 36 (2003), 323–401.
  • O. Kowalski and D. Preiss, Besicovitch-type properties of measures and submanifolds, J. Reine Angew. Math. 379 (1987), 115–151.
  • J. L. Lewis and M. A. M. Murray, The method of layer potentials for the heat equation in time-varying domains, Mem. Amer. Math. Soc. 114 (1995), no. 545.
  • J. L. Lewis and J. Silver, Parabolic measure and the Dirichlet problem for the heat equation in two dimensions, Indiana Univ. Math. J. 37 (1988), 801–839.
  • P. Mattila, M. S. Melnikov, and J. Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1997), 127–136.
  • K. Nyström, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J. 46 (1997), 183–245.
  • D. Preiss, Geometry of measures in $\mathbfR^n$: Distribution, rectifiability, and densities, Ann. of Math. (2) 125 (1987), 537–643.
  • S. Semmes, Chord-arc surfaces with small constant, I, Adv. in Math. 85 (1991), 198–223.
  • –. –. –. –., Chord-arc surfaces with small constant, II: Good parametrizations, Adv. in Math. 88 (1991), 170–199.
  • E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Monogr. Harmon. Anal. 3, Princeton Univ. Press, Princeton, 1993.
  • R. S. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana Univ. Math. J. 29 (1980), 539–558.