Abstract
Harnack’s inequality is one of the most fundamental inequalities for positive harmonic functions and has been extended to positive solutions of general elliptic equations and parabolic equations. This article gives a different generalization; namely, we generalize Harnack chains rather than equations. More precisely, we allow a small exceptional set and yet obtain a similar Harnack inequality. The size of an exceptional set is measured by capacity. The results are new even for classical harmonic functions. Our extended Harnack inequality includes information about the boundary behavior of positive harmonic functions. It yields a boundary Harnack principle for a very nasty domain whose boundary is given locally by the graph of a function with modulus of continuity worse than Hölder continuity.
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H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions, Math. Ann. 312 (1998), 289–318.
H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), 119–145.
H. Aikawa, Equivalence between the boundary Harnack principle and the Carleson estimate, Math. Scand. 103 (2008), 61–76.
H. Aikawa, Boundary Harnack principle and the quasihyperbolic boundary condition, Sobolev Spaces in Mathematics. II, Springer, New York, 2009, pp. 19–30.
H. Aikawa, Modulus of continuity of the Dirichlet solutions, Bull. Lond. Math. Soc. 42 (2010), 857–867.
A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble) 28 (1978), 169–213.
D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer-Verlag London Ltd., London, 2001.
R. Bañuelos, R. F. Bass, and K. Burdzy, Hölder domains and the boundary Harnack principle, Duke Math. J. 64 (1991), 195–200.
R. F. Bass and K. Burdzy, A boundary Harnack principle in twisted Hölder domains, Ann. of Math. (2) 134 (1991), 253–276.
R. F. Bass and K. Burdzy, Lifetimes of conditioned diffusions, Probab. Theory Related Fields 91 (1992), 405–443.
B. E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 275–288.
B. Fuglede, Le théorème du minimax et la théorie fine du potentiel, Ann. Inst. Fourier (Grenoble) 15 (1965), 65–88.
P. Gyrya and L. Saloff-Coste, Neumann and Dirichlet heat kernels in inner uniform domains, Astérisque (2011), no. 336.
T. Itoh, Modulus of continuity of p-Dirichlet solutions in a metric measure space, Ann. Acad. Sci. Fenn. Math. 37 (2012), 339–355.
H. Shiga, Riemann mappings of invariant components of Kleinian groups, J. Lond. Math. Soc. (2) 80 (2009), 716–728.
H. Shiga, Modulus of continuity, a Hardy-Littlewood theorem and its application, Infinite Dimensional Teichmüller Spaces and Moduli Spaces, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010, pp. 127–133.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970.
J. M. G. Wu, Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble) 28 (1978), 147–167.
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This work was supported in part by JSPS KAKENHI (Grant-in-Aid for Scientific Research) (A) 20244007.
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Aikawa, H. Extended Harnack inequalities with exceptional sets and a boundary Harnack principle. JAMA 124, 83–116 (2014). https://doi.org/10.1007/s11854-014-0028-3
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DOI: https://doi.org/10.1007/s11854-014-0028-3