Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
D. R. Adams, Sets and functions of finite L p -capacity, Indiana Univ. Math. J. 27 (1978), 611–627.
-, Weighted nonlinear potential theory, Trans. Amer. Math. Soc. 297 (1986), 73–94.
D. R. Adams and L.-I. Hedberg, Function Spaces and Potential Theory, Springer, 1996.
D. R. Adams and N. G. Meyers, Bessel potentials. Inclusion relations among classes of exceptional sets, Indiana Univ. Math. J. 22 (1973), 873–905.
R. A. Adams, Sobolev Spaces, Academic Press, 1975.
P. Ahern, The Poisson integral of a singular measure, Can. J. Math. 55 (1983), 735–749.
H. Aikawa, Tangential boundary behavior of Green potentials and contractive properties of L p -capacities, Tokyo J. Math. 9 (1986), 223–245.
-, Comparison of L p -capacity and Hausdorff measure, Complex Variables 15 (1990), 223–232.
-, Harmonic functions and Green potentials having no tangential limits, J. Lodon Math. Soc. 43 (1991), 125–136.
-, Quasiadditivity of Riesz capacity Math. Scand. 69 (1991), 15–30.
-, Thin sets at the boundary, Proc. London Math. Soc. (3) 65 (1992), 357–382.
-, Quasiadditivity of capacity and minimal thinness, Ann. Acad. Sci. Fenn. Ser. A. I. mathematica 18 (1993), 65–75.
-, Integrability of superharmonic functions and subharmonic functions, Proc. Amer. Math. Soc. 120 (1994), 109–117.
H. Aikawa, Densities with the mean value property for harmonic functions in a Lipschitz domain, preprint (1995).
H. Aikawa and A. A. Borichev, Quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions, Trans. Amer. Math. Soc. 348 (1996), 1013–1030.
A. Ancona, On strong barriers and an inequality of Hardy for domains in R n, J. London Math. Soc. (2) 34 (1986), 274–290.
D. H. Armitage, On the global integrability of superharmonic functions in balls, J. London Math. Soc. (2) 4 (1971), 365–373.
-, Further result on the global integrability of superharmonic functions, J. London Math. Soc. (2) 6 (1972), 109–121.
J. P. Aubin, Applied Functional Analysis, Wiely, 1979.
J. Bliedtner and W. Hansen, Potential Theory, An Analytic and Probabilistic Approach to Balayage, Springer, 1986, Universitext.
M. Brelot, On Topologies and Boundaries in Potential Theory, Springer, 1971, Lecture Notes in Math. 175.
M. Brelot and J. L. Doob, Limites angularies et limites fines, Ann. Inst. Fourier, Grenoble 13 (1963), 395–415.
L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand, 1967.
B. E. J. Dahlberg, A minimum principle for positive harmonic functions, Proc. London Math. Soc. (3) 33 (1976), 238–250.
J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer, 1984.
M. Essén, On Wiener conditions for minimally thin sets and rarefied sets, 41–50, Birkhäuser, 1988, pp. 41–50, Articles dedicated to A. Pfluger on the occasion of his 80th birthday.
-, On minimal thinness, reduced functions and Green potentials, Proc. Edinburgh Math. Soc. 36 (1992), 87–106.
M. Essén and H. L. Jackson, On the covering properties of certain exceptional sets in a half-space, Hiroshima Math. J. 10 (1980) 233–262.
C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137–193.
B. Fuglede, Le théorème du minimax et la théorie fine du potentiel, Ann. Inst. Fourier 15 (1965), 65–87.
-, A simple proof that certain capacities decrease under contraction, Hiroshima Math. J. 19 (1989), 567–573.
S. J. Gardiner, A short proof of Burdzy’s theorem on the angular derivative, Bull. London Math. Soc. 23 (1991), 575–579.
K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), 77–102.
W. K. Hayman, Subharmonic Functions, Vol. 2, Academic Press, 1989.
L. I. Hedberg and Th. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 4, 161–187.
L. L. Helms, Introduction to Potential Theory, Wiley, 1969.
R. R. Hunt and R. L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 505–527.
D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math. 46 (1982), 80–147.
R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 207–228.
N. S. Landkof, Foundations of Modern Potential Theory, Springer, 1972.
S. Lang, Real Analysis, Second Edition, Addison-Wesley, 1983.
J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positive dans un demi-espace, Ann. Sci. École Norm. Sup. 66 (1949), 125–159.
F. D. Lesley, Conformal mappings of domains satisfying a wedge condition, Proc. Amer. Math. Soc. 93 (1985), 483–488.
J. L. Lewis, Uniformly fat sets, Trans. Amer. Math. Soc. 308 (1988), 177–196.
P. Lindqvist, Global integrability and degenerate quasilinear elliptic equations, J. Analyse Math. 61 (1993), 283–292.
F.-Y. Maeda and N. Suzuki, The integrability of superharmonic functions on Lipschitz domains, Bull. London Math. Soc. 21 (1989), 270–278.
M. Masumoto, A distorsion theorem for conformal mappings with an application to subharmonic functions, Hiroshima Math. J. 20 (1990), 341–350.
-, Integrability of superharmonic functions on plane domains, J. London Math. Soc. (2) 45 (1992), 62–78.
-, Integrability of superharmonic functions on Hölder domains of the plane, Proc. Amer. Math. Soc. 117 (1993), 1083–1088.
V. G. Maz’ya, Beurling’s theorem on a minimum principle for positive harmonic founctions, Zapiski Nauchnykh Seminarov LOMI 30 (1972), 76–90, (Russian).
-, Beurling’s theorem on a minimum principle for positive harmonic functions, J. Soviet Math. 4 (1975), 367–379, (English translation).
V. G. Maz’ya, Sobolev Spaces, Springer, 1985.
N. G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255–292.
Y. Mizuta, On semi-fine limits of potentials, Analysis 2 (1982), 115–139.
A. Nagel, W. Rudin, and J. H. Shapiro, Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. of Math. 116 (1982), 331–360.
A. Nagel and E. M. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 (1984), 83–106.
C. Pérez, Two weight norm inequalities for Riesz potentials and uniform L p -weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), 31–44.
P. J. Rippon, On the boundary behaviour of Green potentials, Proc. London Math. Soc (3) 38 (1979), 461–480.
P. Sjögren, Weak L 1 characterization of Poisson integrals, Green potentials and H p spaces, Trans. Amer. Math. Soc. 233 (1977), 179–196.
W. Smith and D. A. Stegenga, Sobolev imbedding and integrability of harmonic functions on Hölder domains, 303–313, Walter de Gruyter, 1992, pp. 303–313, Proc. Internat. Conf. Potential Theory, Nagoya 1990 (M. Kishi, ed).
D. A. Stegenga and D. C. Ullrich, Superharmonic functions in Hölder domains, preprint.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
N. Suzuki, Nonintegrability of harmonic functions in a domain, Japan. J. Math. 16 (1990), 269–278.
-, Nonintegrability of superharmonic functions, Proc. Amer. Math. Soc. 113 (1991), 113–115.
N. Suzuki, Note on the integrability of superharmonic functions, preprint (1992).
A. Wannebo, Hardy inequalities, Proc. Amer. Math. Soc. 109 (1990), 85–95.
K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37.
P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, 1990.
K. Yosida, Functional Analysis, Sixth Edition, Springer, 1980.
W. P. Ziemer, Weakly Differentiable Functions, Springer, 1989.
Rights and permissions
Copyright information
© 1996 Springer-Verlag
About this chapter
Cite this chapter
Aikawa, H. (1996). Potential theory part II. In: Potential Theory—Selected Topics. Lecture Notes in Mathematics, vol 1633. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0093412
Download citation
DOI: https://doi.org/10.1007/BFb0093412
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61583-5
Online ISBN: 978-3-540-69991-0
eBook Packages: Springer Book Archive