Literature Cited
N. S. Landkof, Foundations of Modern Potential Theory [in Russian], Nauka, Moscow (1966).
M. Brelot, Foundations of Classical Potential Theory [Russian translation], Mir, Moscow (1964).
L. Carleson, Selected Problems of the Theory of Exceptional Sets [Russian translation], Mir, Moscow (1971).
Yu. G. Reshetnyak, “Capacity in the theory of functions with generalized derivatives,” Sib. Mat. Zh.,10, No. 5, 1109–1138 (1969).
Yu. G. Reshetnyak, “Boundary behavior of functions with generalized derivatives,” Sib. Mat. Zh.,13, No. 2, 411–419 (1972).
V. G. Maz'ya and V. P. Khavin, “Nonlinear potential theory,” Usp. Mat. Nauk,27, No. 6, 67–138 (1972).
N. G. Meyers, “A theory of capacities for potentials of functions in Lebesgue classes,” Math. Scand.,26, No. 2, 255–292 (1970).
D. R. Adams and N. G. Meyers, “Thiness and Wiener criteria for nonlinear potentials,” Indiana Univ. Math. J.,22, No. 2, 169–197 (1972).
D. R. Adams and N. G. Meyers, “Bessel potentials. Inclusion relations among classes of exceptional sets,” Indiana Univ. Math. Zh.,22, No. 9, 873–905 (1973).
L. I. Hedberg, “Nonlinear potentials and approximation in the mean by analytic functions,” Math. Zh.,129, 299–319 (1972).
K. Hanson, “Imbedding theorems of Sobolev type in potential theory,” Math. Scand.,45, 77–102 (1979).
T. Ugaheri, “On the general potential and capacity,” Jpn. J. Math.,20, 37–43 (1950).
N. Ninomiya, “Sur le principle de continuité dans la theorie du potentiel,” J. Inst. Polytechn. Osaka City Univ.,8, 51–56 (1957).
S. K. Vodop'yanov, “Geometric properties of domains which satisfy the extension condition for spaces of differentiable functions,” in: Some Applications of Functional Analysis to Problems of Mathematical Physics: Proceedings of the S. L. Sobolev Seminar [in Russian], No. 2, Novosibirsk (1984), pp. 65–95.
S. K. Vodop'yanov, “Isoperimetric relations and extension conditions for differentiable functions,” Dokl. Akad. Nauk SSSR,292, No. 1, 11–16 (1987).
S. K. Vodop'yanov, “Geometric properties of domains and estimates for the norm of the extension operator,” Dokl. Akad. Nauk SSSR, No. 4, 791–796 (1987).
S. K. Vodop'yanov, “Geometric properties of maps and domains lower bounds for the norm of the extension operator,” in: Studies on Geometry and Mathematical Analysis: Proceedings of the Institute of Mathematics, Siberian Section, Academy of Sciences of the USSR [in Russian], Vol. 7, Nauka, Novosibirsk (1987), pp. 70–101.
V. G. Maz'ya, Sobolev Spaces [in Russian], Leningrad State Univ. (1985).
D. R. Adams, Lectures on Lp-Potential Theory, UMEA (Preprint No. 2) (1981).
B. Dahlberg, “Regularity properties of Riesz potentials,” Ind. Univ. Math. J.,28, 257–268 (1979).
V. G. Maz'ya and T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman Advanced Publ. Program, Boston-London-Melbourne (1985).
S. K. Vodop'yanov, “Maximum principle in potential theory,” in: Abstracts of Reports to the Eleventh All-Union School on Operator Theory in Function Spaces (Chelyabinsk, May, 1986) [in Russian], Part 1, Chelyabinsk (1986), p. 29.
S. K. Vodop'yanov, “Anisotropic spaces of differentiable functions and quasiconformal maps,” in: Abstracts of Reports to the Eleventh All-Union School on Operator Theory in Function Spaces (Chelyabinsk, May, 1986) [in Russian], Part II, Chelyabinsk (1986), p. 23.
V. G. Maz'ya, “Removable singularities of bounded solutions of quasilinear elliptic equations of any order,” J. Sov. Math.,3, No. 4 (1975).
M. S. Alborova and S. K. Vodop'yanov, “Removable singularities for bounded solutions of quasielliptic equations,” Novosibirsk, 1987, Dep. in VINITI, Feb. 4, 1987, No. 804-B87.
G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes, No. 28, Princeton Univ. Press (1982).
O. V. Besov, V. P. Il'in, and P. I. Lizorkin, “Lp-estimates of a class of nonisotropically-singular integrals,” Dokl. Akad. Nauk SSSR,169, No. 6, 1250–1253 (1966).
O. V. Besov and P. I. Lizorkin, “Singular integral operators and sequences of convolutions in Lp spaces,” Mat. Sb.,73, No. 1, 65–88 (1967).
E. B. Fabes and N. M. Riviére, “Singular integrals with homogeneity,” Stud. Math.,27, 19–38 (1966).
E. M. Stein and S. Wainger, “Problems in harmonic analysis related to the curvature,” Bull. Am. Math. Soc.,84, 1239–1295 (1978).
V. M. Gol'dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Maps [in Russian], Nauka, Moscow (1983).
A. A. Davtyan, “Anisotropic potentials, their treatment, and some applications,” Dokl. Akad. Nauk SSSR,285, No. 3, 537–541 (1985).
D. R. Adams, “A trace inequality for generalized potentials,” Stud. Math.,48, No. 1, 99–105 (1973).
P. I. Lizorkin, “Generalized Liouville differentiation and the method of multipliers in the theory of imbeddings of classes of differentiable functions,” Tr. Mat. Inst. im. V. A. Steklova AN SSSR,105, 89–167 (1969).
P. I. Lizorkin, “Description of the spaceL (r)p (R n) in terms of differences of singular integrals,” Mat. Sb.,81, No. 1, 79–91 (1970).
H. Dappa and W. Trebels, “On hypersingular integrals and anisotropic Bessel potential spaces,” Trans. Am. Math. Soc.,286, 419–429 (1984).
H. Dappa and W. Trebels, “Pointwise multiplication on anisotropic Bessel potential spaces,” in: Abstracts A. Haar Memorial Conf. (Budapest, Aug. 1985), Budapest (1985), p. 16.
H. Dappa and W. Trebels, “On L1-criteria for quasiradial Fourier multipliers with applications to some anisotropic function spaces,” Anal. Math.,9, No. 4, 275–289 (1983).
H. Dappa and W. Trebels, “A difference quotient norm for anisotropic Bessel potential spaces,” Math. Nachr.,132, 163–174 (1987).
V. I. Yudovich, “Estimates connected with integral operators and solutions of elliptic equations,” Dokl. Akad. Nauk SSSR,138, No. 4, 805–808 (1961).
S. I. Pokhozhaev, “Sobolev imbedding theorem in the case pℓ=n,” in: Reports to the Scientific-Engineering Conference of the Energy Inst., Mathematics Section [in Russian], Moscow Energy Inst., Moscow (1965), pp. 158–170.
N. S. Trudinger, “On imbedding into Orlicz spaces and some applications,” J. Math. Mech.,17, 473–483 (1967).
O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems [in Russian], Nauka, Moscow (1975).
V. A. Solonnikov, “Inequalities for functions from the classesW l p (R n),” J. Sov. Math.,3, No. 4 (1975).
Additional information
Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 2, pp. 17–33, March–April, 1988.
Rights and permissions
About this article
Cite this article
Vodop'yanov, S.K. Maximum principle in potential theory and imbedding theorems for anisotropic spaces of differentiable functions. Sib Math J 29, 176–189 (1988). https://doi.org/10.1007/BF00969729
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00969729