On Maz’ya’s work in potential theory and the theory of function spaces

  • Lars Inge Hedberg
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 109)


A presentation of some of the highlights in Vladimir Maz’ya’s remarkable early work on function spaces, potential theory, and partial differential operators.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Adams, D. R.: Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1971), 203–217.MATHGoogle Scholar
  2. [2]
    —: On the existence of capacitary strong type estimates in RN, Ark. mat 14 (1976), 125–140.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    — and Hedberg, L. I: Function Spaces and Potential Theory, Springer, Berlin Heidelberg, 1996.Google Scholar
  4. [4]
    Agmon, S., Douglis, A. and Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I-II, Comm. Pure Appl. Math. 12 (1959), 623–727, 17 (1964), 35-92.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    BjÖrn, J. and Maz′ya, V. G.: Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form, Preprint, LinkÖping University, 1998.Google Scholar
  6. [6]
    Bombieri, E.: Variational problems and elliptic equations, in Mathematical developments arising from Hilbert problems (F. E. Browder, ed.), 525-535, Proc. Symp. Pure Math. 28:2, Amer. Math. Soc. Providence, R. I., 1976.Google Scholar
  7. [7]
    De Giorgi, E.: Un esempio di estremali discontinue per un problema variazionale di tipo elliptico, Boll. Un. Mat. ltal. (4) 1 (1968), 135–137.MATHGoogle Scholar
  8. [8]
    Eilertsen, S.: On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian, Preprint, LinkÖping University, 1998.Google Scholar
  9. [9]
    Federer, H. and Fleming, W. H.: Normal and integral currents, Ann. of Math. 72 (1960), 458–520.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105, Princeton University Press, Princeton, NJ, 1983.Google Scholar
  11. [11]
    Giusti, E. and Miranda, M.: Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni, Boll. Un. Mat. ltal. (4) 1 (1968), 219–226.MathSciNetMATHGoogle Scholar
  12. [12]
    Havin, V. P. (Khavin, V. P.) and Maz′ya, V. G.: A nonlinear analogue of the Newtonian potential and metric properties of the (p, l)-capacity (Russian), Dokl. Akad. Nauk Sssr 194 (1970), 770–773. English translation: Soviet Math. Dokl. 11 (1970), 1294-1298.MathSciNetGoogle Scholar
  13. [13]
    —: Non-linear potential theory (Russian), Uspekhi Mat Nauk 27:6 (1972), 67–138. English translation: Russian Math. Surveys 27:6 (1972), 71-148.MathSciNetGoogle Scholar
  14. [14]
    —: Use of (p, l)-capacity in problems of the theory of exceptional sets (Russian), Mat. Sb. 90(132) (1973), 558–591. English translation: Math. USSR-Sb. 19 (1973), 547-580.MathSciNetGoogle Scholar
  15. [15]
    Heinonen, J., Kilpeläinen, T. and Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993.MATHGoogle Scholar
  16. [16]
    Kilpeläinen, T. and Malý, J.: The Wiener test and potential estimates for quasi-linear elliptic equations, Acta Math. 172, 137–161, 1994.Google Scholar
  17. [17]
    Malý, J. and Ziemer, W. P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations, Amer. Math. Soc, Providence, R.I., 1997.Google Scholar
  18. [18]
    Maz′ya, V. G.: Classes of domains and embedding theorems for function spaces (Russian), Dokl. Akad. Nauk SSSR 133 (1960), 527–530. English translation: Soviet Math. 1 (1961), 882-885.Google Scholar
  19. [19]
    —: The p-conductivity and theorems on embedding certain function spaces into a C-space (Russian), Dokl. Akad. Nauk SSSR 140 (1961), 299–302. English translation: Soviet Math. 2 (1961), 1200-1203.MathSciNetGoogle Scholar
  20. [20]
    —: The negative spectrum of the n-dimensional Schrödinger operator (Russian), Dokl. Akad. Nauk SSSR 144 (1962), 721–722. English translation: Soviet Math. 3 (1962), 808-810.MathSciNetGoogle Scholar
  21. [21]
    —: The Dirichlet problem for elliptic equations of arbitrary order in unbounded regions (Russian), Dokl. Akad. Nauk SSSR 150 (1963), 1221–1224. English translation: Soviet Math. 4 (1963), 860-863.MathSciNetGoogle Scholar
  22. [22]
    —: Regularity at the boundary of solutions of elliptic equations and conformai mapping (Russian), Dokl. Akad. Nauk SSSR 152 (1963), 1297–1300. English translation: Soviet Math. 4 (1963), 1547-1551.MathSciNetGoogle Scholar
  23. [23]
    —: On the theory of the n-dimensional Schrödinger operator (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 1145–1172.MathSciNetMATHGoogle Scholar
  24. [24]
    —: Polyharmonic capacity in the theory of the first boundary value problem (Russian), Sibirsk. Mat. Zh. 6 (1965), 127–148.MATHGoogle Scholar
  25. [25]
    —: On the modulus of continuity of a solution of the Dirichlet problem near an irregular boundary (Russian), Problemy Mat. Anal. 1, Izdat. Leningrad. Univ., Leningrad, 1966, 45–58. English translation: Problems in Math. Anal. 1, Plenum Press, New York, 1968, 41-54.Google Scholar
  26. [26]
    —: The behavior near the boundary of the solution of the Dirichlet problem for a second order elliptic equation in divergence form (Russian), Mat. Zametki 2 (1967), 209–220. English translation: Math. Notes 2 (1967), 610-617.MathSciNetGoogle Scholar
  27. [27]
    —: Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients (Russian), Funkcional. Anal, i Prilozhen. 2:3 (1968), 53–57. English translation: Funkt. Anal. Appl. 2 (1968), 230-234.Google Scholar
  28. [28]
    —: Classes of sets and measures connected with embedding theorems (Russian), in Embedding Theorems and Their Applications (Russian), Proc, Baku 1966 (L. D. Kudryavtsev, ed.), 142–159, Nauka, Moscow, 1970.Google Scholar
  29. [29]
    —: On the continuity at a boundary point of solutions of quasilinear equations (Russian), Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 25:13 (1970), 42–55. Correction, ibid. 27:1 (1972), 160. English translation: Vestnik Leningrad Univ. Math. 3 (1976), 225-242.MathSciNetGoogle Scholar
  30. [30]
    —: On certain integral inequalities for functions of many variables (Russian), Problemy Matematicheskogo Analiza, Leningrad. Univ. 3 (1972), 33–68. English translation: J. Soviet Math. 1 (1973), 205-234.Google Scholar
  31. [31]
    —: Removable singularities of bounded solutions of quasilinear elliptic equations of any order (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 26 (1972), 116–130. English translation: J. Soviet Math. 3 (1975), 480-492.Google Scholar
  32. [32]
    —: Behaviour of solutions to the Dirichlet problem for the biharmonic operator at a boundary point, in Equadiff IV, Proc, Prague, 1977 (J. Fábera, Ed.) Lecture Notes in Math. 703, 250–262, Springer-Verlag, Berlin-Heidelberg, 1979.Google Scholar
  33. [33]
    —: The modulus of continuity of a harmonic function at a boundary point (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 135 (1984), 87–95. English translation: J. Soviet Math..MathSciNetMATHGoogle Scholar
  34. [34]
    —: Sobolev Spaces, Springer-Verlag, Berlin-New York, 1985. Russian edition, Izd. Leningrad. Univ., Leningrad, 1985.Google Scholar
  35. [35]
    —: Classes of domains, measures and capacities in the theory of differentiable functions, in Encyclopaedia of Mathematical Sciences, Vol. 26, Analysis III, S. M. Nikolskiĭ ed., Springer, Berlin Heidelberg, 1991, 141–211.Google Scholar
  36. [36]
    —: Unsolved problems connected with the Wiener criterion, in The Legacy of Norbert Wiener: A Centennial Symposium (Proc, Cambridge, Massachusetts, 1994) 199–208, Proc. Sympos. Pure Math., 60, Amer. Math. Soc, Providence, Rhode Island, 1997.Google Scholar
  37. [37]
    —and Donchev, T.: On Wiener regularity at a boundary point for a poly-harmonic operator (Russian), C. R. Acad. Bulgare Sci 36 (1983), 177–179. English translation: Amer. Math. Soc. Transl. (2) 137 (1987), 53-55.MathSciNetMATHGoogle Scholar
  38. [38]
    —and Netrusov, YU. V.: Some counterexamples for the theory of Sobolev spaces on bad domains, Potential Analysis 4 (1995), 47–65.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    — and Poborchi, S. V.: Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997.MATHGoogle Scholar
  40. [40]
    —and Shaposhnikova, T. O.: The Theory of Multipliers in Spaces of Differentiable Functions, Pitman, Boston-London, 1985. Russian edition (with additions), Izd. Leningrad. Univ., Leningrad, 1986.Google Scholar
  41. [41]
    —and Verbitsky, I. E.: Capacitary inequalities for fractional integrals with applications to partial differential equations and Sobolev multipliers, Ark. mat. 33 (1995), 81–115.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1999

Authors and Affiliations

  • Lars Inge Hedberg
    • 1
  1. 1.Department of MathematicsLinköping UniversityLinköpingSweden

Personalised recommendations