# Potential Theory and Function Theory for Irregular Regions by Yu. D. Burago; V. G. Maz'ya

By Yu. D. Burago; V. G. Maz'ya

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122-123 of the Russian translation) and we carry it out merely for completeness of presentation. Here, without saying so particularly, we shall rely on well-known facts from the general theory of integral equations. LEMMA20. Let \CIC) be a sequence of linearly independent operators in C (oE) satisfying the following conditions: 24 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS 1) CK* takes the class 2) if ¢E (! , then v 0 00 Pro of. C=~ ~ 0 0 c. ,... G"". y~· is continuous, the series E .

E. Kamke, Das Lebesgue-Stieltjes Integral. B. G. Teubner Verlagsgesellschaft, Leipzig (1956). Yu. D. Burago and V. G. Maz'ya, "On the space of functions whose derivatives are measures," Part 2 of this monograph. V. A. Zalgaller, "The variations of curves along fixed directions," Izv. Akad, Nauk SSSR, Ser. , 15(5): 463-476 (1951). H. Federer, "Curvature measures," Trans. Am. Math. , 93:418-491 (1959). H. Hahn and A. Rosenthal, Set Functions, University of New Mexico Press, Albuquerque, New Mexico (1948).

IJ(_ (£r) (Rh ), which, by Lemma 28, ensures the regular convergence of The theorem is proved. tr-' . Thus, f" to r . LITERATURE CITED 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. Yu. D. Burago, V. G. Maz'ya, and V. D. Sapozhnikova, "Double-layer potential for irregular regions," Doklady Akad. Nauk SSSR, 147(3): 523-525 (1962). [In English: Soviet Math. ] V. G. Maz'ya and V. D. Sapozhnikova, "The solution of the Dirichlet and Neumann problems by potential theory methods for irregular regions," Doklady Akad.