Geodesic
Hardy spaces, approximation issues and boundary value problems
Eurasian mathematical journal, Tome 9 (2018) no. 3, pp. 85-94.

Voir la notice de l'article provenant de la source Math-Net.Ru

The weighted Hardy spaces $e_p(\mathscr{B};\rho)$ of harmonic functions are introduced on simply connected domains $\mathscr{B}$ with rectifiable boundaries. Boundary properties of functions in these spaces are investigated, the solvability of the Dirichlet problem is established, while its solution with its derivatives are estimated. Approximation properties of the system of harmonic polynomials in $e_p(\mathscr{B};\rho)$ are studied. 
@article{EMJ_2018_9_3_a6,
     author = {V. I. Vlasov},
     title = {Hardy spaces, approximation issues and boundary value problems},
     journal = {Eurasian mathematical journal},
     pages = {85--94},
     publisher = {mathdoc},
     volume = {9},
     number = {3},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2018_9_3_a6/}
}
TY  - JOUR
AU  - V. I. Vlasov
TI  - Hardy spaces, approximation issues and boundary value problems
JO  - Eurasian mathematical journal
PY  - 2018
SP  - 85
EP  - 94
VL  - 9
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2018_9_3_a6/
LA  - en
ID  - EMJ_2018_9_3_a6
ER  - 
%0 Journal Article
%A V. I. Vlasov
%T Hardy spaces, approximation issues and boundary value problems
%J Eurasian mathematical journal
%D 2018
%P 85-94
%V 9
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2018_9_3_a6/
%G en
%F EMJ_2018_9_3_a6
V. I. Vlasov. Hardy spaces, approximation issues and boundary value problems. Eurasian mathematical journal, Tome 9 (2018) no. 3, pp. 85-94. http://geodesic.mathdoc.fr/item/EMJ_2018_9_3_a6/

[1] D.L. Burgholder, R.F. Gundy, M.L. Silverstein, “A maximal function characterisation of the class $H_p$”, Trans. Amer. Math. Soc., 157 (1971), 137–153 | MR

[2] G. Cimmino, “Nuovo tipo di condizione al contorno e nuovo metodo di trattazione per il problema generalizzato di Dirichlet”, Rend. Circolo Math. Palermo, 61:2 (1937), 117–221

[3] J.B. Garnett, Bounded analytic functions, Academic Press, New York, 1981 | MR | Zbl

[4] J. Garsia-Cuerva, Weighted $H^p$-spaces, Rospr. Mat., 162, 1985, 62 pp. | MR

[5] C. Gattegno, A. Ostrowski, Representation conforme a' la frontiere; domaines particuliers, Mem. or Sc. Math., CX, Paris, 1949 | MR | Zbl

[6] G.M. Goluzin, Geometric theory of functions of a complex variable, Amer. Math. Soc., Providence, 1969 | MR | Zbl

[7] A.K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Matem. Sb., 203:1 (2012), 1–27 (in Russian) | DOI | MR | Zbl

[8] A.K. Gushchin, V.P. Mikhailov, “On the existence of boundary values of solutions of en elliptic equation”, Matem. Sb., 73:1 (1991), 171–194 (in Russian) | DOI | MR

[9] B.E.J. Dahlberg, “Harmonic functions in Lipshitz domains”, Harm. Anal. Euclidean Spaces (Williamstone, Mass., 1978), v. 1, Proc. Symp. Pure Math., Amer. Math. Soc., Providence R. I., 1979, 313–332 | DOI | MR

[10] B.E. Dahlberg, C.E. Kenig, “Hardy spaces and the Neumann problem in $L_p$ for Laplace's equation in Lipshitz domains”, Ann. of Math., 125:3 (1987), 437–455 | DOI | MR

[11] P.L. Duren, Theory of $H^p$ spaces, Academic Press, N.Y.–London, 1970 | MR

[12] D.S. Jerison, C.E. Kenig, “The Neuman problem om Lipschitz domains”, Bull. (New Series) American Math. Soc., 4:2 (1981), 201–207 | DOI | MR

[13] M.V. Keldysh, M.A. Lavrentieff, “Sur la representation conforme des domaines limites par les curves rectifiables”, Ann. sci. Ecole norm. supér., 54:1 (1937), 1–38 | MR | Zbl

[14] C.E. Kenig, “Weighted $H^p$ spaces on Lipshitz domains”, Amer. J. Math., 102:1 (1980), 129–163 | DOI | MR | Zbl

[15] C.E. Kenig, “Recente progress on boundary–value problems on Lipshitz domains”, Proc. Symp. Pure Math. Amer. Soc., 43 (1985), 185–205 | MR

[16] E. Magenes, G. Stampacchia, “II problemi al contorno per le equazioni differeziali di tipo ellittico”, Ann. Scuola normale super. Pisa, 12:3 (1958), 247–357 | MR

[17] A.I. Markushevich, Theory of functions of a complex variable, Translated and edited by Richard A. Silverman, v. III, Second English edition, Chelsea Publishing Co., New York, 1977 | MR | Zbl

[18] V.G. Maz'ya, “The degenerate problem with an oblique derivative”, Matem. Sb. (N.S.), 87(129):3 (1972), 417–454 (in Russian) | MR | Zbl

[19] C. Miranda, Partial differential equations of elliptic type, Springer, Berlin–Heidelberg–New York, 1970 | MR | Zbl

[20] I.I. Privalov, The boundary properties of analitic functions, Gostehizdat, M.–L., 1950 (in Russian) | MR

[21] M. Riesz, “Sur les functions conjugees”, Math. Zeitshr., 27:2 (1927), 218–244 | MR | Zbl

[22] F. Riesz, “Uber die Randwerte einer analytischen Funcktion”, Math. Zeitshr., 18 (1923), 87–96 | DOI | MR

[23] Differ. Equat., 38:6 (2002), 855–864 | DOI | MR | Zbl

[24] Russian Acad. Sci. Dokl. Math., 78:2 (2007) | MR | Zbl

[25] Russian Acad. Sci. Dokl. Math., 418:2 (2008), 162–167 | MR | MR | Zbl

[26] E.M. Stein, G. Weiss, “On the theory of harmonic functions of several variables. The theory of $H^p$-spaces”, Acta Math., 103 (1960), 25–62 | DOI | MR | Zbl

[27] V.I. Vlasov, Boundary value problems in domains with curvilinear boundary, D. Sc. thesis, Computing Center of the Russ. Acad. Sci., M., 1990 (in Russian) | MR

[28] Russian Acad. Sci. Dokl. Math., 47:1 (1993), 57–61 | MR | Zbl

[29] S. Warschawski, “Über das Ranwerhalten der Ableitung der Abbildungsfunction bei conformer Abbildung”, Math. Zeitshr., 35:3–4 (1932), 321–456 | DOI | MR