Weighted estimates for tangential boundary behaviour

V. G. Krotov   ; L. V. Smovzh

Sbornik. Mathematics, Tome 197 (2006) no. 2, pp. 193-211

Résumé

Let $(X,\mu,d)$ be a space of homogeneous type (here $d$ is a quasimetric and $\mu$ a measure). A function $\varepsilon$ of modulus of continuity kind gives rise to approach regions $\Gamma_{\varepsilon}(x)$ at the boundary of $\mathbf{X}$, $\mathbf{X}=X\times[0,1)$, where for a point $x\in X$, $$ \Gamma_{\varepsilon}(x)=\{(y,t)\in\mathbf{X}:d(x,y)<\varepsilon(1-t)\}. $$ These are ‘tangential’ regions if $\lim_{t\to+0}\varepsilon(t)/t=\infty$. Weighted $L^p$-estimates are proved for the corresponding maximal functions of integral operators. Applications of these estimates to potentials in $\mathbb{R}^n$ and to multipliers of homogeneous expansions of holomorphic functions in the Hardy classes in the unit ball of $\mathbb{C}^n$ are presented. Bibliography: 20 titles.

Détail

Keywords: space of homogeneous type, tangential boundary behaviour, weighted inequalities.

Bibliographie

[1] P. Fatou, “Séries trigonométriques et séries de Taylor”, Acta Math., 30 (1906), 335–400 | DOI | MR | Zbl

[2] G. H. Hardy, J. E. Littlewood, “A maximal theorem with function-theoretic applications”, Acta Math., 54 (1930), 81–116 | DOI | MR | Zbl

[3] C. Fefferman, E. M. Stein, “$H^p$ spaces of several variables”, Acta Math., 129:3–4 (1972), 137–193 | DOI | MR | Zbl

[4] I. Stein, G. Veis, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | MR | Zbl

[5] U. Rudin, Teoriya funktsii v edinichnom share v $\mathbb{C}^n$, Mir, M., 1984 | MR | Zbl

[6] A. B. Aleksandrov, “Teoriya funktsii v share”, Itogi nauki i tekhniki. Sovr. problemy matem. Fundament. napravleniya, 8, VINITI, M., 1985, 115–190

[7] A. Nagel, W. Rudin, J. Shapiro, “Tangential boundary behavior of function in Dirichlet-type spaces”, Ann. of Math. (2), 116:2 (1982), 331–360 | DOI | MR | Zbl

[8] A. Nagel, E. M. Stein, “On certain maximal functions and approach regions”, Adv. Math., 54:1 (1984), 83–106 | DOI | MR | Zbl

[9] P. Ahern, A. Nagel, “Strong $L^p$-estimates for maximal functions with respect to singular measure with applications to exceptional sets”, Duke Math. J., 53:2 (1986), 359–393 | DOI | MR | Zbl

[10] V. G. Krotov, “O granichnom povedenii drobnykh integralov golomorfnykh funktsii v edinichnom share v $\mathbb{C}^N$”, Izv. vuzov. Ser. matem., 1988, no. 4, 73–75 | MR

[11] V. G. Krotov, “Otsenki dlya maksimalnykh operatorov, svyazannykh s granichnym povedeniem, i ikh prilozheniya”, Trudy MIAN, 190, 1989, 117–138 | MR

[12] J. Sueiro, “Tangential boundary limits and exceptional sets for harmonic function in Dirichlet-type spaces”, Math. Ann., 286:4 (1990), 661–678 | DOI | MR | Zbl

[13] P. Cifuentes, J. Dorronsoro, J. Sueiro, “Boundary tangential convergence in spaces of homogeneous type”, Trans. Amer. Math. Soc., 332:1 (1992), 331–350 | DOI | MR | Zbl

[14] V. G. Krotov, “Tochnaya otsenka granichnogo povedeniya funktsii iz klassov Khardi–Soboleva v kriticheskom sluchae”, Matem. zametki, 62:4 (1997), 527–539 | MR | Zbl

[15] I. N. Katkovskaya, V. G. Krotov, “O kasatelnom granichnom povedenii potentsialov”, Trudy Inst. matem. NAN Belarusi, 5 (2000), 80–83 | MR | Zbl

[16] V. G. Krotov, “Kasatelnoe granichnoe povedenie funktsii mnogikh peremennykh”, Matem. zametki, 68:2 (2000), 230–248 | MR | Zbl

[17] A. M. Sedletskii, “Kasatelnye granichnye znacheniya preobrazovanii Laplasa. Primenenie k approksimatsii tipa Myuntsa–Sasa”, Izv. RAN. Ser. matem., 67:1 (2003), 177–198 | MR

[18] R. R. Coifman, G. Weiss, Analyse harmonique non-commutative sur certain espaces homogenés, Lecture Notes in Math., 242, Springer-Verlag, Berlin, 1971 | MR | Zbl

[19] P. Sjögren, “Une remarque sur la convergence des fonctions propres du Laplasian à valeur propre critique”, Lecture Notes in Math., 1096, 1984, 544–548 | MR | Zbl

[20] J.-O. Rönning, “Convergence results for the square root of the Poisson kernel”, Math. Scand., 81:2 (1997), 219–235 | MR | Zbl