Previous |  Up |  Next

Article

Keywords:
square root of the Poisson kernel; approach regions; almost everywhere convergence; maximal functions; Orlicz spaces
Summary:
If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of $L^{p}$ and weak $L^{p}$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces $L^{\Phi }$ having the property $L^{\infty }\subset L^{\Phi }\subset L^{p}$, $1\le p<\infty $. The second contains spaces $L^{\Phi }$ that resemble $L^{p}$ spaces.
References:
[1] H. Aikawa: Harmonic functions having no tangential limits. Proc. Am. Math. Soc. 108 (1990), 457–464. DOI 10.1090/S0002-9939-1990-0990410-X | MR 0990410 | Zbl 0694.31001
[2] A. Bellow, R. L. Jones: A Banach principle for $L^\infty $. Adv. Math. 120 (1996), 155–172. DOI 10.1006/aima.1996.0035 | MR 1392277
[3] M. Brundin: Approach regions for the square root of the Poisson kernel and weak $L^p$  boundary functions. Preprint 1999:56, Department of Mathematics, Göteborg University and Chalmers University of Technology, 1999.
[4] M. Brundin: Approach Regions for $L^p$  potentials with respect to fractional powers of the Poisson rernel of the halfspace. Department of Mathematics, Göteborg University and Chalmers University of Technology, 2001.
[5] M. Brundin: Approach regions for $L^p$  potentials with respect to the square root of the Poisson kernel. Math. Scand (to appear). MR 2153413
[6] F. Di Biase: Fatou Type Theorems: Maximal Functions and Approach Regions. Birkhäuser-Verlag, Boston, 1998. MR 1483892 | Zbl 0889.31002
[7] P. Fatou: Séries trigonométriques et séries de Taylor. Acta Math. 30 (1906), 335–400. DOI 10.1007/BF02418579 | MR 1555035
[8] J. E.  Littlewood: On a theorem of Fatou. J.  London Math. Soc. 2 (1927), 172–176. DOI 10.1112/jlms/s1-2.3.172
[9] A. Nagel, E. M.  Stein: On certain maximal functions and approach regions. Adv. Math. 54 (1984), 83–106. DOI 10.1016/0001-8708(84)90038-0 | MR 0761764
[10] M. M. Rao, Z. D.  Ren: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 146. Marcel Dekker, New York, 1991. MR 1113700
[11] J.-O.  Rönning: Convergence results for the square root of the Poisson kernel. Math. Scand. 81 (1997), 219–235. MR 1613784
[12] H. A. Schwarz: Zur Integration der partiellen Differentialgleichung $\frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial y^{2}} =0$. J.  Reine Angew. Math. 74 (1872), 218–253.
[13] P.  Sjögren: Une remarque sur la convergence des fonctions propres du laplacien à valeur propre critique. In: Théorie du potentiel (Orsay,  1983), Springer-Verlag, Berlin, 1984, pp. 544–548. MR 0890377
[14] P. Sjögren: Approach regions for the square root of the Poisson kernel and bounded functions. Bull. Austral. Math. Soc. 55 (1997), 521–527. DOI 10.1017/S0004972700034183 | MR 1456282
Partner of
EuDML logo