Access the full text.
Sign up today, get DeepDyve free for 14 days.
(1963)
On the minimal boundary of A(E)
A. O’Farrell (1977)
Hausdorff content and rational approximation in fractional Lipschitz normsTransactions of the American Mathematical Society, 228
A. Browder (1967)
Point derivations on function algebrasJournal of Functional Analysis, 1
A. O’Farrell (1974)
Analytic capacity, Hölder conditions and -spikesTransactions of the American Mathematical Society, 196
A. O’Farrell (1975)
Lip 1 Rational ApproximationJournal of The London Mathematical Society-second Series
A. Hallstrom (1969)
On bounded point derivations and analytic capacityJournal of Functional Analysis, 4
AA Gonchar (1963)
949Isv. Akad. Nauk. SSSR ser. mat., 27
D. Lord, A. O’Farrell (1994)
Boundary smoothness properties of Lipα analytic functionsJournal d’Analyse Mathématique, 63
J. Wermer (1967)
Bounded point derivations on certain Banach algebrasJournal of Functional Analysis, 1
P. Curtis (1969)
Peak points for algebras of analytic functionsJournal of Functional Analysis, 3
E. Bishop (1959)
A minimal boundary for function algebras.Pacific Journal of Mathematics, 9
Let U be an open subset of C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}$$\end{document} with boundary point x0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x_0$$\end{document} and let Aα(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{\alpha }(U)$$\end{document} be the space of functions analytic on U that belong to lipα(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha (U)$$\end{document}, the “little Lipschitz class”. We consider the condition S=∑n=1∞2(t+λ+1)nM∗1+α(An\U)<∞,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S= \sum _{n=1}^{\infty }2^{(t+\lambda +1)n}M_*^{1+\alpha }(A_n \setminus U)< \infty ,$$\end{document} where t is a non-negative integer, 0<λ<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<\lambda <1$$\end{document}, M∗1+α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M_*^{1+\alpha }$$\end{document} is the lower 1+α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1+\alpha $$\end{document} dimensional Hausdorff content, and An={z:2-n-1<|z-x0|<2-n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_n = \{z: 2^{-n-1}<|z-x_0|<2^{-n}$$\end{document}. This is similar to a necessary and sufficient condition for bounded point derivations on Aα(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{\alpha }(U)$$\end{document} at x0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x_0$$\end{document}. We show that S=∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S= \infty $$\end{document} implies that x0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x_0$$\end{document} is a (t+λ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t+\lambda )$$\end{document}-spike for Aα(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{\alpha }(U)$$\end{document} and that if S<∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S<\infty $$\end{document} and U satisfies a cone condition, then the t-th derivatives of functions in Aα(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{\alpha }(U)$$\end{document} satisfy a Hölder condition at x0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x_0$$\end{document} for a non-tangential approach.
Analysis and Mathematical Physics – Springer Journals
Published: Mar 13, 2021
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.