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THE APPROXIMATION NUMBERS OF HARDY-TYPE OPERATORS ON TREES

Published online by Cambridge University Press:  23 August 2001

W. D. EVANS
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, evanswd@cardiff.ac.uk
D. J. HARRIS
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, evanswd@cardiff.ac.uk
J. LANG
Affiliation:
Mathematics Department, 202 Mathematical Sciences Building, University of Missouri, Columbia, MO 65211, USA, langjan@math.missouri.edu Permanent address: Mathematics Institute AV CR Zitna 25 Prague 1 Czech Republic
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Abstract

The Hardy operator $T_a$ on a tree $\Gamma$ is defined by \[ (T_af)(x):=v(x) \int^x_a f(t)u(t)\,dt \quad \mbox{for } a, x\in \Gamma. \] Properties of $T_a$ as a map from $L^p(\Gamma)$ into itself are established for $1\le p \le \infty$. The main result is that, with appropriate assumptions on $u$ and $v$, the approximation numbers $a_n(T_a)$ of $T_a$ satisfy \begin{equation*} \tag{$*$} \lim_{n\rightarrow \infty} na_n(T_a) = \alpha_p\int_{\Gamma}|uv|\,dt \end{equation*} for a specified constant $\alpha_p$ and $1 p<\infty$. This extends results of Naimark, Newman and Solomyak for $p=2$. Hitherto, for $p\neq 2$, $(*)$ was unknown even when $\Gamma$ is an interval. Also, upper and lower estimates for the $l^q$ and weak-$l^q$ norms of $\{a_n(T_a)\}$ are determined.

2000 Mathematical Subject Classification: 47G10, 47B10.

Type
Research Article
Copyright
2001 London Mathematical Society

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