Memoirs of the American Mathematical Society 2002; 87 pp; softcover Volume: 157 ISBN10: 082182791X ISBN13: 9780821827918 List Price: US$59 Individual Members: US$35.40 Institutional Members: US$47.20 Order Code: MEMO/157/745
 We consider the Volterra integral operator \(T_{\rho,\psi}:L_p(0,\infty)\to L_q(0,\infty)\) for \(1\leq p,q\leq \infty\), defined by \((T_{\rho,\psi}f)(s) =\rho(s)\int_0^s \psi(t) f(t) dt\) and investigate its degree of compactness in terms of properties of the kernel functions \(\rho\) and \(\psi\). In particular, under certain optimal integrability conditions the entropy numbers \(e_n(T_{\rho,\psi})\) satisfy \(c_1\Vert{\rho\,\psi}\Vert_r\leq \liminf_{n\to\infty} n\, e_n(T_{\rho,\psi}) \leq \limsup_{n\to\infty} n\, e_n(T_{\rho,\psi})\leq c_2\Vert{\rho\,\psi}\Vert_r\) where \(1/r = 1 1/p +1/q >0\). We also obtain similar sharp estimates for the approximation numbers of \(T_{\rho,\psi}\), thus extending former results due to Edmunds et al. and Evans et al.. The entropy estimates are applied to investigate the small ball behaviour of weighted Wiener processes \(\rho\, W\) in the \(L_q(0,\infty)\)norm, \(1\leq q\leq \infty\). For example, if \(\rho\) satisfies some weak monotonicity conditions at zero and infinity, then \(\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2\). Readership Graduate students and research mathematicians interested in operator theory, probability theory, and stochastic processes. Table of Contents  Introduction
 Main results
 Scale transformations
 Upper estimates for entropy numbers
 Lower estimates for entropy numbers
 Approximation numbers
 Small ball behaviour of weighted Wiener processes
 Appendix
 Bibliography
