Description

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 $\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = -k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2$.