The asymptotic behavior of approximation numbers and estimates of the Schatten-von Neumann norms for the Hardy integral operator (Q1594483)

Suppose that the operator defined by \[ Tf(x)= v(x) \int^x_0 f(y) u(y) dy, \] where \(u\in L^{p'}(0, t)\) and \(v\in L^q(t,\infty)\) for any \(t> 0\), is a compact operator from \(L^p(\mathbb{R}^+)\) to \(L^q(\mathbb{R}^+)\). The approximation numbers \(a_m(T)\) of \(T\) are defined by \[ a_m(T)= \inf\{\|T-P\|;\text{ rank }P< m\},\quad m= 1,2,\dots\;. \] The author establishes the estimates from below or above \(\liminf_{N\to\infty} Na_N(T)\), \(\limsup_{N\to\infty}N^{1/r}a_N(T)\) and \(\liminf_{N\to\infty} N^{1/r} a_N(T)\) in terms \((\int^\infty_0|u(t)|^r|v(t)|^r dt)^{1/r}\), where \(r= qp'/(q+ p')\) (\(\geq 1\) or \(<1\)), and those of \(\sum_{k\in\mathbb{N}} [a_k(T) k^{1/p- 1/q}]^s\) and \(\sum_{k\in\mathbb{N}} a_k(T)^s\) in terms of some value depending on \(u\), \(v\) and \(s\). The corollary of the last result is \[ \Biggl(\sum_{k\in\mathbb{N}} a_k(T)^s\Biggr)^{1/s}\asymp \Biggl(\int^\infty_0 \Biggl(\int^x_0|u(y)|^{p'} dy\Biggr)^{s/p'} \Biggl(\int^\infty_x|v(y)|^p dy\Biggr)^{s/p-1}|v(x)|^p dx\Biggr)^{1/s}. \]