On estimates for norms of some integral operators with Oinarov's kernel

In this work, we give estimates for the norm of the integral operator
\begin{equation} H: L_{p, v}\to L_{q, u}, \quad (Hf)(x):=\int_a^x k(x, t)f(t)dt \tag{0.1} \end{equation}
with the so-called Oinarov's kernel $k(x, t)$ in the weighted Lebesgue spaces
$$ L_{p, v}=\{f: ||f||_{p, v}^p:=\int_a^b |f(t)|^p v(t)dt<\infty\} $$
and
$$ L_{q, u}=\{f: ||f||_{q, u}^q:=\int_a^b |f(t)|^q u(t)dt<\infty\}, $$
in the case $1<q<p<\infty$.