On weighted estimates for a class of integral operators

The weighted estimates of the form
\begin{equation} \biggl(\int_0^\infty|T_\varphi f(x)u(x)|^q\,dx\biggr)^{1/q}\le C\biggl(\int_0^\infty|f(x)v(x)|^p\,dx\biggr)^{1/p}, \tag{1} \end{equation}
are considered with
$$ T_\varphi f(x)=\int_0^x\varphi(t/x)f(t)\,dt, $$
where the measurable function $\varphi$ satisfies the conditions:
a) $\varphi(t)\geqslant0$ and $\varphi(t)$ is nonincreasing for $t\in[0,1]$,
b) $\varphi(t_1,t_2)\leqslant D(\varphi(t_1)+\varphi(t_2))$, $0<t_1$, $t_2<1$ and $D$ independent of $t_1$, $t_2$.
We state necessary and/or sufficient conditions for (1) to hold if $1<p$, $q<\infty$ or $0<q<1<p<\infty$.