Weighted estimates for the averaging integral operator

Let $1
satisfying:
\hskip-5cm
\begin{align}
(\star)\qquad &v(x)x^{\rho} \text{is equivalent to a non-decreasing function on}\ (0,+\infty)\nonumber\\
&\text{for some}\ \rho \ge 0;\nonumber
\end{align}\[
\qquad [w(x)x]^{1/q}\approx [v(x)x]^{1/p} \quad \text{for all } x\in(0,+\infty).
\]

We prove that if the averaging operator $(Af)(x):=\frac1x\int_0^x
f(t)\,dt$, $x \in (0,+\infty)$, is bounded from the weighted Lebesgue
space $L^p((0,+\infty);v)$ into the weighted Lebesgue
space $L^q((0,+\infty);w)$, then there exists
$\varepsilon_0\in(0,p-1)$ such that the operator $A$ is also bounded
from the space
$L^{p-\varepsilon}((0,+\infty);v(x)^{1+\delta}x^\gamma)$ into the space
$L^{q-\varepsilon q/p}((0,+\infty);w(x)^{1+\delta}x^{\delta(1-q/p)}x^{\gamma q/p})$
for all $\varepsilon, \delta, \gamma\in[0,\varepsilon_0)$.

Conversely, assuming that the operator
\[A : L^{p-\varepsilon}((0,+\infty);v(x)^{1+\delta}x^\gamma)\rightarrow
L^{q-\varepsilon q/p}((0,+\infty);w(x)^{1+\delta}x^{\delta(1-q/p)}x^{\gamma q/p})
\]
is bounded for some $\varepsilon\in[0,p-1)$, $\delta\ge0$ and $\gamma\ge0$, we prove that
the operator $A$ is also bounded from the space $L^p((0,+\infty);v)$
into the space $L^q((0,+\infty);w)$.

In particular, our results imply that the class of weights $v$ for which ($\star$) holds and the operator $A$ is bounded on the space $L^p((0,+\infty);v)$ possesses similar properties to those of the $A_p$-class of B. Muckenhoupt.