Hilbert's Inequality

Could you help me to show the following:

The operator
$$
T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy
$$
satisfies
$$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p
$$
for $1 <p< \infty$ where
$$
C_p = \int_0^\infty \frac{t^{-1/p}}{t+1}dt
$$

Thanks a lot!

Solutions Collecting From Web of "Hilbert's Inequality"

Note that substitution $y=tx$ gives
$$
\int\limits_0^\infty\frac{f(y)}{x+y}dy=\int\limits_0^\infty\frac{f(tx)}{1+t}dt
$$
then from Minkowski’s integral inequality you get
$$
\begin{align}
\Vert T(f)\Vert_p
&=\left(\int\limits_0^\infty\left|\int\limits_0^\infty\frac{f(tx)}{1+t}dt\right|^p dx\right)^{1/p}\\
&\leq\int\limits_0^\infty\left(\int\limits_0^\infty\left|\frac{f(tx)}{(1+t)}\right|^pdx\right)^{1/p}dt\\
&=\int\limits_0^\infty\frac{1}{1+t}\left(\int\limits_0^\infty|f(tx)|^p dx\right)^{1/p}dt\\
&=\int\limits_0^\infty\frac{1}{1+t}\left(\int\limits_0^\infty|f(x)|^p \frac{dx}{t}\right)^{1/p}dt\\
&=\int\limits_0^\infty\frac{1}{(1+t)t^{1/p}}\left(\int\limits_0^\infty|f(x)|^p dx\right)^{1/p}dt\\
&=\int\limits_0^\infty\frac{ t^{-1/p}dt}{1+t}\Vert f\Vert_p dt\\
&=C_p\Vert f\Vert_p
\end{align}
$$
Q.E.D.