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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

$$

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Thanks a lot!

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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.

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