# 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!

## 1 Answer

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.