I came across this question in an old qualifying exam, but I am stumped on how to approach it:
For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2
, let $(Vf)(x) = \frac{1}{x} \int_x^{10x} \frac{f(t)}{t^{1/4}} dt$ Prove that $||Vf||_{L^2} \leqslant C_p ||f||_{L^p}$ for some finite number $C_p$, which depends on $p$ but not on $f$.