Proving that $\displaystyle\int_0^\infty f(t)\frac{\sin(xt)}{t}dt$ exist for $f\in\mathcal{L}^p(0,\infty)$, $1<p<\infty$ and $x>0$

Question

Show that $\displaystyle\int_0^\infty f(t)\frac{\sin(xt)}{t}dt$ exist for $f\in\mathcal{L}^p(0,\infty)$, $1<p<\infty$ and $x>0$.

I try to show this exercise using the Holder inequality but i don't use this inequality adecually.

I wait can help me.

Answer 1

Hint: $\dfrac{\sin(xt)}{t}$ is bounded on $(0,1),$ and in absolute value is bounded above by $\dfrac{1}{t}$ on $[1,\infty).$ Now try Holder.