Show that the operators given by convolution with the $\sin t / t$ and the distribution p.v. $\cos t / t$ are bounded on $L^{p}(\mathbb{R})$

Question

Question:Show that the operators given by convolution with the smooth function $\sin t / t$ and the distribution p.v. $\cos t / t$ are bounded on $L^{p}(\mathbb{R})$ whenever $1<p<\infty$.

My try is Minkowski's integral inequality, but $|\sin t/t|$ is not integrable on $\mathbb R$, any hint will be appreciated.

Answer 1

Notice $\sin t=\sin(x-(x-t))$, and $H$ is of type $(p,p)$.