Lp functions regularity

Let $g(x,\theta) = \int_{-a}^0 f(x+t\theta)dt$. If $f$ is in $L^p$ then $g$ is in $L^p$.

Proof.

$||g(x,\theta)||_{L^p(\Omega \times S^1)}^p = \displaystyle\int_{\Omega \times S^1}|g(x,\theta)|^p dxd\theta \leq |a|^{p/q} \int_{\Omega \times S^1}\int_{-a}^0 |f(x+t\theta)|^p dt dxd\theta.$