$\mathcal{L}^p$ spaces and convolution

Question

Suppose that $f \in \mathcal{L}^p$ and $g \in \mathcal{L}^q$, and $p,q$ are conjugate exponents. Then prove that

(a) $h(x) = \int_{-\infty}^{\infty} f(t) g(x+t) \, dt$ defines a bounded continuous function on $\mathbb{R}$ that satisfies $\Vert h \Vert_\infty \leq \Vert f \Vert_p \Vert g \Vert_q$

(b) $h$ is differentiable, if either $f$ or $g$ is differentiable and find $h'$ in terms of $f'$ or $g'$.

Any help is appreciated.

Answer 1

a) Holder is the answer.

b) if $g$ is differentiable, then $h'(x) = \int f(t)g'(x+t)dt$, otherwise let $t' = x + t$, then $h(x) = \int f(t' - x)g(t')dt'$ and therefore if $f$ is differentiable $h'(x) = -\int f'(t - x)g(t)dt$