I am trying to get the derivative of
$$H(t) = \int_{- \infty}^\infty |f(x) + t g(x)|^p \mathrm{d} x$$
where $f,g \int L^p(\mathbb{R}) : p > 1$ and $t \in \mathbb{R}$
From a theorem in Folland, I can differentiate this through the integral, if the absolute value of the partial derivative of the integrand with respect to $t$ is bounded by some integrable function $h(x)$. My problem is finding this function.
The derivative is easy to obtain. Let $I(x,t)$ be the integrand of the above integral:
$$\left|\frac{\partial I(x,t)}{\partial t} \right| = \left|p|f(x) + tg(x)|^{p-1} \cdot \mathrm{sign}(f(x)+t g(x)) \cdot g(x)\right|$$
$$\leq \left|p|f(x) + tg(x)|^{p-1} g(x) \right|$$
I don't have any more information on the functions $f$ and $g$ so I am stuck on how to show the bound.
Hint: By Young's inequality we get $|g(x)||f(x)+tg(x)|^{p-1} \leq \frac {|g(x)|^{p}} p+ \frac {|f(x)+tg(x)|^{(p-1)q}} q$. Since $(p-1)q=p$ this gives the bound $\frac {|g(x)|^{p}} p+ \frac {|f(x)+tg(x)|^{p}} q$. For differentiability of $H$ we only have to consider $t$ in a bounded set. Can you finish now?
For Young's inequality see https://en.wikipedia.org/wiki/Young%27s_inequality_for_products