Differentiability of function defined as integral form

Question

Let $H(t)=\displaystyle\int_{\Bbb R}\lvert\, f(x)+tg(x)\rvert^p\mathrm dx$ and $f,g\in L^p(\Bbb R)$.

Then, how to prove that $H$ is differentiable and find its derivative?

I think it's impossible to find it by $\dfrac{H(t+h)-H(t)}{h}$.

Should I show that $H$ is of bounded variation?

Answer 1

Hint: $$ \frac{|f(x)+(t+h)g(x)|^p-|f(x)+tg(x)|^p}{h}=p|f(x)+(t+\vartheta)g(x)|^{p-1}g(x), $$ for some $\vartheta\in(0,h)$, if $p>1$. The right-hand side is bounded by $$ p\big(|f(t)|+s|g(t)|\big)^{p-1}|g(x)| \in L^{1}(\mathbb R), $$ where $s\ge |t+\vartheta h|$, for all small $h$. Then use Lebesgue Dominated Convergence Theorem.