$H(t) = \int_{-\infty}^\infty |f(x) + tg(x)|^p\,dx$ differentiable or not?

Question

Let $p > 1$ and $f$, $g \in L^p(\mathbb{R})$. Define$$H(t) = \int_{-\infty}^\infty |f(x) + tg(x)|^p\,dx$$for $t \in \mathbb{R}$. I have two questions.

1. Is $H$ a differentiable or not?
2. If so, what is its derivative?

Answer 1

For any $h\in\mathbb{R}$ $$ \frac{H(t+h)-H(t)}{h}=\int_\mathbb{R}\frac{|f(x)+t\,g(x)+h\,g(x)|^p-|f(x)+t\,g(x)|^p}{h}\,dx. $$ The integrand converges pointwise to $p(g(x)+t\,g(x))|f(x)+t\,g(x)|^{p-2}$. Moreover, it is bounded by $$ |h|\,|g|\,\max(|f|,t\,|g|)^{p-1}. $$ Now use the Dominated Convergence Theorem.