How can I compute $\lim_{|r|\to\infty}\int_{\Bbb R}|f(x+r)+f(x)|^p\ dx$ with $f\in L^p$? This is an exercise in real analysis:  

Suppose $f\in L^p(\Bbb{R})$ for $1\leq p<\infty$. Compute
  $$
\lim_{|r|\to\infty}\int_{\Bbb R}|f(x+r)+f(x)|^p\ dx. 
$$


For $p=1$ and nonnegative $f$, one gets $2\|f\|_{L^1}^1$. For $p=2$,
$$
\begin{align}
\int |f(x+r)+f(x)|^2\ dx&=\int f^2(x+r)+2f(x+r)f(x)+f^2(x)\ dx\\
&=2\|f\|_{L^2}^2+2\int f(x+r)f(x)dx.
\end{align}
$$
Then one can deal with the second term in the last equality with the dominated convergence theorem. But I don't see how to do the general case. $2\|f\|_p^p$ might be a reasonable guess though. 
 A: The limit is indeed $2\|f\|_p^p$. Let $\varepsilon > 0$. We can decompose $f$ as $g + h$, where $g$ has compact support on $\Bbb R$ and $\|h\|_p < \varepsilon$. Define the translation operator $T_rh(x) := h(x + r)$. Choose a positive integer $k$ such that for $|r| > k$, the support of $T_rg$ is disjoint from the support of $g$. Hence, for $|r| > k$,
\begin{align}\int_{\Bbb R} |g(x + r) + g(x)|^p\, dx 
&= \int_{\operatorname{supp}(g)}|g(x+r) + g(x)|^p\, dx + \int_{\operatorname{supp}(T_rg)} |g(x + r) + g(x)|^p\, dx\\
&= \int_{\operatorname{supp}(g)} |g(x)|^p\, dx + \int_{\operatorname{supp}(T_rg)} |g(x + r)|^p\\
&= \int_{\Bbb R} |g(x)|^p + \int_{\Bbb R} |g(x + r)|^p\\
&= \int_{\Bbb R} |g(x)|^p + \int_{\Bbb R} |g(x)|^p\\
&= 2\|g\|_p^p.
\end{align}
In other words, $$\|T_rg + g\|_p = 2^{1/p}\|g\|_p\quad \text{if}\quad |r| > k.$$ By the triangle inequality, if $|r| > k$, then 
\begin{align}
\|T_rf + f\|_p - 2^{1/p}\|f\|_p & = \|(T_r g + g) + (T_rh + h)\|_p - 2^{1/p}\|g + h\|_p\\
& \le \|T_rg + g\|_p + \|T_rh + h\|_p - 2^{1/p}(\|g\|_p - \|h\|_p)\\
& \le (\|T_rg + g\|_p - 2^{1/p}\|g\|_p) + (\|T_rh + h\|_p + 2^{1/p}\|h\|_p)\\
& \lesssim_p \varepsilon
\end{align}
and 
\begin{align}
\|T_rf + f\|_p - 2^{1/p}\|f\|_p &= \|(T_rg + g) + (T_rh + h)\|_p - 2^{1/p}\|g + h\|_p\\
&\ge \|T_rg + g\|_p - \|T_rh + h\|_p - 2^{1/p}(\|g\|_p + \|h\|_p)\\
&= (\|T_rg + g\|_p - 2^{1/p}\|g\|_p) - (\|T_rh + h\|_p + 2^{1/p}\|h\|_p)\\
&\gtrsim_p -\varepsilon.
\end{align}
Thus 
$$\Bigl|\|T_rf + f\|_p - 2^{1/p}\|f\|_p\Bigr| \lesssim_p \varepsilon \quad (|r| > k),$$ which shows that $\|T_rf + f\|_p \to 2^{1/p}\|f\|_p$ as $|r| \to \infty$. This implies
$$\lim_{|r|\to \infty} \int_{\Bbb R} |f(x + r) + f(x)|^p = 2\|f\|_p^p.$$
Note. Using this result, one can show the Schroedinger operator $e^{it\Delta} : L^p(\Bbb R) \to L^q(\Bbb R)$ is not continuous for any triple $(p,q,t)$ satisfying $1 \le q < p < \infty$ and $t\in \Bbb R\setminus\{0\}$.
