$L^p$ space and map $T_h(f)(x)=f(x+h)$ on $L^p(\mathbb R^n)$

Question

I am trying to show these two statements:

Let $f \in L^p(\mathbb R^d)$, $1 \leq p < \infty$, then

\begin{align}(a) & \quad \left(\int_{\mathbb R^d}|f(x-h)-f(x)|^p\,dx\right)^{\frac{1}{p}} \to 0 \text{ when } \|h\| \to 0 \\ (b) & \quad \left(\int_{\mathbb R^d}|f(x-h)+f(x)|^p\,dx\right)^{\frac{1}{p}} \to 2^{\frac{1}{p}}\|f\|_p \text{ when } \|h\| \to \infty \end{align}

I am having some difficulty with this problem.

For (a), it is sufficient to show that $\|f(x+h)-f(x)\|_p^p \to 0 $ when $\|h\| \to 0 $. First I've tried to prove this for continuous functions of compact support. So take $f$ such that $f(x)=0$ for all $x \in K^c$ with $K$ a compact set.

Now, $\int_{\mathbb R^d}|f(x-h)-f(x)|^p\,dx=\int_{K^c}|f(x-h)-f(x)|^p\,dx+\int_K|f(x-h)-f(x)|^p\,dx$.

If I could show that for $\|h\|$ sufficiently small, $x-h \in K^c$ for all $x \in K^c$ and $x-h \in K$ for all $x \in K$, then I can use the fact that $f$ is uniformly continuous on $K$ and that $f=0$ on $K^c$ to prove that $\int_{\mathbb R^d}|f(x-h)-f(x)|^p\,dx$ tends to $0$. I don't know what to do to show this.

As for part (b) I am completely lost.

Any help would be greatly appreciated. Thanks in advance.

Answer 1

Hint: Suppose first $f$ is continuous with compact support. For (a) note the integral in question is over a fixed bounded set if $|h|<1.$ So the result will hold by the uniform continuity of $f.$ For (b) observe that the supports of $f(x-h), f(x)$ are disjoint for $|h|$ large. Thus the integral equals, exactly, $2^{1/p}\|f\|_p$ for $|h|$ large. For the general $f\in L^p$ use the fact that continuous functions with compact support are dense in $L^p.$

Answer 2

This has a very strong flavor of Lebesgue dominated convergence theorem...

In (a), we might say $h_n\longrightarrow0$, and for $f_n(x)=f(x-h_n)$, we have pointwise-a.e. convergence to $f$. Further it shouldn't be too difficult to find a dominating function?

For (b), we can ask what happens to $f_n$ as $h_n\longrightarrow\infty$? In particular, $f\in L^p(\mathbb{R}^d)$ so has to decay to zero at infinity, so as we shift $f_n$ "further away" from $f$, the sum $f_n+f$ should be approaching something that looks like two separate copies of $f$?