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If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$

where $g_{(t)}(x):=g(t+x)$.

Any hints? Try to give me only hints/outlines not complete solutions

Not sure where to go from there?

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You can try to prove it for characteristic function of hypercube , and then for characteristic function of any set, with using iterated limits, starting from t. An example of using iterated limits is here: math.stackexchange.com/questions/677108/… ,you can also consider inequality described here: math.stackexchange.com/questions/458230/…, or some tricks from this document: fractal.math.unr.edu/~ejolson/761/notes/761sep12.pdf –  Darius Apr 23 at 18:44

3 Answers 3

Hint: Prove this for compactly supported functions.

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This $2^{1/p}$ thing is throwing me off...if it was a plain $2$ it would have been easier i think. –  said Oct 29 '12 at 1:21
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said: This might be because you forgot that the $L^p$ norm is the power $1/p$ of an integral. Hence the factor $2^{1/p}$ is only natural. –  Did Oct 29 '12 at 8:59

Hint:

  1. First show it for characteristic functions $\chi_I$ where $I$ is some interval.

  2. Using 1. prove this for simple functions (i.e. finite sums $\sum \alpha_I\chi_I$).

  3. Prove the general statement by approximating a general function $g\in L^p$ by those of 2.

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Hint:

Let's introduce $B_r(P)$ as it is defined and $\|\ f\|_{L_{p}{\{\text{Area of integrating}\}}}=\left(\int\limits_{\text{Area of integrating}}|f(x)|^p dx\right)^{\frac{1}{p}}$. The key to solving the problem are triangle inequalities as it is shown below: $$\|g\|_{L_{p}\{B_{0.5|t|}(0)\}}\leqslant\|g_{(t)}+g\|_{L_{p}\{B_{0.5|t|}(0)\}}+\|g_{(t)}\|_{L_{p}\{B_{0.5|t|}(0)\}}$$

and: $$\|g_{(t)}+g\|_{L_{p}\{B_{0.5|t|}(0)\}}\leqslant \|g\|_{L_{p}\{B_{0.5|t|}(0)\}}+\|g_{(t)}\|_{L_{p}\{B_{0.5|t|}(0)\}}$$

Notice that: $$\|g_{(t)}+g\|_{L_p}=\left(\|g_{(t)}+g\|_{L_{p}\{B_{0.5|t|}(0)\}}^p+\|g_{(t)}+g\|_{L_{p}\{B_{0.5|t|}(-t)\}}^p+\|g_{(t)}+g\|_{L_{p}\{\overline{B_{0.5|t|}(0)\cup B_{0.5|t|}(-t)}\}}^p\right)^{\frac{1}{p}}$$

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