Convergence a.e of a sequence of functions in $L^{p}$

Question

Let $(f_{k})_{k}$ a sequence of functions on $L^{p}(\mathbb{R}^{n})$, $1<p<\infty$ such that $||f_{k}||_{p}\leq Ck^{\frac{\alpha}{p}}$, for all $k \in\mathbb{N}$ where $\alpha,C$ are positive constants. Prove that $k^{-(1+\alpha.p)}f_{k}(x) \to 0$ almost everywhere $x \in \mathbb{R}^{n}$.

I could not think of an efficient tool to solve this problem. Any tips?

Answer 1

Put $g_k = k^{-(1+\alpha/ p)} f_k.$ By homogeneity, we have $$\| g_k \|_p \le C k^{-1}.$$ In particular, by the Chebyshev inequality, we see that for any $\epsilon > 0$, $$\lvert \{x \in \mathbb R^n : \lvert g_k(x) \rvert > \epsilon\} \rvert \le \frac{\| g_k \|^p_p}{\epsilon^p} \le \frac{C^p}{k^p\epsilon^p} \to 0 \,\,\,\, \text{ as } k \to \infty.$$ This shows that $g_k \to 0$ in measure. Here is a proof that convergence in measure implies convergence a.e. along a subsequence: Convergence in $L^p$ and convergence almost everywhere.

Answer 2

Hint: Suppose $\epsilon>0$ and $\int_{\mathbb R} |g_k| < 1/k^{1 +\epsilon}.$ Then

$$\tag 1 \sum_{k=1}^{\infty} \int_{\mathbb R} |g_k| <\infty.$$

Switching the order of summation and integration will then show $\sum_{k=1}^{\infty} |g_k(x)| <\infty$ for a.e. $x,$ hence $|g_k(x)| \to 0$ for a.e. $x.$ This will be useful in your problem.