An example of a sequence which satisfies a number of conditions

Question

I was studying on the completeness of $L^p$ Space. Could someone give an example of a sequence $\{f_k\}$ satisfying the following?

1. $f_k \in L^p (1 \le p \lt \infty)$

2. $f_k \to 0$ in $L^p$ (in the sense of p-norm)

   However,

3. $f_k$ does not converge in $L^q$ for $q>p$

Any idea would be appreciated.

Answer 1

Let the underlying space be $X = (0,1)$ with the usual Lebesgue measure. For convenience choose $p = 1$.

Let

$$ f_k = k^{1 - \alpha(k)}\mathbf{1}_{(0,1/k)} $$

where $\mathbf{1}_A$ is the indicator function of the set $A$. If $\alpha$ satisfies:

1. $\alpha$ is positive
2. $\alpha$ is monotonically decreasing in $k$, with limit 0 as $k\to\infty$
3. $\lim_{k \to \infty} k^{-\alpha(k)} = 0$

Then 2. implies that for any $q > 1$, $\|f_k\|_q \nearrow +\infty$. And 3. implies that for $p = 1$, $\|f_k\|_1 \searrow 0$.

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A possible choice of $\alpha$ is $1 / \log\log k$. Since we have

$$ \log (k^{-\alpha(k)}) = - \frac{\log k}{\log \log k} $$

with limit $-\infty$, this implies the desired result 3.