Analysis Problem of the Day 21

Today’s problem appeared as Problem 7 on the Stanford Spring 2023 Analysis Qual:

Problem 7: Let $f \in L^1([0,1])$ and $1<p<\infty.$ Show that $f \in L^p([0,1])$ if and only if

$\sup_{\{I_j\}} \sum_j |I_j| \left(\frac{1}{|I_j|}\int_{I_j} |f| \right)^p} < \infty,$

where the supremum is taken over all partitions of $[0,1]$ into finitely many intervals.

Solution: Note that the given expression is the $L^p$ norm of the function where $|f|$ is replaced by a locally constant function defined on $I_j$ to be the average of $|f|$ on $I_j.$ For the forward direction, assume $f \in L^p([0,1]).$ Since we have an inequality where one is taking the integral to the power of $p,$ one might wonder if you can somehow bring the power inside the integral. This is exactly what Jensen’s inequality allows us to do – it says that for any convex function, the function applied to an average is bounded by the average of the function. In other words, since $x^p$ is convex for $p>1,$

$\left(\frac{1}{|I_j|} \int_{I_j} |f| dx\right)^p \leq \frac{1}{|I_j|} \int_{I_j}|f|^p dx.$

Using this inequality, we see that the supremum above is bounded by $\sum_{I_j} \|f\|^p_{L^p(I_j)} = \|f\|_p^p<\infty.$

For the reverse inequality, we must necessarily take a different approach. Recall that the span $S$ of characteristic functions of intervals is dense in simple functions in the $L^p$ norm, and thus dense in $L^p$ for $p<\infty.$ Moreover, for $f \in S,$ the expression in the problem statement is precisely $\|f\|_p^p.$ In particular, for any simple function $\chi \leq f,$ there exists a function $g \in S$ such that $\|g-\chi\|_p < \epsilon.$ Consequently,

$\|f\|_p^p = \sup\{\|\chi\|_p^p: 0 \leq \chi \leq f \text{ simple}\} < \sup\{\|g\|_p^p: g \in S\} + \epsilon < \infty,$

so $f \in L^p([0,1]).$

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