Analysis Problem of the Day 73

Today’s problem appeared as Problem 5 on the UCLA Spring 2024 Analysis Qual:

Problem 5. Fix $1 \leq p<q <\infty.$

a) Suppose $f \in L^p(\mathbb{R})$ and $\int_A |f(x)|^q dx <\infty$ for all $A$ with Lebesgue measure $|A|=1.$ Show that $f \in L^q(\mathbb{R}).$

b) Show that there exists $f \in L^p(\mathbb{R})$ such that $\int_a^{a+1} |f(x)|^q dx < \infty$ for all $a \in \mathbb{R}$ but $f \not \in L^q(\mathbb{R}).$

Solution: a) Split $|f|$ into where it is greater than and less than or equal to $1.$ On $\{x:|f(x)| \leq 1\},$ since $p<q,$ $|f(x)|^q \leq |f(x)|^p,$ so $\int_{|f| \leq 1} |f(x)|^q dx \leq \int_{|f| \leq 1} |f(x)|^p dx < \infty.$ Conversely, when $|f| \geq 1,$ apply Chebyshev’s inequality to note that $|\{x:|f(x)|>1\}| \leq \|f\|_p^p < \infty.$ In particular, $\{x:|f(x)|>1\}$ has finite measure, and thus may be covered by finitely many Borel sets $A_1,A_2,...,A_k$ of measure $1.$ Then,

$\int_{|f|>1} |f(x)|^q dx \leq \sum_{i=1}^k \int_{A_i} |f(x)|^q dx < \infty.$

It follows that $\int |f(x)|^q dx = \int_{|f| \leq 1} |f(x)|^q dx + \int_{|f| >1} |f(x)|^q dx <\infty,$ so that $f \in L^q(\mathbb{R}).$

b) For simplicity, suppose that $f$ is a simple function of the form $f(x):=\sum_{i=1}^\infty a_i \chi_{B_i}(x),$ where $B_i \subset [i,i+1], |B_i| = b_i,$ and $a_i>0.$ Since $f \in L^\infty_{loc},$ it follows that $f \in L^q_{loc},$ so that $\int_a^{a+1}|f(x)|^q dx < \infty$ for all $a \in \mathbb{R}.$ Moreover, since $f \in L^p(\mathbb{R}),$ one must have $\sum_{i=1}^\infty b_i a_i^p <\infty.$ Finally, for $f \not \in L^q(\mathbb{R})$ to hold, one must have $\sum_{i=1}^\infty b_i a_i^q = \infty.$ Now, note that if we set $a_i=n$ and $b_i = n^c$ for $-1-q \leq c < -1-p,$ then

$\sum b_i a_i^p \leq \sum n^{-1-\epsilon} < \infty$

for some $\epsilon>0,$ and

$\sum b_i a_i^q \geq \sum n^{-1} = \infty,$

and $b_i \leq 1,$ so $f$ as defined above with $a_i$ and $b_i$ as given satisfies the required properties.

Remark: The crucial distinction between (a) and (b) is that the set where $|f|$ is large may not be compact, and thus may not be covered by finitely many intervals of length 1 (i.e. the mass of $|f|^p$ escapes to both horizontal and vertical infinity). In fact, one may relax the assumption to $f \in L^q_{loc}$ and still obtain the conclusion in (a).

Leave a Reply Cancel reply