Analysis Problem of the Day 91

Today’s problem appeared as Problem 5 on the UCLA Fall 2021 Analysis Qual:

Problem 5. Let $\phi,\psi$ be bounded linear functionals on $L^p(\mathbb{R}).$ Find all the values of $1 \leq p \leq \infty$ for which $\|\phi + \psi\| = \|\phi\|+\|\psi\|$ implies that $\phi$ and $\psi$ are linearly dependent.

Solution: Recall by the theory of $L^p$ spaces that the dual of $L^p$ for $p < \infty$ is $L^q,$ where $q$ is the Hölder conjugate satisfying $\frac{1}{p}+\frac{1}{q}=1.$ For now, we restrict ourselves to the case $1<p<\infty.$ Thus, the norms above are $L^q$ norms. Recall Minkowski’s inequality, which states that $\|f+g\|_p \leq \|f\|_p+\|g\|_p$ for all $1 \leq p \leq \infty.$ The proof of Minkowski’s inequality is based on the proof of Hölder’s inequality, which itself is based on Young’s inequality. We are interested in the equality case for these inequalities. Recall that Young’s inequality is proven by integrating the inverse functions $x^{p-1}$ and $x^{q-1}$ on the intervals $x \in [0,a], y \in [0,b],$ respectively. The inequality then bounds the area of the rectangle $[0,a] \times [0,b]$ by the sum of the integrals of these two functions. In particular, we note that equality occurs iff $a^{p-1}=b^{q-1}.$ Now, the proof of Hölder’s inequality involves applying Young’s inequality to the normalized product $a= \frac{f}{\|f\|_p}, b=\frac{g}{\|g\|_q}.$ Thus, equality holds iff $\frac{f^{p-1}}{\|f\|_p^{p-1}} = \frac{g^{q-1}}{\|g\|_q^{q-1}}.$ Finally, the proof of Minkowski’s inequality involves applying triangle inequality and Hölder’s to $|f|,|g|,$ and $|f+g|^{p-1},$ placing $f$ and $g$ in $L^p,$ respectively. Thus, equality occurs iff $\text{sign}(f)=\text{sign}(g)$ and

$\frac{f^{p-1}}{\|f\|_p^{p-1}} = \frac{|f+g|^{(p-1)(q-1)}}{\||f+g|^{p-1}\|_q^{q-1}}, \quad \frac{g^{p-1}}{\|g\|_p^{p-1}} = \frac{|f+g|^{(p-1)(q-1)}}{\||f+g|^{p-1}\|_q^{q-1}}.$

This implies that $\frac{f^{p-1}}{\|f\|_p^{p-1}} = \frac{g^{p-1}}{\|g\|_p^{p-1}},$ i.e. $f$ and $g$ are scalar multiples of each other (since they have the same sign). Thus, we conclude that the given equality implies linear dependence of $f$ and $g$ for $1 < p < \infty.$

We now separately analyze the case $p=1$ and $p=\infty.$ Clearly, the statement does not have to hold for $p=1,q=\infty$ as $\|\chi_{[0,1]}+\chi_{[\frac12,\frac32]}\|_\infty = \|\chi_{[0,1]}\|_\infty+\|\chi_{[\frac12,\frac32]}\|_\infty = 2$ but $\chi_{[0,1]}$ and $\chi_{[\frac12,\frac32]}$ are not linearly dependent. Finally, for the case $p=\infty,$ if $\phi,\psi$ are given by $L^1$ functions, the statement implies $\int |\phi+\psi| dx= \int |\phi|+\int |\psi| dx,$ so with the same counterexample as above, we see that the statement does not have to hold. Thus, the range of values for which the statement holds is $p \in (1,\infty).$

Remark: The dual of $L^\infty$ is the space $\text{ba}$ of bounded finitely additive signed measures, so in particular not every bounded linear functional on $L^\infty$ is given by integration against an element of $L^1.$

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