Orals Weekly Readings (09/02-09/23)

Over the past few weeks, I have been reading about various results from the theory of functional analysis, harmonic analysis, and (non)linear wave equations (as mentioned in the first blog post). I figured that it would be a valuable exercise to summarize what I have read each week in a blog post. This post is the first in such a series. Since the gap between my first two meetings with my advisor was three weeks apart, this blog post covers three weeks of readings, rather than the usual one.

The Riesz-Thorin Interpolation Theorem

On the functional and harmonic analysis side, the major result I read about in the past three weeks is the Riesz-Thorin Interpolation Theorem.

Theorem. (Riesz-Thorin Interpolation Theorem) Suppose $T$ is a linear mapping from $L^{p_{0}} + L^{p_{1}}$ to $L^{q_{0}} + L^{q_{1}}$. Assume that $T$ is bounded from $L^{p_{0}}$ to $L^{q_{0}}$ and from $L^{p_{1}}$ to $L^{q_{1}}$:

Then $T$ is bounded from $L^{p}$ to $L^{q}$: $|Tf|_{L^{q}} \leq M|f|_{L^{p}}$, whenever the pair $(p, q)$ can be written as

for some $t$ with $0 \leq t \leq 1$. Moreover, the bound $M$ satisfies $M \leq M_{0}^{1 - t}M_{1}^{t}$.

The underlying motivation behind this theorem is the notion of interpolation. Broadly speaking, suppose that we have some collection of “estimates”: $A_{p_{0}} \lesssim B_{p_{0}}$ and $A_{p_{1}} \lesssim B_{p_{1}}$. Here $A$ and $B$ can be any quantity that we’re interested in (in a while, we will study the case when $A$ and $B$ are the $L^{p}$ norms on Banach spaces). Then an interesting question to ask is that if $\theta$ is some continuous parameter between $p_{0}$ and $p_{1}$, is it necessarily true that $A_{\theta} \lesssim B_{\theta}$? There are different reasons why we would care about this question. For instance, maybe we are interested obtaining the estimates $A_{\theta_{0}} \lesssim B_{\theta_{0}}$ where $\theta_{0} \in [p_{0}, p_{1}]$. If we can “interpolate” the estimates in the above sense, then the effort of showing the above estimates reduces to showing the estimates hold for the endpoints. This might not necessarily be an easy task, by any means, but it is certainly far easier than showing an estimate for every $\theta_{0}$ in the interval.

In the specific case of $L^{p}$ spaces, the interpolation theorem enables us to study linear operators on $L^{p}$ spaces. For example, consider the Fourier transform operator $\mathcal{F}$ defined by $\mathcal{F}(f)(\xi) = \int_{\mathbb{R}^{d}}f(x)e^{-2\pi i x \cdot \xi}\;dx.$ We know by Plancherel’s theorem that $\mathcal{F}$ extends to a unitary mapping on $L^{2}(\mathbb{R}^{d})$, and is a bounded operator from $L^{1}(\mathbb{R}^{d}) \to L^{\infty}(\mathbb{R}^{d})$: $|Tf|_{L^{\infty}(\mathbb{R}^{d})} \leq |f|_{L^{1}(\mathbb{R}^{d})}$. Then the Riesz-Thorin Interpolation theorem enables us to say that the Fourier transform extends to a bounded map from any $L^{p}$ to $L^{q}$, where $1 \leq p \leq 2$, and $p, q$ are conjugate exponents.

This is not all. The theorem also enables us to establish estimates on convolutions of functions via Young’s inequality for convolutions: for any $(p, q, r)$ with $q \in [1, \infty]$ and $q^{-1} = p^{-1} + r^{-1} - 1$, $|f \ast g|_{L^{q}} \leq |f|_{L^{p}}|g|_{L^{r}}$, where $f \in L^{p}(\mathbb{R}^{d})$ and $g \in L^{r}(\mathbb{R}^{d})$. As we will see eventually, interpolation will play an important role when it comes to establishing estimates for PDEs.