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As discussed in previous notes, a function space norm can be viewed as a means to rigorously quantify various statistics of a function ${f: X \rightarrow {\bf C}}$. For instance, the “height” and “width” can be quantified via the ${L^p(X,\mu)}$ norms (and their relatives, such as the Lorentz norms ${\|f\|_{L^{p,q}(X,\mu)}}$). Indeed, if ${f}$ is a step function ${f = A 1_E}$, then the ${L^p}$ norm of ${f}$ is a combination ${\|f\|_{L^p(X,\mu)} = |A| \mu(E)^{1/p}}$ of the height (or amplitude) ${A}$ and the width ${\mu(E)}$.

However, there are more features of a function ${f}$ of interest than just its width and height. When the domain ${X}$ is a Euclidean space ${{\bf R}^d}$ (or domains related to Euclidean spaces, such as open subsets of ${{\bf R}^d}$, or manifolds), then another important feature of such functions (especially in PDE) is the regularity of a function, as well as the related concept of the frequency scale of a function. These terms are not rigorously defined; but roughly speaking, regularity measures how smooth a function is (or how many times one can differentiate the function before it ceases to be a function), while the frequency scale of a function measures how quickly the function oscillates (and would be inversely proportional to the wavelength). One can illustrate this informal concept with some examples:

• Let ${\phi \in C^\infty_c({\bf R})}$ be a test function that equals ${1}$ near the origin, and ${N}$ be a large number. Then the function ${f(x) := \phi(x) \sin(Nx)}$ oscillates at a wavelength of about ${1/N}$, and a frequency scale of about ${N}$. While ${f}$ is, strictly speaking, a smooth function, it becomes increasingly less smooth in the limit ${N \rightarrow \infty}$; for instance, the derivative ${f'(x) = \phi'(x) \sin(Nx) + N \phi(x) \cos(Nx)}$ grows at a roughly linear rate as ${N \rightarrow \infty}$, and the higher derivatives grow at even faster rates. So this function does not really have any regularity in the limit ${N \rightarrow \infty}$. Note however that the height and width of this function is bounded uniformly in ${N}$; so regularity and frequency scale are independent of height and width.
• Continuing the previous example, now consider the function ${g(x) := N^{-s} \phi(x) \sin(Nx)}$, where ${s \geq 0}$ is some parameter. This function also has a frequency scale of about ${N}$. But now it has a certain amount of regularity, even in the limit ${N \rightarrow \infty}$; indeed, one easily checks that the ${k^{th}}$ derivative of ${g}$ stays bounded in ${N}$ as long as ${k \leq s}$. So one could view this function as having “${s}$ degrees of regularity” in the limit ${N \rightarrow \infty}$.
• In a similar vein, the function ${N^{-s} \phi(Nx)}$ also has a frequency scale of about ${N}$, and can be viewed as having ${s}$ degrees of regularity in the limit ${N \rightarrow \infty}$.
• The function ${\phi(x) |x|^s 1_{x > 0}}$ also has about ${s}$ degrees of regularity, in the sense that it can be differentiated up to ${s}$ times before becoming unbounded. By performing a dyadic decomposition of the ${x}$ variable, one can also decompose this function into components ${\psi(2^n x) |x|^s}$ for ${n \geq 0}$, where ${\psi(x) := (\phi(x)-\phi(2x)) 1_{x>0}}$ is a bump function supported away from the origin; each such component has frequency scale about ${2^n}$ and ${s}$ degrees of regularity. Thus we see that the original function ${\phi(x) |x|^s 1_{x > 0}}$ has a range of frequency scales, ranging from about ${1}$ all the way to ${+\infty}$.
• One can of course concoct higher-dimensional analogues of these examples. For instance, the localised plane wave ${\phi(x) \sin(\xi \cdot x)}$ in ${{\bf R}^d}$, where ${\phi \in C^\infty_c({\bf R}^d)}$ is a test function, would have a frequency scale of about ${|\xi|}$.

There are a variety of function space norms that can be used to capture frequency scale (or regularity) in addition to height and width. The most common and well-known examples of such spaces are the Sobolev space norms ${\| f\|_{W^{s,p}({\bf R}^d)}}$, although there are a number of other norms with similar features (such as Hölder norms, Besov norms, and Triebel-Lizorkin norms). Very roughly speaking, the ${W^{s,p}}$ norm is like the ${L^p}$ norm, but with “${s}$ additional degrees of regularity”. For instance, in one dimension, the function ${A \phi(x/R) \sin(Nx)}$, where ${\phi}$ is a fixed test function and ${R, N}$ are large, will have a ${W^{s,p}}$ norm of about ${|A| R^{1/p} N^s}$, thus combining the “height” ${|A|}$, the “width” ${R}$, and the “frequency scale” ${N}$ of this function together. (Compare this with the ${L^p}$ norm of the same function, which is about ${|A| R^{1/p}}$.)

To a large extent, the theory of the Sobolev spaces ${W^{s,p}({\bf R}^d)}$ resembles their Lebesgue counterparts ${L^p({\bf R}^d)}$ (which are as the special case of Sobolev spaces when ${s=0}$), but with the additional benefit of being able to interact very nicely with (weak) derivatives: a first derivative ${\frac{\partial f}{\partial x_j}}$ of a function in an ${L^p}$ space usually leaves all Lebesgue spaces, but a first derivative of a function in the Sobolev space ${W^{s,p}}$ will end up in another Sobolev space ${W^{s-1,p}}$. This compatibility with the differentiation operation begins to explain why Sobolev spaces are so useful in the theory of partial differential equations. Furthermore, the regularity parameter ${s}$ in Sobolev spaces is not restricted to be a natural number; it can be any real number, and one can use fractional derivative or integration operators to move from one regularity to another. Despite the fact that most partial differential equations involve differential operators of integer order, fractional spaces are still of importance; for instance it often turns out that the Sobolev spaces which are critical (scale-invariant) for a certain PDE are of fractional order.

The uncertainty principle in Fourier analysis places a constraint between the width and frequency scale of a function; roughly speaking (and in one dimension for simplicity), the product of the two quantities has to be bounded away from zero (or to put it another way, a wave is always at least as wide as its wavelength). This constraint can be quantified as the very useful Sobolev embedding theorem, which allows one to trade regularity for integrability: a function in a Sobolev space ${W^{s,p}}$ will automatically lie in a number of other Sobolev spaces ${W^{\tilde s,\tilde p}}$ with ${\tilde s < s}$ and ${\tilde p > p}$; in particular, one can often embed Sobolev spaces into Lebesgue spaces. The trade is not reversible: one cannot start with a function with a lot of integrability and no regularity, and expect to recover regularity in a space of lower integrability. (One can already see this with the most basic example of Sobolev embedding, coming from the fundamental theorem of calculus. If a (continuously differentiable) function ${f: {\bf R} \rightarrow {\bf R}}$ has ${f'}$ in ${L^1({\bf R})}$, then we of course have ${f \in L^\infty({\bf R})}$; but the converse is far from true.)

Plancherel’s theorem reveals that Fourier-analytic tools are particularly powerful when applied to ${L^2}$ spaces. Because of this, the Fourier transform is very effective at dealing with the ${L^2}$-based Sobolev spaces ${W^{s,2}({\bf R}^d)}$, often abbreviated ${H^s({\bf R}^d)}$. Indeed, using the fact that the Fourier transform converts regularity to decay, we will see that the ${H^s({\bf R}^d)}$ spaces are nothing more than Fourier transforms of weighted ${L^2}$ spaces, and in particular enjoy a Hilbert space structure. These Sobolev spaces, and in particular the energy space ${H^1({\bf R}^d)}$, are of particular importance in any PDE that involves some sort of energy functional (this includes large classes of elliptic, parabolic, dispersive, and wave equations, and especially those equations connected to physics and/or geometry).

We will not fully develop the theory of Sobolev spaces here, as this would require the theory of singular integrals, which is beyond the scope of this course. There are of course many references for further reading; one is Stein’s “Singular integrals and differentiability properties of functions“.