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In harmonic analysis and PDE, one often wants to place a function $f: {\bf R}^d \to {\bf C}$ on some domain (let’s take a Euclidean space ${\bf R}^d$ for simplicity) in one or more function spaces in order to quantify its “size” in some sense.  Examples include

• The Lebesgue spaces $L^p$ of functions $f$ whose norm $\|f\|_{L^p} := (\int_{{\bf R}^d} |f|^p)^{1/p}$ is finite, as well as their relatives such as the weak $L^p$ spaces $L^{p,\infty}$ (and more generally the Lorentz spaces $L^{p,q}$) and Orlicz spaces such as $L \log L$ and $e^L$;
• The classical regularity spaces $C^k$, together with their Hölder continuous counterparts $C^{k,\alpha}$;
• The Sobolev spaces $W^{s,p}$ of functions $f$ whose norm $\|f\|_{W^{s,p}} = \|f\|_{L^p} + \| |\nabla|^s f\|_{L^p}$ is finite (other equivalent definitions of this norm exist, and there are technicalities if $s$ is negative or $p \not \in (1,\infty)$), as well as relatives such as homogeneous Sobolev spaces $\dot W^{s,p}$, Besov spaces $B^{s,p}_q$, and Triebel-Lizorkin spaces $F^{s,p}_q$.  (The conventions for the superscripts and subscripts here are highly variable.)
• Hardy spaces ${\mathcal H}^p$, the space BMO of functions of bounded mean oscillation (and the subspace VMO of functions of vanishing mean oscillation);
• The Wiener algebra $A$;
• Morrey spaces $M^p_q$;
• The space $M$ of finite measures;
• etc., etc.

As the above partial list indicates, there is an entire zoo of function spaces one could consider, and it can be difficult at first to see how they are organised with respect to each other.  However, one can get some clarity in this regard by drawing a type diagram for the function spaces one is trying to study.  A type diagram assigns a tuple (usually a pair) of relevant exponents to each function space.  For function spaces $X$ on Euclidean space, two such exponents are the regularity $s$ of the space, and the integrability $p$ of the space.  These two quantities are somewhat fuzzy in nature (and are not easily defined for all possible function spaces), but can basically be described as follows.  We test the function space norm $\|f\|_X$ of a modulated rescaled bump function

$f(x) := A e^{i x \cdot \xi} \phi( \frac{x-x_0}{R} )$ (1)

where $A > 0$ is an amplitude, $R > 0$ is a radius, $\phi \in C^\infty_c({\bf R}^d)$ is a test function, $x_0$ is a position, and $\xi \in {\bf R}^d$ is a frequency of some magnitude $|\xi| \sim N$.  One then studies how the norm $\|f\|_X$ depends on the parameters $A, R, N$.  Typically, one has a relationship of the form

$\|f\|_X \sim A N^s R^{d/p}$ (2)

for some exponents $s, p$, at least in the high-frequency case when $N$ is large (in particular, from the uncertainty principle it is natural to require $N \gtrsim 1/R$, and when dealing with inhomogeneous norms it is also natural to require $N \gtrsim 1$).  The exponent $s$ measures how sensitive the $X$ norm is to oscillation, and thus controls regularity; if $s$ is large, then oscillating functions will have large $X$ norm, and thus functions in $X$ will tend not to oscillate too much and thus be smooth.    Similarly, the exponent $p$ measures how sensitive the $X$ norm is to the function $f$ spreading out to large scales; if $p$ is small, then slowly decaying functions will have large norm, so that functions in $X$ tend to decay quickly; conversely, if $p$ is large, then singular functions will tend to have large norm, so that functions in $X$ will tend to not have high peaks.

Note that the exponent $s$ in (2) could be positive, zero, or negative, however the exponent $p$ should be non-negative, since intuitively enlarging $R$ should always lead to a larger (or at least comparable) norm.  Finally, the exponent in the $A$ parameter should always be $1$, since norms are by definition homogeneous.  Note also that the position $x_0$ plays no role in (1); this reflects the fact that most of the popular function spaces in analysis are translation-invariant.

The type diagram below plots the $s, 1/p$ indices of various spaces.  The black dots indicate those spaces for which the $s, 1/p$ indices are fixed; the blue dots are those spaces for which at least one of the $s, 1/p$ indices are variable (and so, depending on the value chosen for these parameters, these spaces may end up in a different location on the type diagram than the typical location indicated here).

(There are some minor cheats in this diagram, for instance for the Orlicz spaces $L \log L$ and $e^L$ one has to adjust (1) by a logarithmic factor.   Also, the norms for the Schwartz space ${\mathcal S}$ are not translation-invariant and thus not perfectly describable by this formalism. This picture should be viewed as a visual aid only, and not as a genuinely rigorous mathematical statement.)

The type diagram can be used to clarify some of the relationships between function spaces, such as Sobolev embedding.  For instance, when working with inhomogeneous spaces (which basically identifies low frequencies $N \ll 1$ with medium frequencies $N \sim 1$, so that one is effectively always in the regime $N \gtrsim 1$), then decreasing the $s$ parameter results in decreasing the right-hand side of (1).  Thus, one expects the function space norms to get smaller (and the function spaces to get larger) if one decreases $s$ while keeping $p$ fixed.  Thus, for instance, $W^{k,p}$ should be contained in $W^{k-1,p}$, and so forth.  Note however that this inclusion is not available for homogeneous function spaces such as $\dot W^{k,p}$, in which the frequency parameter $N$ can be either much larger than $1$ or much smaller than $1$.

Similarly, if one is working in a compact domain rather than in ${\bf R}^d$, then one has effectively capped the radius parameter $R$ to be bounded, and so we expect the function space norms to get smaller (and the function spaces to get larger) as one increases $1/p$, thus for instance $L^2$ will be contained in $L^1$.  Conversely, if one is working in a discrete domain such as ${\Bbb Z}^d$, then the radius parameter $R$ has now effectively been bounded from below, and the reverse should occur: the function spaces should get larger as one decreases $1/p$.  (If the domain is both compact and discrete, then it is finite, and on a finite-dimensional space all norms are equivalent.)

As mentioned earlier, the uncertainty principle suggests that one has the restriction $N \gtrsim 1/R$.  From this and (2), we expect to be able to enlarge the function space by trading in the regularity parameter $s$ for the integrability parameter $p$, keeping the dimensional quantity $d/p - s$ fixed.  This is indeed how Sobolev embedding works.   Note in some cases one runs out of regularity before p goes all the way to infinity (thus ending up at an $L^p$ space), while in other cases p hits infinity first.  In the latter case, one can embed the Sobolev space into a Holder space such as $C^{k,\alpha}$.

On continuous domains, one can send the frequency $N$ off to infinity, keeping the amplitude $A$ and radius $R$ fixed.  From this and (1) we see that norms with a lower regularity $s$ can never hope to control norms with a higher regularity $s' > s$, no matter what one does with the integrability parameter.   Note however that in discrete settings this obstruction disappears; when working on, say, ${\bf Z}^d$, then in fact one can gain as much regularity as one wishes for free, and there is no distinction between a Lebesgue space $\ell^p$ and their Sobolev counterparts $W^{k,p}$ in such a setting.

When interpolating between two spaces (using either the real or complex interpolation method), the interpolated space usually has regularity and integrability exponents on the line segment between the corresponding exponents of the endpoint spaces.  (This can be heuristically justified from the formula (2) by thinking about how the real or complex interpolation methods actually work.)  Typically, one can control the norm of the interpolated space by the geometric mean of the endpoint norms that is indicated by this line segment; again, this is plausible from looking at (2).

The space $L^2$ is self-dual.  More generally, the dual of a function space $X$ will generally have type exponents that are the reflection of the original exponents around the $L^2$ origin.  Consider for instance the dual spaces $H^s, H^{-s}$ or ${\mathcal H}^1, BMO$ in the above diagram.

Spaces whose integrability exponent $p$ is larger than 1 (i.e. which lie to the left of the dotted line) tend to be Banach spaces, while spaces whose integrability exponent is less than 1 are almost never Banach spaces.  (This can be justified by covering a large ball into small balls and considering how (1) would interact with the triangle inequality in this case).  The case $p=1$ is borderline; some spaces at this level of integrability, such as $L^1$, are Banach spaces, while other spaces, such as $L^{1,\infty}$, are not.

While the regularity $s$ and integrability $p$ are usually the most important exponents in a function space (because amplitude, width, and frequency are usually the most important features of a function in analysis), they do not tell the entire story.  One major reason for this is that the modulated bump functions (1), while an important class of test examples of functions, are by no means the only functions that one would wish to study.  For instance, one could also consider sums of bump functions (1) at different scales.  The behaviour of the function space norms on such spaces is often controlled by secondary exponents, such as the second exponent $q$ that arises in Lorentz spaces, Besov spaces, or Triebel-Lizorkin spaces.  For instance, consider the function

$f_M(x) := \sum_{m=1}^M 2^{-md} \phi(x/2^m)$, (3)

where $M$ is a large integer, representing the number of distinct scales present in $f_M$.  Any function space with regularity $s=0$ and $p=1$ should assign each summand $2^{-md} \phi(x/2^m)$ in (3) a norm of O(1), so the norm of $f_M$ could be as large as $O(M)$ if one assumes the triangle inequality.  This is indeed the case for the $L^1$ norm, but for the weak $L^1$ norm, i.e. the $L^{1,\infty}$ norm,  $f_M$ only has size $O(1)$.  More generally, for the Lorentz spaces $L^{1,q}$, $f_M$ will have a norm of about $O(M^{1/q})$.   Thus we see that such secondary exponents can influence the norm of a function by an amount which is polynomial in the number of scales.  In many applications, though, the number of scales is a “logarithmic” quantity and thus of lower order interest when compared against the “polynomial” exponents such as $s$ and $p$.  So the fine distinctions between, say, strong $L^1$ and weak $L^1$, are only of interest in “critical” situations in which one cannot afford to lose any logarithmic factors (this is for instance the case in much of Calderon-Zygmund theory).

We have cheated somewhat by only working in the high frequency regime.  When dealing with inhomogeneous spaces, one often has a different set of exponents for (1) in the low-frequency regime than in the high-frequency regime.  In such cases, one sometimes has to use a more complicated type diagram to  genuinely model the situation, e.g. by assigning to each space a convex set of type exponents rather than a single exponent, or perhaps having two separate type diagrams, one for the high frequency regime and one for the low frequency regime.   Such diagrams can get quite complicated, and will probably not be much use to a beginner in the subject, though in the hands of an expert who knows what he or she is doing, they can still be an effective visual aid.

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“.