As discussed in previous notes, a function space norm can be viewed as a means to rigorously quantify various statistics of a function . For instance, the “height” and “width” can be quantified via the
norms (and their relatives, such as the Lorentz norms
). Indeed, if
is a step function
, then the
norm of
is a combination
of the height (or amplitude)
and the width
.
However, there are more features of a function of interest than just its width and height. When the domain
is a Euclidean space
(or domains related to Euclidean spaces, such as open subsets of
, 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
be a test function that equals
near the origin, and
be a large number. Then the function
oscillates at a wavelength of about
, and a frequency scale of about
. While
is, strictly speaking, a smooth function, it becomes increasingly less smooth in the limit
; for instance, the derivative
grows at a roughly linear rate as
, and the higher derivatives grow at even faster rates. So this function does not really have any regularity in the limit
. Note however that the height and width of this function is bounded uniformly in
; so regularity and frequency scale are independent of height and width.
- Continuing the previous example, now consider the function
, where
is some parameter. This function also has a frequency scale of about
. But now it has a certain amount of regularity, even in the limit
; indeed, one easily checks that the
derivative of
stays bounded in
as long as
. So one could view this function as having “
degrees of regularity” in the limit
.
- In a similar vein, the function
also has a frequency scale of about
, and can be viewed as having
degrees of regularity in the limit
.
- The function
also has about
degrees of regularity, in the sense that it can be differentiated up to
times before becoming unbounded. By performing a dyadic decomposition of the
variable, one can also decompose this function into components
for
, where
is a bump function supported away from the origin; each such component has frequency scale about
and
degrees of regularity. Thus we see that the original function
has a range of frequency scales, ranging from about
all the way to
.
- One can of course concoct higher-dimensional analogues of these examples. For instance, the localised plane wave
in
, where
is a test function, would have a frequency scale of about
.
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 , 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
norm is like the
norm, but with “
additional degrees of regularity”. For instance, in one dimension, the function
, where
is a fixed test function and
are large, will have a
norm of about
, thus combining the “height”
, the “width”
, and the “frequency scale”
of this function together. (Compare this with the
norm of the same function, which is about
.)
To a large extent, the theory of the Sobolev spaces resembles their Lebesgue counterparts
(which are as the special case of Sobolev spaces when
), but with the additional benefit of being able to interact very nicely with (weak) derivatives: a first derivative
of a function in an
space usually leaves all Lebesgue spaces, but a first derivative of a function in the Sobolev space
will end up in another Sobolev space
. 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
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 will automatically lie in a number of other Sobolev spaces
with
and
; 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
has
in
, then we of course have
; but the converse is far from true.)
Plancherel’s theorem reveals that Fourier-analytic tools are particularly powerful when applied to spaces. Because of this, the Fourier transform is very effective at dealing with the
-based Sobolev spaces
, often abbreviated
. Indeed, using the fact that the Fourier transform converts regularity to decay, we will see that the
spaces are nothing more than Fourier transforms of weighted
spaces, and in particular enjoy a Hilbert space structure. These Sobolev spaces, and in particular the energy space
, 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“.
— 1. Hölder spaces —
Throughout these notes, is a fixed dimension.
Before we study Sobolev spaces, let us first look at the more elementary theory of Hölder spaces , which resemble Sobolev spaces but with the aspect of width removed (thus Hölder norms only measure a combination of height and frequency scale). One can define these spaces on many domains (for instance, the
norm can be defined on any metric space) but we shall largely restrict attention to Euclidean spaces
for sake of concreteness.
We first recall the spaces, which we have already been implicitly using in previous lectures. The space
is the space of bounded continuous functions
on
, with norm
This norm gives the structure of a Banach space. More generally, one can then define the spaces
for any non-negative integer
as the space of all functions which are
times continuously differentiable, with all derivatives of order
bounded, and whose norm is given by the formula
where we view as a rank
, dimension
tensor with complex coefficients (or equivalently, as a vector of dimension
with complex coefficients), thus
(One does not have to use the norm here, actually; since all norms on a finite-dimensional space are equivalent, any other means of taking norms here will lead to an equivalent definition of the
norm. More generally, all the norms discussed here tend to have several definitions which are equivalent up to constants, and in most cases the exact choice of norm one uses is just a matter of personal taste.)
Remark 1 In some texts,
is used to denote the functions which are
times continuously differentiable, but whose derivatives up to
order are allowed to be unbounded, so for instance
would lie in
for every
under this definition. Here, we will refer to such functions (with unbounded derivatives) as lying in
(i.e. they are locally in
), rather than
. Similarly, we make a distinction between
(smooth functions, with no bounds on derivatives) and
(smooth functions, all of whose derivatives are bounded). Thus, for instance,
lies in
but not
.
Exercise 2 Show that
is a Banach space.
Exercise 3 Show that for every
and
, the
norm is equivalent to the modified norm
in the sense that there exists a constant
(depending on
and
) such that
for all
. (Hint: use Taylor series with remainder.) Thus when defining the
norms, one does not really need to bound all the intermediate derivatives
for
; the two extreme terms
suffice. (This is part of a more general interpolation phenomenon; the extreme terms in a sum often already suffice to control the intermediate terms.)
Exercise 4 Let
be a bump function, and
. Show that if
with
,
, and
, then the function
has a
norm of at most
, where
is a constant depending only on
,
and
. Thus we see how the
norm relates to the height
, width
, and frequency scale
of the function, and in particular how the width
is largely irrelevant. What happens when the condition
is dropped?
We clearly have the inclusions
and for any constant-coefficient partial differential operator
of some order , it is easy to see that
is a bounded linear operator from
to
for any
.
The Hölder spaces are designed to “fill up the gaps” between the discrete spectrum
of the continuously differentiable spaces. For
and
, these spaces are defined as the subspace of functions
whose norm
is finite. To put it another way, if
is bounded and continuous, and furthermore obeys the Hölder continuity bound
for some constant and all
.
The space is easily seen to be just
(with an equivalent norm). At the other extreme,
is the class of Lipschitz functions, and is also denoted
(and the
norm is also known as the Lipschitz norm).
Exercise 5 Show that
is a Banach space for every
.
Exercise 6 Show that
for every
, and that the inclusion map is continuous.
Exercise 7 If
, show that the
norm of a function
is finite if and only if
is constant. This explains why we generally restrict the Hölder index
to be less than or equal to
.
Exercise 8 Show that
is a proper subspace of
, and that the restriction of the
norm to
is equivalent to the
norm. (The relationship between
and
is in fact closely analogous to that between
and
, as can be seen from the fundamental theorem of calculus.)
Exercise 9 Let
be a distribution. Show that
if and only if
, and the distributional derivative
of
also lies in
. Furthermore, for
, show that
is comparable to
.
We can then define the spaces for natural numbers
and
to be the subspace of
whose norm
is finite. (As before, there are a variety of ways to define the norm of the tensor-valued quantity
, but they are all equivalent to each other.)
Exercise 10 Show that
is a Banach space which contains
, and is contained in turn in
.
As before, is equal to
, and
contains
when
. The space
is slightly larger than
, but is fairly close to it, thus providing a near-continuum of spaces between the sequence of spaces
. The following examples illustrates this:
Exercise 11 Let
be a test function, let
be a natural number, and let
.
- Show that the function
lies in
whenever
.
- Conversely, if
is not an integer,
, and
, show that
does not lie in
.
- Show that
lies in
, but not in
.
This example illustrates that the quantity
can be viewed as measuring the total amount of regularity held by functions in
:
full derivatives, plus an additional
amount of Hölder continuity.
Exercise 12 Let
be a test function, let
be a natural number, and let
. Show that for
with
, the function
has a
norm of at most
, for some
depending on
.
By construction, it is clear that continuously differential operators of order
will map
continuously to
.
Now we consider what happens with products.
Exercise 13 Let
be natural numbers, and
.
- If
and
, show that
, and that the multiplication map is continuous from
to
. (Hint: reduce to the case
and use induction.)
- If
and
, and
, show that
, and that the multiplication map is continuous from
to
.
It is easy to see that the regularity in these results cannot be improved (just take
). This illustrates a general principle, namely that a pointwise product
tends to acquire the lower of the regularities of the two factors
.
As one consequence of this exercise, we see that any variable-coefficient differential operator of order
with
coefficients will map
to
for any
and
.
We now briefly remark on Hölder spaces on open domains in Euclidean space
. Here, a new subtlety emerges; instead of having just one space
for each choice of exponents
, one actually has a range of spaces to choose from, depending on what kind of behaviour one wants to impose at the boundary of the domain. At one extreme, one has the space
, defined as the space of
times continuously differentiable functions
whose Hölder norm
is finite; this is the “maximal” choice for the . At the other extreme, one has the space
, defined as the closure of the compactly supported functions in
. This space is smaller than
; for instance, functions in
must converge to zero at the endpoints
, while functions in
do not need to do so. An intermediate space is
, defined as the space of restrictions of functions in
to
. For instance, the restriction of
to
, where
is a cutoff function non-vanishing at the origin, lies in
, but is not in
or
(note that
itself is not in
, as it is not continuously differentiable at the origin). It is possible to clarify the exact relationships between the various flavours of Hölder spaces on domains (and similarly for the Sobolev spaces discussed below), but we will not discuss these topics here.
Exercise 14 Let
and
. Show that
is a dense subset of
if one places the
topology on the latter space. (Hint: To approximate a compactly supported
function by a
one, convolve with a smooth, compactly supported approximation to the identity.) What happens in the endpoint case
?
Hölder spaces are particularly useful in elliptic PDE, because tools such as the maximum principle lend themselves well to the suprema that appear inside the definition of the norms; see for instance the book of Gilbarg and Trudinger for a thorough treatment. For simple examples of elliptic PDE, such as the Poisson equation
, one can also use the explicit fundamental solution, through lengthy but straightforward computations. We give a typical example here:
Exercise 15 (Schauder estimate) Let
, and let
be a function supported on the unit ball
. Let
be the unique bounded solution to the Poisson equation
(where
is the Laplacian), given by convolution with the Newton kernel:
- (i) Show that
.
- (ii) Show that
, and rigorously establish the formula
for
.
- (iii) Show that
, and rigorously establish the formula
for
, where
is the Kronecker delta. (Hint: first establish this in the two model cases when
, and when
is constant near
.)
- (iv) Show that
, and establish the Schauder estimate
where
depends only on
.
- (v) Show that the Schauder estimate fails when
. Using this, conclude that there eixsts
supported in the unit ball such that the function
defined above fails to be in
. (Hint: use the closed graph theorem.) This failure helps explain why it is necessary to introduce Hölder spaces into elliptic theory in the first place (as opposed to the more intuitive
spaces).
Remark 16 Roughly speaking, the Schauder estimate asserts that if
has
regularity, then all other second derivatives of
have
regularity as well. This phenomenon – that control of a special derivative of
at some order implies control of all other derivatives of
at that order – is known as elliptic regularity, and relies crucially on
being an elliptic differential operator. We will discus ellipticity a little bit more later in Exercise 45. The theory of Schauder estimates is by now extremely well developed, and applies to large classes of elliptic operators on quite general domains, but we will not discuss these estimates and their applications to various linear and nonlinear elliptic PDE here.
Exercise 17 (Rellich-Kondrachov type embedding theorem for Hölder spaces) Let
. Show that any bounded sequence of functions
that are all supported in the same compact subset of
will have a subsequence that converges in
. (Hint: use the Arzelá-Ascoli theorem to first obtain uniform convergence, then upgrade this convergence.) This is part of a more general phenomenon: sequences bounded in a high regularity space, and constrained to lie in a compact domain, will tend to have convergent subsequences in low regularity spaces.
— 2. Classical Sobolev spaces —
We now turn to the “classical” Sobolev spaces , which involve only an integral amount
of regularity.
Definition 18 Let
, and let
be a natural number. A function
is said to lie in
if its weak derivatives
exist and lie in
for all
. If
lies in
, we define the
norm of
by the formula
(As before, the exact choice of convention in which one measures the
norm of
is not particularly relevant for most applications, as all such conventions are equivalent up to multiplicative constants.)
The space is also denoted
in some texts.
Example 19
is of course the same space as
, thus the Sobolev spaces generalise the Lebesgue spaces. From Exercise 9 we see that
is the same space as
, with an equivalent norm. More generally, one can see from induction that
is the same space as
for
, with an equivalent norm. It is also clear that
contains
for any
.
Example 20 The function
lies in
, but is not everywhere differentiable in the classical sense; nevertheless, it has a bounded weak derivative of
. On the other hand, the Cantor function (aka the “Devil’s staircase”) is not in
, despite having a classical derivative of zero at almost every point; the weak derivative is a Cantor measure, which does not lie in any
space. Thus one really does need to work with weak derivatives rather than classical derivatives to define Sobolev spaces properly (in contrast to the
spaces).
Exercise 21 Let
be a bump function,
, and
. Show that if
with
,
, and
, then the function
has a
norm of at most
, where
is a constant depending only on
,
and
. (Compare this with Exercise 4 and Exercise 12.) What happens when the condition
is dropped?
Exercise 22 Show that
is a Banach space for any
and
.
The fact that Sobolev spaces are defined using weak derivatives is a technical nuisance, but in practice one can often end up working with classical derivatives anyway by means of the following lemma:
Lemma 23 Let
and
. Then the space
of test functions is a dense subspace of
.
Proof: It is clear that is a subspace of
. We first show that the smooth functions
is a dense subspace of
, and then show that
is dense in
.
We begin with the former claim. Let , and let
be a sequence of smooth, compactly supported approximations to the identity. Since
, we see that
converges to
in
. More generally, since
is in
for
, we see that
converges to
in
. Thus we see that
converges to
in
. On the other hand, as
is smooth,
is smooth; and the claim follows.
Now we prove the latter claim. Let be a smooth function in
, thus
for all
. We let
be a compactly supported function which equals
near the origin, and consider the functions
for
. Clearly, each
lies in
. As
, dominated convergence shows that
converges to
in
. An application of the product rule then lets us write
. The first term converges to
in
by dominated convergence, while the second term goes to zero in the same topology; thus
converges to
in
. A similar argument shows that
converges to
in
for all
, and so
converges to
in
, and the claim follows.
As a corollary of this lemma we also see that the space of Schwartz functions is dense in
.
Exercise 24 Let
. Show that the closure of
in
is
(the space of
functions whose first
derivatives all go to zero at infinity), thus Lemma 23 fails at the endpoint
.
Now we come to the important Sobolev embedding theorem, which allows one to trade regularity for integrability. We illustrate this phenomenon first with some very simple cases. First, we claim that the space embeds continuously into
, thus trading in one degree of regularity to upgrade
integrability to
integrability. To prove this claim, it suffices to establish the bound
for all test functions and some constant
, as the claim then follows by taking limits using Lemma 23. (Note that any limit in either the
or
topologies, is also a limit in the sense of distributions, and such limits are necessarily unique. Also, since
is the dual space of
, the distributional limit of any sequence bounded in
remains in
, by Exercise 28 of Notes 3.) To prove (1), observe from the fundamental theorem of calculus that
for all ; in particular, from the triangle inequality
Also, taking to be sufficiently large, we see (from the compact support of
) that
and (1) follows.
Since the closure of in
is
, we actually obtain the stronger embedding, that
embeds continuously into
.
Exercise 25 Show that
embeds continuously into
, thus there exists a constant
(depending only on
) such that
for all
.
Now we turn to Sobolev embedding for exponents other than and
.
Theorem 26 (Sobolev embedding theorem for one derivative) Let
be such that
, but that one is not in the endpoint cases
. Then
embeds continuously into
.
Proof: By Lemma 23 and the same limiting argument as before, it suffices to establish the Sobolev embedding inequality
for all test functions , and some constant
depending only on
, as the inequality will then extend to all
. To simplify the notation we shall use
to denote an estimate of the form
, where
is a constant depending on
(the exact value of this constant may vary from instance to instance).
The case is trivial. Now let us look at another extreme case, namely when
; by our hypotheses, this forces
. Here, we use the fundamental theorem of calculus (and the compact support of
) to write
for any and any direction
. Taking absolute values, we conclude in particular that
We can average this over all directions :
Switching from polar coordinates back to Cartesian (multiplying and dividing by ) we conclude that
thus is pointwise controlled by the convolution of
with the fractional integration
. By the Hardy-Littlewood-Sobolev theorem on fractional integration (Corollary 7 of Notes 1) we conclude that
and the claim follows. (Note that the hypotheses are needed here in order to be able to invoke this theorem.)
Now we handle intermediate cases, when . (Many of these cases can be obtained from the endpoints already established by interpolation, but unfortunately not all such cases can be, so we will treat this case separately.) Here, the trick is not to integrate out to infinity, but instead to integrate out to a bounded distance. For instance, the fundamental theorem of calculus gives
for any , hence
What value of should one pick? If one picks any specific value of
, one would end up with an average of
over spheres, which looks somewhat unpleasant. But what one can do here is average over a range of
‘s, for instance between
and
. This leads to
averaging over all directions and converting back to Cartesian coordinates, we see that
Thus one is bounding pointwise (up to constants) by the convolution of
with the kernel
, plus the convolution of
with the kernel
. A short computation shows that both kernels lie in
, where
is the exponent in Young’s inequality, and more specifically that
(and in particular
). Applying Young’s inequality (Exercise 25 of Notes 1), we conclude that
and the claim follows.
Remark 27 It is instructive to insert the example in Exercise 21 into the Sobolev embedding theorem. By replacing the
norm with the
norm, one trades one factor of the frequency scale
for
powers of the width
. This is consistent with the Sobolev embedding theorem so long as
, which is essentially one of the hypotheses in that exercise. Thus, one can view Sobolev embedding as an assertion that the width of a function must always be greater than or comparable to the wavelength scale (the reciprocal of the frequency scale), raised to the power of the dimension; this is a manifestation of the uncertainty principle.
Exercise 28 Let
. Show that the Sobolev endpoint estimate fails in the case
. (Hint: experiment with functions
of the form
, where
is a test function supported on the ball
and equal to one on
.) Conclude in particular that
is not a subset of
. (Hint: Either use the closed graph theorem, or use some variant of the function
used in the first part of this exercise.) Note that when
, the Sobolev endpoint theorem for
follows from the fundamental theorem of calculus, as mentioned earlier. There are substitutes known for the endpoint Sobolev embedding theorem, but they involve more sophisticated function spaces, such as the space
of spaces of bounded mean oscillation, which we will not discuss here.
The case of the Sobolev inequality cannot be proven via the Hardy-Littlewood-Sobolev inequality; however, there are other proofs available. One of these (due to Gagliardo and Nirenberg) is based on
Exercise 29 (Loomis-Whitney inequality) Let
, let
for some
, and let
be the function
Show that
(Hint: induct on
, using Hölder’s inequality and Fubini’s theorem.)
Lemma 30 (Endpoint Sobolev inequality)
embeds continuously into
.
Proof: It will suffice to show that
for all test functions . From the fundamental theorem of calculus we see that
and thus
where
From Fubini’s theorem we have
and hence by the Loomis-Whitney inequality
and the claim follows.
Exercise 31 (Connection between Sobolev embedding and isoperimetric inequality) Let
, and let
be an open subset of
whose boundary
is a smooth
-dimensional manifold. Show that the surface area
of
is related to the volume
of
by the isoperimetric inequality
for some constant
depending only on
. (Hint: Apply the endpoint Sobolev theorem to a suitably smoothed out version of
.) It is also possible to reverse this implication and deduce the endpoint Sobolev embedding theorem from the isoperimetric inequality and the coarea formula, which we will do in later notes.
Exercise 32 Use dimensional analysis to argue why the Sobolev embedding theorem should fail when
. Then create a rigorous counterexample to that theorem in this case.
Exercise 33 Show that
embeds into
whenever
and
are such that
, and such that at least one of the two inequalities
,
is strict.
Exercise 34 Show that the Sobolev embedding theorem fails whenever
. (Hint: experiment with functions of the form
, where
is a test function and the
are widely separated points in space.)
Exercise 35 (Hölder-Sobolev embedding) Let
. Show that
embeds continuously into
, where
is defined by the scaling relationship
. Use dimensional analysis to justify why one would expect this scaling relationship to arise naturally, and give an example to show that
cannot be improved to any higher exponent.
More generally, with the same assumptions on
, show that
embeds continuously into
for all natural numbers
.
Exercise 36 (Sobolev product theorem, special case) Let
,
, and
be such that
. Show that whenever
and
, then
, and that
for some constant
depending only on the subscripted parameters. (This is not the most general range of parameters for which this sort of product theorem holds, but it is an instructive special case.)
Exercise 37 Let
be a differential operator of order
whose coefficients lie in
. Show that
maps
continuously to
for all
and all integers
.
— 3. -based Sobolev spaces —
It is possible to develop more general Sobolev spaces than the integer-regularity spaces
defined above, in which
is allowed to take any real number (including negative numbers) as a value, although the theory becomes somewhat pathological unless one restricts attention to the range
, for reasons having to do with the theory of singular integrals.
As the theory of singular integrals is beyond the scope of this course, we will illustrate this theory only in the model case , in which Plancherel’s theorem is available, which allows one to avoid dealing with singular integrals by working purely on the frequency space side.
To explain this, we begin with the Plancherel identity
which is valid for all functions and in particular for Schwartz functions
. Also, we know that the Fourier transform of any derivative
of
is
. From this we see that
for all and so on summing in
we have
A similar argument then gives
and so on summing in we have
for all and all Schwartz functions
. Since the Schwartz functions are dense in
, a limiting argument (using the fact that
is complete) then shows that the above formula also holds for all
.
Now observe that the quantity is comparable (up to constants depending on
) to the expression
, where
(this quantity is sometimes known as the “Japanese bracket” of
). We thus conclude that
where we use here to denote the fact that
and
are comparable up to constants depending on
, and
denotes the variable of independent variable on the right-hand side. If we then define, for any real number
, the space
to be the space of all tempered distributions
such that the distribution
lies in
, and give this space the norm
then we see that embeds into
, and that the norms are equivalent.
Actually, the two spaces are equal:
Exercise 38 For any
, show that
is a dense subspace of
. Use this to conclude that
for all non-negative integers
.
It is clear that , and that
whenever
. The spaces
are also (complex) Hilbert spaces, with the Hilbert space inner product
It is not hard to verify that this inner product does indeed give the structure of a Hilbert space (indeed, it is isomorphic under the Fourier transform to the Hilbert space
which is isomorphic in turn under the map
to the standard Hilbert space
).
Being a Hilbert space, is isomorphic to its dual
(or more precisely, to the complex conjugate of this dual). There is another duality relationship which is also useful:
Exercise 39 (Duality between
and
) Let
, and
. Show also for any continuous linear functional
there exists a unique
such that
for all
, where the inner product
is defined via the Fourier transform as
Also show that
for all
.
The Sobolev spaces also enjoy the same type of embedding estimates as their classical counterparts:
Exercise 40 (Sobolev embedding for
, I) If
, show that
embeds continuously into
whenever
. (Hint: use the Fourier inversion formula and the Cauchy-Schwarz inequality.)
Exercise 41 (Sobolev embedding for
, II) If
, show that
embeds continuously into
whenever
. (Hint: it suffices to handle the extreme case
. For this, first reduce to establishing the bound
to the case when
is a Schwartz function whose Fourier transform vanishes near the origin (and
depends on
), and write
for some
which is bounded in
. Then use Exercise 35 from Notes 3 and Corollary 7 from Notes 1.
Exercise 42 In this exercise we develop a more elementary variant of Sobolev spaces, the
Hölder spaces. For any
and
, let
be the space of functions
whose norm
is finite, where
is the translation of
by
. Note that
(with equivalent norms).
- (i) For any
, establish the inclusions
for any
. (Hint: take Fourier transforms and work in frequency space.)
- (ii) Let
be a bump function, and let
be the approximations to the identity
. If
, show that one has the equivalence
where we use
to denote the assertion that
and
are comparable up to constants depending on
. (Hint: To upper bound
for
, express
as a telescoping sum of
for
, plus a final term
where
is comparable to
.)
- (iii) If
and
are such that
, show that
embeds continuously into
. (Hint: express
as
plus a telescoping series of
, where
is as in the previous exercise. The additional convolution is in place in order to apply Young’s inequality.)
The functions
are crude versions of Littlewood-Paley projections, which play an important role in harmonic analysis and nonlinear wave and dispersive equations.
Exercise 43 (Sobolev trace theorem, special case) Let
. For any
, establish the Sobolev trace inequality
where
depends only on
and
, and
is the restriction of
to the standard hyperplane
. (Hint: Convert everything to
-based statements involving the Fourier transform of
, and use either the Cauchy-Schwarz inequality or Schur’s test, see Lemma 5 of Notes 1.)
- (i) Show that if
for some
, and
, then
(note that this product has to be defined in the sense of tempered distributions if
is negative), and the map
is continuous from
to
. (Hint: First prove this when
is a non-negative integer using an argument similar to that in Exercise 36, then exploit duality to handle the case of negative integer
. To handle the remaining cases, decompose the Fourier transform of
into annular regions of the form
for
, as well as the ball
, and use the preceding cases to estimate the
norm of the Fourier transform of
these annular regions and on the ball.)
- (ii) Let
be a partial differential operator of order
with coefficients in
for some
. Show that
maps
continuously to
for all
.
Now we consider a partial converse to Exercise 44.
Exercise 45 (Elliptic regularity) Let
, and let
be a constant-coefficient homogeneous differential operator of order
. Define the symbol
of
to be the homogeneous polynomial of degree
, defined by the formula
We say that
is elliptic if one has the lower bound
for all
and some constant
. Thus, for instance, the Laplacian is elliptic. Another example of an elliptic operator is the Cauchy-Riemann operator
in
. On the other hand, the heat operator
, the Schrödinger operator
, and the wave operator
are not elliptic on
.
- (i) Show that if
is elliptic of order
, and
is a tempered distribution such that
, then
, and that one has the bound
for some
depending on
. (Hint: Once again, rewrite everything in terms of the Fourier transform
of
.)
- (ii) Show that if
is a constant-coefficient differential operator of
which is not elliptic, then the estimate (2) fails.
- (iii) Let
be a function which is locally in
, and let
be an elliptic operator of order
. Show that if
, then
is smooth. (Hint: First show inductively that
for every test function
and every natural number
.)
Remark 46 The symbol
of an elliptic operator (with real coefficients) tends to have level sets that resemble ellipsoids, hence the name. In contrast, the symbol of parabolic operators such as the heat operator
has level sets resembling paraboloids, and the symbol of hyperbolic operators such as the wave operator
has level sets resembling hyperboloids. The symbol in fact encodes many important features of linear differential operators, in particular controlling whether singularities can form, and how they must propagate in space and/or time; but this topic is beyond the scope of this course.
185 comments
Comments feed for this article
23 February, 2018 at 5:31 am
Sobolev Holder question
Dear Prof. Tao,
is it true that C^s \subset H^s = W^{s,2}? I imagine that it is true but I cannot provide an argument. Thanks in advance.
24 February, 2018 at 8:08 am
Terence Tao
Yes if one has a compact domain, but not in general. One can already see this at the
level where the question is whether
embeds into
.
26 June, 2018 at 7:12 am
Paul Hager
I was confused by the definition “… to be the space of all tempered distributions
such that the distribution
lies in
…”
A distribution that lies in
implicitly means that the distribution is an “ordinary” function?
26 June, 2018 at 11:29 am
Terence Tao
Yes, we view
(or more generally,
) as a subspace of the space of distributions (see the remarks near Exercise 8 of the previous set of notes).
26 July, 2018 at 10:43 pm
Rajesh
Prof Tao,
“…as the very useful Sobolev embedding theorem, which allows one to trade regularity for integrability…”. Thats the use of Sobolev spaces. In this context, Fourier and Plancheral methods come very handy, when the corresponding Sobolev space is also a Hilbert space. But that is not always the case…. Only L^2 based Sobolev spaces are Hilbert spaces. According to Sobolev emebdding, if the function in R^d need to be holder continuous, then its gardient needs to be L^p integrable with p >= d+1. So for d >1, we need p > 2, so the associated Sobolev space cannot have a Hilbert space structure. So in this context, we cannot use Fourier Plancheral techniques. “But if” (stress If)… I say, that I can always find a Hilbert space, for any d, (even for cases when d>1), how useful a tool that it would be, in the context of PDE. What would the impact be? Any examples of PDE, on which there would be impact? Appreciate your valuable comments…
Thanks and Regards
Rajesh
6 May, 2021 at 3:30 am
Anonymus
Is there any example of a function that is in a sobolev space intersection some gevrey space?
10 August, 2021 at 5:46 pm
Anonymous
I think in exercise 28, phi should be supported in B(2) and identically 1 in B(1) to create a “layer cake”
[Corrected, thanks – T.]
15 March, 2022 at 3:59 pm
Anonymous
At the beginning of the notes, the function
and the function
both have a frequency scale of about
. Why does the function
also have a “frequency scale” of about
?
Don’t we need the periodicity of the sine function as in the previous two examples?
16 March, 2022 at 9:52 am
Terence Tao
The Fourier transform of the function
is concentrated in the range
, which is why one should heuristically think of this function as having a frequency scale of
. Alternatively, this function has a “wavelength” of
(albeit with only one oscillation of this wavelength being present, rather than many), and frequency is inversely proportional to wavelength.
16 March, 2022 at 11:01 am
J
Why in the function
,
is not considered as the “width” of
as for instance Example 4 does?
[I’m sorry, I was not able to parse this question. -T]
16 March, 2022 at 1:00 pm
J
Sorry, I meant to say *Exercise* 4. In that exercise, the function
is said to have “width”
. So I was wondering if one should also say that the function
has with
, where you interprets
as the frequency scale.
16 March, 2022 at 2:14 pm
Terence Tao
Yes, the function
would informally have frequency
, wavelength
, amplitude
, and width
, and consist of
oscillations. Drawing a graph of this function should make these statistics clear.
16 March, 2022 at 10:58 am
Anonymous
In Exercise 4, whenever one differentiates the function once, one has a factor of
and
; so by the assumption on
, it is bounded by totally one factor of
. Since
, so all
,
, are dominated by
. On the other hand, the constant
comes from the max of
,
.
How would you organize a neat formal argument? The higher-order mixed partial derivatives are messy. Even if one focuses on the case when
, more and more terms appear by the product rule as
increases.
16 March, 2022 at 12:32 pm
Terence Tao
An induction on
would work here. (Here one has to invest a little bit of thought into developing a suitable induction hypothesis. For instance, it would be good to have the induction hypothesis cover both the sine and cosine cases.)
16 March, 2022 at 12:54 pm
J
It is said in the notes that
are designed to “fill up the gaps” between the discrete spectrum
of the continuously differentiable spaces.
The Hölder spaces
Exercise 8 seems to suggest that the gaps are not perfectly filled up yet:

Is there a family of spaces filling the gaps between
and
?
16 March, 2022 at 1:17 pm
J
As before,
I think you want to switch
and
if they are meant to be consistent with Exercise 6.
[Corrected, thanks – T.]
16 March, 2022 at 1:47 pm
J
As Exercise 10 and 11 show, the Holder spaces do not fill the gaps between
and
completely.
One may want to make some kind of room of
argument for
so that one can conclude something for
. But this does not work because of the gaps. (Perhaps this is not how the Holder spaces are supposed to be used at all.)
Naively, one can simply assume stronger differentiability when needed. How much more flexibility does one gain if one has already at least the regularity of, say,
by using
instead of simply
?
16 March, 2022 at 4:06 pm
Terence Tao
If one really wants to work with a continuum of spaces that does not experience discontinuities at integer regularities, then one can work with Holder-Zygmund spaces
(a special type of Besov space), which agree with the Holder spaces
when
and
but disagree with the classical spaces
when
. These are discussed for instance in Stein’s “Singular integrals and differentiability properties of functions”.
In elliptic and parabolic PDE one often expends significant effort to improve the regularity of a solution from some “critical” regularity (which for instance could be
) to a slightly “subcritical” regularity, such as
; this is often non-trivial and uses tools such as Moser iteration. Once one has some subcritical regularity, though, it is often relatively easy to gain even more regularity.
even more regularity.
16 March, 2022 at 2:12 pm
J
Can one use Lemma 23 to define the Sobolev spaces, i.e., completion of the space of test functions with respect to the Sobolev norm
? Then one would have Lemma 23 for free; would this make other originally “easy” propositions more difficult to prove?
16 March, 2022 at 4:23 pm
Terence Tao
Yes, this is an alternate route to setting up the foundations of the theory; Definition 18 will now need to be a theorem, rather than a definition, but one ends up in the same place at the end. (The situation is more delicate in the presence of a boundary; cf. the comments after Exercise 13.)
16 February, 2023 at 6:32 am
Anonymous
At the beginning of the discussion of Sobolev embeddings, is the fact that
is a subset of
a consequence of the inequality (1) or something one assumes true when proving the inclusion map from
to
is continuous?
16 February, 2023 at 9:59 am
Anonymous
Inequality (1) shows
is a subset of
.
19 February, 2023 at 5:06 am
Anonymous
1. One first establishes the inequality (1) for
. If
, Lemma 23 shows
in
for some sequence
. The inequality (1) (for
)shows that
is a Cauchy sequence in
and by completeness,
converges to some
in
. The uniqueness of limits in the sense of distributions implies that
and one thus shows (1) for every
, which implies
. Does this argument work?
2. It seems that one does not need to use the fact that
is the dual space of
as in:
“Also, since
is the dual space of
, the distributional limit of any sequence bounded in
remains in
“.
(One way to show completeness of
is to use duality, though.)