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 1In 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 1Show that is a Banach space.

Exercise 2Show that for every and , the norm is equivalent to the modified normin 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 3Let 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 4Show that is a Banach space for every .

Exercise 5Show that for every , and that the inclusion map is continuous.

Exercise 6If , 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 7Show 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 8Let 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 9Show that is a Banach space which contains , and is contained in turn in .

As before, is equal to , and is contained in . 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 10Let 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
notlie 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 11Let 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 12Let 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

lowerof 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 13Show that is a dense subset of for any and . (Hint:To approximate a compactly supported function by a one, convolve with a smooth, compactly supported approximation to the identity.)

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 14 (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 estimatewhere 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 2Roughly 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 aselliptic regularity, and relies crucially on being anellipticdifferential operator. We will discus ellipticity a little bit more later in Exercise 36. 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 15 (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 1Let , 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 1is of course the same space as , thus the Sobolev spaces generalise the Lebesgue spaces. From Exercise 8 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 2The 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 16Let 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 3 and Exercise 11.) What happens when the condition is dropped?

Exercise 17Show 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 2Let 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 18Let . Show that the closure of in is contained in , thus Lemma 2 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 2. (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 19Show that embeds continuously into , thus there exists a constant (depending only on ) such thatfor all .

Now we turn to Sobolev embedding for exponents other than and .

Theorem 3 (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 2 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 3It is instructive to insert the example in Exercise 16 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 20Let . 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 annulus .) 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 21 (Loomis-Whitney inequality)Let , let for some , and let be the functionShow that

(

Hint:induct on , using Hölder’s inequality and Fubini’s theorem.)

Lemma 4 (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 22 (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 theisoperimetric inequalityfor 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 23Use dimensional analysis to argue why the Sobolev embedding theorem should fail when . Then create a rigorous counterexample to that theorem in this case.

Exercise 24Show that embeds into whenever and are such that , and such that at least one of the two inequalities , is strict.

Exercise 25Show 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 26 (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 27 (Sobolev product theorem, special case)Let , , and be such that . Show that whenever and , then , and thatfor 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 28Let 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 29For 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 30 (Duality between and )Let , and . Show also for any continuous linear functional there exists a unique such thatfor 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 31 (Sobolev embedding for , I)If , show that embeds continuously into whenever . (Hint:use the Fourier inversion formula and the Cauchy-Schwarz inequality.)

Exercise 32 (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 33In this exercise we develop a more elementary variant of Sobolev spaces, theHölder spaces. For any and , let be the space of functions whose normis 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 34 (Sobolev trace theorem, special case)Let . For any , establish theSobolev trace inequalitywhere 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 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:As with the previous exercise, convert everything to -based statements involving the Fourier transform of , and use Schur’s test.)- (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 35.

Exercise 36 (Elliptic regularity)Let , and letbe a constant-coefficient homogeneous differential operator of order . Define the

symbolof to be the homogeneous polynomial of degree , defined by the formulaWe say that is

ellipticif one has the lower boundfor 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
notelliptic, 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 4The symbol of an elliptic operator (with real coefficients) tends to have level sets that resemble ellipsoids, hence the name. In contrast, the symbol ofparabolicoperators such as the heat operator has level sets resembling paraboloids, and the symbol ofhyperbolicoperators 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.

## 82 comments

Comments feed for this article

1 May, 2009 at 8:50 am

abcCan the statement that H^{1/2}-functions don’t have “jumps” (I don’t know what this means for an L^p-function) be made precise?

1 May, 2009 at 10:01 am

Terence TaoDear abc,

Well, it is true that no piecewise smooth function with a jump discontinuity can lie in ; the Fourier transform will decay like , causing a logarithmic divergence in the norm. (This fact is loosely related to the Gibbs phenomenon, though that is more about the failure of absolute integrability of the Fourier transform, rather than the weighted square-integrability.)

Also, it is known that functions, while possibly being unbounded or discontinuous, must at least lie in VMO, the space of functions with vanishing mean oscillation; functions with jump discontinuities do not lie in this class.

Finally, there are a number of results known that are consistent with the philosophy of having no jump discontinuities, for instance it is known (by

~~Brezis and Nirenberg, I think~~Boutet de Monvel and Gabber) that there is a well-defined notion of a degree (i.e. winding number) of an map from the unit circle to itself, despite the fact that such functions can be arbitrarily oscillatory (and in particular, discontinuous).20 April, 2011 at 2:15 pm

StudentDear Professor Terrence,

do you know an easy example of a function u in the space H^1/2(0,1) but not in the H^1(0,1) space?

Thanks!

6 August, 2011 at 3:25 pm

AnonymousA triangle in (0,1).

27 April, 2014 at 6:30 am

Fanuh-huh? the Fourier transform of a triangle decays like , so if we multiply it by we still have decay, which is square integrable.

1 May, 2009 at 12:39 pm

AnonymousDear Professor Tao:

The entire blog entry instead of the preview is showing on the home page.

[Corrected, thanks – T]1 May, 2009 at 8:39 pm

lutfuDear Prof. Tao,

In exercise 29, in the definition of inner product you write ”…..is defined via the Fourier transform as …..”

on the right hand side do you mean Fourier transform of f and g or f,g themselves?

thanks

[Corrected, thanks – T.]1 May, 2009 at 9:50 pm

timurThanks a lot for these notes! I find the insights, exercises and examples to be invaluable to play with.

I think some hats also missing in the definition of Hs-inner product just after Exercise 28.

1 May, 2009 at 9:52 pm

timurRemark 4 (at the end) has a “yellow” formula.

2 May, 2009 at 7:27 pm

PolamIn Exercise 21, do you just mean that the Hardy-Littlewood-Sobolev theorem on fractional integration fails when p = 1? The Sobolev embedding theorem should be true for p = 1, i.e. $W^{1,1}$ should embed into $L^{d/d-1}$ in d dimensions for all d, and in fact this is equivalent to the isoperimetric inequality.

2 May, 2009 at 8:47 pm

Terence TaoOh, good point! I had forgotten all about this connection. I’ve corrected the exercise and added some relevant material about the endpoint Sobolev inequality and the isoperimetric inequality.

2 May, 2009 at 9:56 pm

lutfuDear Prof. Tao,

in the section 2. classical sobolev spaces, in example 2, you say ” cantor function despite having classical derivative a.e zero, the weak derivative is cantor measure….”

I did not understand this sentence. if a function has classical derivative, isn’t its weak derivative the same?

Could you please clarify this point?

and after exercise 12, you define Holder norm and in the second sup of the norm definition, should it be k or j?

3 May, 2009 at 7:34 am

Terence TaoDear Lutfu,

The classical and weak derivatives only agree for absolutely continuous functions (in the one dimensional setting), because one needs the fundamental theorem of calculus to apply to equate the two.

I suppose I should use k instead of j here in order to retain the claim of being the “maximal” concept of a Holder space, though for any reasonable domain the two notions would be equivalent.

12 May, 2009 at 7:09 am

bkDear Terry,

1. are you sure you want to this notation:

” and for any constant-coefficient partial differential operator

”

instead of

”

2. “For instance, the restriction of to lies in , but is not in…”

do you focus on the origin, or do you really want to say

[Corrected, thanks – T]12 May, 2009 at 10:58 am

bkDear Terry,

You were in a hurry.

1. also take a look at the ‘denominator’ of the operator

2. according to your definition if |x| in C^{1,0}, it should be bounded, right?

I don’t enjoy being so wordy but I like your stuff so much that I want them to be typoless :).

[Corrected again – T.]12 May, 2009 at 2:31 pm

future topics « 逝去日子[…] 五月 12, 2009 作者为 曾经话说要如何 1. Schauder estimate/Rellich-Kondrakov type embedding theorem for Holder space—terry’s. […]

24 May, 2009 at 3:53 am

AntonioDear Terry,

there is a “formula does not parse” yellow message in the third paragraph (just after the list of examples).

Thanks for your work.

25 May, 2009 at 11:15 am

studentDear Prof Tao,

Are sobolev embedding theorems also true for any open set?(bounded or unbounded)

And when we say fundamental solution, do we mean that a tempered distribution which satisfy the equation?(not just distribution, tempered distribution? right?)

thanks

25 May, 2009 at 10:17 pm

Terence TaoDear Student,

The situation with the Sobolev embedding theorems on domains is somewhat subtle. For one thing, one has to decide upon exactly how one is to define the Sobolev spaces and norms (cf. the discussion after Exercise 12). If these spaces and norms are defined on domains as restrictions of their counterparts (or as the closure of smooth, compactly supported functions), then the embedding theorem on domains is a tautological consequence of that for ; but if they are defined differently then one has to take more care. Generally speaking, though, for sufficiently “nice” domains (e.g. bounded, with smooth boundary) the embedding theorems continue to hold no matter how one defines the Sobolev spaces and norms, but care would have to be taken if the domain starts developing very thin necks or has a fractal boundary. There is a close relationship between Sobolev inequalities and isoperimetric inequalities (see Exercise 22), so as a rule of thumb, any domain for which the isoperimetric inequality continues to hold, will also have a good Sobolev inequality. (Note that thin necks and fractal boundaries can allow a small boundary to enclose a large volume, thus leading to possible degradation of the isoperimetric inequality and hence Sobolev inequalities.)

Fundamental solutions can be either tempered or non-tempered, though in the latter case they can only be used to solve PDE with test function data (rather than PDE with Schwartz data).

26 May, 2009 at 4:07 am

studentThank you very much Prof. Tao,

in exc 16 hint, I think there should be x in front of the integral.

26 May, 2009 at 4:09 am

studentexcuse me, not for this post. exc 16 of Distributions.

14 June, 2009 at 4:00 pm

lutfuDear Prof. Tao

the conditions for Exc 24 is given different in some other sources.

for example like this:

and

are both equivalent?

thanks

14 June, 2009 at 4:54 pm

Terence TaoUp to endpoints, yes. The and endpoints are often treated differently (and here it becomes important whether k, l are integers or not); I am including the endpoints but not the endpoints in my formulation, but the reverse is being stated in your example.

21 June, 2009 at 2:56 pm

StudentDear Professor Tao:

I have a question about trying to extend Poincare’s inequality (or Sobolev Embedding) to negative derivative. In other words, if one has D^{\alpha-1}f where D is a differentiation, \alpha \in (0,1), and f some smooth function, how can we estimate such term? If one could extend Poincare’s inequality to negative derivative, then I suppose this norm can be bounded by that of f, analogously to how the norm of f can be bounded by the norm of its gradient in any Lp norm for p even 1 or infinity, but I do not know if this is true.

I also looked at Riesz potential from Stein’s book looking for answer to this question, but such inequality seems to be true only in Lp norm but not when p=1 or infinity.

Is there any way to estimate the negative derivative of f in sup norm? Can we at least say that the sup norm of negative derivative of f is bounded by the that of gradient of f?

Any share of thoughts would be appreciated.

Thank you.

22 June, 2009 at 8:39 pm

Terence TaoDear Student,

I am not exactly sure what kind of inequalities you are looking for, but generally speaking the Hardy-Littlewood-Sobolev fractional integration inequality provides all the relevant estimates, except possibly at endpoints , for which the fractional integration estimates tend to fail.

Note also that in any of these inequalities, the right-hand side norm needs to have at least as many derivatives as the left-hand side, thus for instance one can hope to control a norm of f by a norm of , but not conversely. This can be seen by considering high-frequency test inputs for f, such as (in one dimension) where is a bump function and N is a large frequency.

29 June, 2009 at 12:29 am

vedadiDear Prof. Tao,

is the following definition right?

for and we say that a function

is in if

1) is continuous and bounded

2)all partial derivatives upto and including order k are continuous and bounded

3) derivative is Holder continuous (just kth order derivatives)

thanks

29 June, 2009 at 9:23 am

Terence TaoDear Vedadi,

Yes, this will give an equivalent definition of the Holder spaces. (If a higher derivative is bounded, then the lower derivatives are automatically Lipschitz continuous and thus Holder continuous also.) In fact there are quite a few useful equivalent formulations of both the Holder norms and the Sobolev norms.

14 July, 2009 at 10:55 am

sjtIn the first sentence you refer to some previous notes containing a discussion of function space norms. Are those notes available? If so, where?

14 July, 2009 at 11:47 am

Terence TaoDear sjt,

I was referring to my 245B lecture notes,

https://terrytao.wordpress.com/category/teaching/245b-real-analysis/

with Lectures 3,5,6,9 being perhaps the most relevant.

21 August, 2009 at 3:45 am

StudentDear Professor Tao:

I would like to know if the change of sign upon integration by parts still holds when it is fractional integration by parts. In other words, instead of

D^{1}, where D is a differentiation operator,

with

D^{\alpha} for some \alpha \in (0, 1)

and for simplicity assuming f and g are compactly supported smooth functions,

do we have

– \int f D^{\alpha}g = \int(D^{\alpha}f)g ?

Related to this question,

do we have the usual product rule for fractional derivative?

I.e. do we have

D^{\alpha}(fg) = (D^{\alpha}f)g + f(D^{\alpha}g)?

Having read a little about Riesz potential, I have a feeling it is not so simple. But I could not definitively figure this out.

21 August, 2009 at 8:50 am

Terence TaoFractional derivatives should be more accurately denoted rather than ; their Fourier symbol is , whereas the symbol for the ordinary derivative D is (give or take a factor of , depending on one’s Fourier conventions). In particular, these operators are self-adjoint (as is any other Fourier multiplier with real symbol).

The Leibnitz rule for fractional derivatives only holds approximately (up to lower order terms); this “fractional Leibnitz rule” is one of the foundational results of paradifferential calculus. Taylor’s book “Tools for PDE” is a good reference here; see also my lecture notes at

http://www.math.ucla.edu/~tao/247b.1.07w/

21 August, 2009 at 2:09 pm

StudentDear Professor Tao:

Thank you very much for your reply as well as suggested readings.

Best regards,

29 September, 2009 at 7:34 am

StudnetDear Professor Tao,

Is it true that $W^{k,2}\times W^{-k,2}\rightarrow L^1$ through the product map?

21 October, 2009 at 3:26 pm

Terence TaoNo; a product, in general, cannot become smoother than its factors (in sharp contrast to convolution), and since one of the factors has a negative regularity, the product will have negative regularity also. For a more concrete example, consider the product of and in one dimension, where is a bump function. The first factor is bounded in , while the second factor goes to zero in norm as . The product, however, does not go to zero in norm, and so the product map is not continuous in these topologies. (One can use the closed graph theorem to then show that the product map will not stay inside , if one interprets products in the distributional sense.)

21 October, 2009 at 2:47 pm

AnonymousDear Professor Tao:

Is there a connection of http://en.wikipedia.org/wiki/Fractional_calculus to Sobolev spaces with non-integer-regularity? E.g., is there an easy solution to finding the largest s so that for a given ?

This is a great resource, many thanks!

21 October, 2009 at 3:22 pm

Terence TaoYes; for instance, a function lies in for some if and only if and both lie in , where is the fractional derivative of . There are similar statements for other -based Sobolev spaces (though there are technical issues at the endpoints and ).

The functions are not globally in for any , but one can show by various means (e.g. Fourier transform, Littlewood-Paley type decomposition, or the various potential theory characterisations of the Sobolev norms) that they lie locally in for whenever .

21 October, 2009 at 5:57 pm

AnonymousI mean, is it correct that the main term using fractional calculus and this is simply why for $\alpha-s>-1/2$ around x=0, since $ |x|^{\alpha-s}\in L_2 (0,1)$ ?

21 October, 2009 at 7:01 pm

Terence TaoYes; this can be done for instance using the machinery in the preceding lecture notes,

https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/

21 October, 2009 at 7:43 pm

AnonymousDear Professor Tao:

Let us define function f = the Heaviside step function. Your answer https://terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/#comment-38496 implies that f is not in , correct? Is there a direct reference?

Let’s the fun continue. Is it in with s<1/2? What is ?

These are simple questions with very hard to find or figure out answers for non-experts, so your help is extremely appreciated.

20 December, 2009 at 12:46 pm

PDEbeginnerDear Prof. Tao,

In exercise 3, it seems we need to time the function by the height . In exercise 16, it seems that is .

[Corrected, thanks – T.]5 January, 2010 at 3:38 pm

PDEbeginnerDear Prof. Tao,

I think the condition in Ex 32 should be .

I got stuck on prove the first step in your hint. I tried to use the Fourier inversion formula and the Cauchy-Schwartz inequalitie as in Ex 31, but this seems not to work. Could you give a little detailed hint.

Thanks in advance!

5 January, 2010 at 6:00 pm

Terence TaoThanks for the correction!

For Exercise 32, one uses the Hardy-Littlewood-Sobolev inequality (Corollary 7 from Notes 1) rather than Cauchy-Schwarz. I’ll put in a link to that corollary in the notes.

11 March, 2010 at 10:01 pm

anaonThe product rule for weak derivatives could be proven with the help of lemma 3 (i.e. C^\infty_c is dense in W^{k,p}) but here the proof uses the product rule :)

16 May, 2010 at 6:47 am

SimionDear Prof. Tao,

Perhaps I’m terribly wrong, but how can in Exercise 13 compactly supported functions be dense in , given that for a function in such a space, there’s no decay requirement? For example, a constant function is in , but can’t be approximated by a compactly supported one.

16 May, 2010 at 7:39 am

Terence TaoOops, you’re right. I’ve replaced with in the exercise.

23 May, 2010 at 11:39 pm

wrfDear Prof. Tao,

I have thought about a problem about the gagliaedo-nirenberg interpolation inequation. I want to know if it holds for the frational

derivative.

a chinese student

6 November, 2010 at 1:46 pm

xuhmathI think there is a typo on Exercise 16: did you mean “the function has a norm” rather than “the function has a norm” to emphasize the contrast with Exercise 3?

[Corrected, thanks – T.]18 November, 2010 at 12:25 pm

anthonyDear Prof. Tao,

it should be Rellich-Kondrashov not Kondrakov.

[Corrected, thanks. The exact Romanisation of Kondrachov seems to be somewhat in dispute, but “ch” seems to be slightly more common than “sh” in this case. – T.]19 November, 2010 at 6:58 am

anthonyI’m pretty sure this is because of the influence of the classical book of Adams. As someone who actually speaks Russian I still think that Kondrashov is the correct english Romanisation because it produces the right pronunciation of russian “ш” (like in SHow) whereas “ch” (like in CHeck) corresponds to russian “ч”. The actual russian spelling is “Кондрашов”.

See http://en.wikipedia.org/wiki/Sobolev_inequality

[Thanks for the clarification. I confirmed this with a Russian friend of mine, and corrected the spelling. -T.]12 January, 2011 at 1:20 pm

AbhishekDear Prof. Tao,

I am wondering if there is any inverse theorem for Sobolev embedding, namely when is embedded in .

12 January, 2011 at 1:23 pm

AbhishekSorry, I meant when is embedded in . Can something like this be said in high dimensions?

13 January, 2011 at 3:40 am

Terence TaoIn general, spaces with lower regularity cannot be embedded into spaces with higher regularity (the computation in Exercise 16 already shows this). I talk a bit about the general embedding relationships between function spaces in

https://terrytao.wordpress.com/2010/03/11/a-type-diagram-for-function-spaces/

There is however something called an inverse theorem for Sobolev embedding, but it is a bit different, referring to a classification of those functions f for which the Sobolev inequality is close to equality (roughly speaking, this occurs when f looks like a rescaled bump function). This is inverse theorem important in the theory of concentration compactness; see e.g. Proposition 4.9 of http://www.math.ucla.edu/~visan/ClayLectureNotes.pdf

21 April, 2011 at 7:26 am

timurJust to complement what prof. Tao said; you can of course embed certain subspaces (e.g. with compactly supported Fourier transform) of L^q into e.g. H^k, with the constant blowing up as the subspace gets larger. These are called inverse- or Bernstein-type inequalities, and very useful in approximation theory and numerical analysis. The complementary inequalities are called direct- or Jackson-type estimates, and these concern subspaces e.g. with Fourier transform supported in a complement of a compact set, and the inequality sign is reversed (so compatible with the usual embedding theorems) as compared to the inverse estimates. The difference between direct-type estimates and the usual embedding theorems is that in direct estimates the constant tends to 0 as one removes larger and larger compact sets from the support of the Fourier transform.

22 February, 2011 at 1:26 pm

Sobolev spaces on manifolds « Secret Blogging Seminar[…] Tao has a very comprehensive blog post on this, so I’ll be […]

21 April, 2011 at 3:36 pm

anonymousDear Prof. Tao,

What is known for Sobolev inequality on T^d (d-dimensional torus)? And, what would be a good reference? Also, on Riemannian manifolds?

Thank you.

22 April, 2011 at 11:29 am

timurEmmanuel Hebey has a book on Sobolev spaces on manifolds, and I found that Palais’ seminar notes on the index theorem has many details worked out. But I would say the manifold case is not much different once you figure out how to transfer from local to global by using partitions of unity, and how this is equivalent to defining Sobolev spaces via covariant derivatives for integer indices. This is my quick answer; others will surely have deeper answers than mine.

1 September, 2011 at 6:15 pm

254A, Notes 1: Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula « What’s new[…] functions, but much stronger than the class of singly continuously differentiable functions. See this previous blog post for more on these sorts of regularity classes. The reason for the terminology in the above […]

16 December, 2011 at 10:51 am

254B, Notes 3: Quasirandom groups, expansion, and Selberg’s 3/16 theorem | t1u[…] See for instance this blog post for a very brief introduction to Riemannian geometry, and these two previous posts for an introduction to distributions and Sobolev […]

20 December, 2011 at 7:11 am

254B, Notes 3: Quasirandom groups, expansion, and Selberg’s 3/16 theorem « What’s new[…] See for instance this blog post for a very brief introduction to Riemannian geometry, and these two previous posts for an introduction to distributions and Sobolev […]

19 January, 2012 at 7:33 pm

Seungly OhConcerning Exercise 18, shouldn’t closure of should be something like ?

[Corrected, thanks – T.]28 April, 2012 at 3:47 pm

RexIn the statement of exercise 3, do you mean to call your frequency scale instead of ? I don’t see any in the function.

28 April, 2012 at 4:24 pm

Terence TaoYes, in this example the frequency scale will be .

30 April, 2012 at 8:50 pm

JackDear Prof. Tao,

I can’t find the definition for either in the note or in the index of your book. Do you mean ?

1 May, 2012 at 5:21 am

Terence Taois the closure of in the uniform topology, or equivalently the set of continuous functions that vanish at infinity. (It’s introduced in Exercise 1.10.7, but I guess I didn’t add an index entry for it there.)

30 April, 2012 at 9:16 pm

JackThis might be a stupid question:

As the wikipedia said, embedding is a structure-preserving map. What structure is preserved in the “Sovolev embedding”? Is it the structure of normed vector space? Or other “structures”?

1 May, 2012 at 5:25 am

Terence TaoThe Sobolev embedding preserves the convergence of sequences (i.e. it is a continuous map) and the vector space structure (i.e. it is linear), but it does not preserve the norm (i.e. it is not an isometric embedding). Note though that the topology of the embedded space is a bit stronger than that of the ambient space; when , there are sequences in that converge in norm but not in norm. So, strictly speaking, it doesn’t fully preserve the topological structure (instead, it makes the topology a little weaker upon embedding), but it is still customary to abuse notation slightly and refer to this map as an embedding.

4 May, 2012 at 11:17 am

JackHmm, any hint for finding sequences in that converge in norm but not in norm?

29 July, 2012 at 1:03 pm

karabasovDear Prof. Tao,

can you, please, give references about Sobolev product theorems, like one in Exercise 27?

Thank you very much,

karabasov

29 July, 2012 at 10:05 pm

Terence TaoTaylor’s “Tools for PDE” develops the theory of paraproducts, from which Sobolev product theorems can be deduced as corollaries.

22 August, 2012 at 2:38 pm

hassan jolanyWhat is the Schauder estimate on the space time Reimannian manifold in parabolic differential equation, ? (I mean on usual Hölder space )

29 December, 2012 at 1:04 pm

A mathematical formalisation of dimensional analysis « What’s new[…] In fact, this condition turns out to be sufficient as well as necessary, although this is a non-trivial fact that cannot be proven purely by dimensional analysis; see e.g. these notes. […]

31 March, 2013 at 6:02 pm

anonCowardI found the notation employed throughout this post to be a departure from the norm. Did you intend to write

?

[Corrected, thanks – T.]7 November, 2013 at 5:36 am

functions in Holder space | Question and Answer[…] I came up with this while I was reading this https://terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/ […]

2 December, 2013 at 2:37 am

zuchongzhiReblogged this on Diary of a budding mathematican and commented:

Very nice article!

5 December, 2013 at 4:25 pm

zuchongzhiDear Professor Tao:

May I ask a hint to prove (III) in Exercise 14? After establishing (II), I found I could not establish (III) rigorously at the singularity . Since I have , I cannot cancel out the singularity as I have at the bottom for after introducing a cut off function . I am hoping the part within the ball should vanish as .

5 December, 2013 at 8:16 pm

Terence TaoAs suggested in the hint, you should first try the cases when f vanishes at x, or when f is constant in a neighbourhood of x. (The easiest case of all is when f vanishes in a neighbourhood of x, although this case is not large enough to be of much use by itself.)

6 December, 2013 at 3:26 pm

zuchongzhiI understand. I think you mean to expand f(y)=f(x)+g(y-x), with g(y-x) vanishes at x, and f(x) a constant. The integration then gives me 0 by symmetricity.

27 April, 2014 at 6:44 am

FanThere is comment “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” in the proof of the embedding theorem. I can’t see what case isn’t covered by interpolation. If we fix , then the allowed is an interval, and we have already established two endpoints. Shouldn’t interpolation give us the whole interval?

27 April, 2014 at 7:09 am

Terence TaoIn some cases, the relevant interval is open (or half-open) rather than closed, so at least one of the endpoint estimates are not available.

27 April, 2014 at 1:43 pm

FanOn the RHS of the last equation of Exercise 21, is it norm instead of norm?

[Corrected, thanks – T.]27 April, 2014 at 5:39 pm

FanSorry, but there is another which should be on the LHS of the last but one equation in Lemma 4.

[Corrected, thanks – T.]24 November, 2014 at 4:02 am

eliYour notes are really interesting. I’m wondering if you could suggest me a book or something similar for taking a look deep in this argument.

Thanks for your attention.