247A, Notes 3: Singular integral operators

— 1. The Hilbert transform —

In this set of notes we begin the theory of singular integral operators – operators which are almost integral operators, except that their kernel just barely fails to be integrable near the diagonal . (This is in contrast to, say, fractional integral operators such as , whose kernels go to infinity at the diagonal but remain locally integrable.) These operators arise naturally in complex analysis, Fourier analysis, and also in PDE (in particular, they include the zeroth order pseudodifferential operators as a special case, about which more will be said later). It is here for the first time that we must make essential use of cancellation: cruder tools such as Schur’s test which do not take advantage of the sign of the kernel will not be effective here.

Before we turn to the general theory, let us study one of the simplest non-trivial singular integral operators, the Hilbert transform

It is not immediately obvious that is well-defined even for nice functions . But if is and compactly supported, then we can restrict the integral to a compact interval for some large (depending on and ), and use symmetry to write

The mean-value theorem then shows that is uniformly bounded on the interval for fixed , and so the limit actually exists from the dominated convergence theorem. A variant of this argument shows that is also well-defined for in the Schwartz class, though it does not map the Schwartz class to itself. Indeed it is not hard to see that we have the asymptotic

for such functions, and so if has non-zero mean then only decays like at infinity. In particular, we already see that is not bounded on . However, we do see that at least maps the Schwartz class to .

Let us first make comment on some algebraic properties of the Hilbert transform. It commutes with translations

and -normalized dilations

In fact, it is essentially the only such operator:

Exercise 1 Let be a continuous linear operator which maps Schwartz functions to bounded continuous functions, and which commutes with translations and dilations as in . Show that is a linear multiple of the Hilbert transform and the identity. (Hint: first establish the existence of a distribution such that for all , and such that is homogeneous of degree . Restrict the action of to Schwartz functions on or and show that is a constant multiple of on these functions.) (The claim is also true if we relax to , the space of distributions, but this is a little trickier to prove.)

It is also formally skew-adjoint:

Exercise 2 If are compactly supported, show that

The Hilbert transform is connected to complex analysis (and in particular to Cauchy integrals) by the Plemelj formulae.

Proposition 3 (Plemelj formulae) Let obey a qualitative decay bound (say – these conditions are needed just to make well-defined). Then for any

Proof: By translation invariance we can take . By taking complex conjugates we may assume that the sign is . Our task is then to show that

Multiplying by and making the change of variables , we reduce to showing

Direct computation shows that the expression in parentheses is absolutely integrable and

So we reduce to showing that

The claim then follows from dominated convergence.

Now suppose that not only obeys the hypotheses of the Plemelj formulae, but also extends holomorphically to the upper half-plane and obeys the decay bound in this region. Then Cauchy’s theorem gives

and

and thus by either of the Plemelj formulae we see that in this case. In particular, comparing real and imaginary parts we conclude that and . Thus for reasonably decaying holomorphic functions on the upper half-plane, the real and imaginary parts of the boundary value are connected via the Hilbert transform. In particular this shows that such functions are uniquely determined by just the real part of the boundary value.

The above discussion also strongly suggests the identity . This can be made more manifest by the following Fourier representation of the Hilbert transform.

Proposition 4 If , then for (almost every) . (Recall that lies in and so its Fourier transform is defined via density by Plancherel’s theorem.)

Proof: Let us first give a non-rigorous proof of this identity. Morally speaking we have

and so since Fourier transforms ought to interchange convolution and product

On the other hand we have

so on integrating this (using the relationship between differentiation and multiplication) we get

and the claim follows.

Now we turn to the rigorous proof. As the Hilbert transform is odd, a symmetry argument allows one to reduce to the case . It then suffices to show that has vanishing Fourier transform in this half-line. Define the Cauchy integral operator

The Plemelj formulae show that these converge pointwise to as ; since is Schwartz, it is also not hard to show via dominated convergence that they also converge in . Thus by the boundedness of the Fourier transform it suffices to show that each of the also have vanishing Fourier transform on the half-line.

This is part of a more general phenomenon, that functions which extend holomorphically to the upper half-space in a controlled manner tend to have vanishing Fourier transform on , while those which extend to the lower half-space have vanishing Fourier transform on . Thus there is a close relationship between holomorphic extension and distribution of the Fourier transform.

Fix (we could rescale to be, say, , but we will not have need of this normalization). We can truncate and define

more dominated convergence shows that converges to in as and so it will suffice to show that the Fourier coefficients of converge pointwise to zero as on the half-line . From Fubini’s theorem we easily compute

but then by shifting the contour to the lower semicircle of radius and then letting we obtain the claim.

From this proposition and Plancherel’s theorem we conclude that is an isometry:

Because of this, has a unique dense extension to , and (by another application of Plancherel’s theorem to justify taking the limit) the formula is valid for all . There is still the question of whether the original definition is valid for , or in other words whether the sequence of functions

(note that Cauchy–Schwarz ensures that the integrand is absolutely integrable) converges as to for , and in what sense this convergence holds. It turns out that the convergence is true both pointwise almost everywhere and in the metric sense, and in fact it follows from variants of the Calderón–Zygmund theory presented in this notes, but we shall not do so here (as it may be covered in the participating student seminar).

From we also see that we do indeed have the identity on , and that is indeed skew-adjoint on (); in particular, is unitary.

Let us briefly mention why the Hilbert transform is also connected to the theory of partial Fourier integrals, via the modulation operators

Proposition 5 Let and . Then

Proof: By limiting arguments (and because everything is continuous on ) it suffices to verify this identity for in the Schwartz class. But then the claim follows by taking the Fourier transform of both sides and using .

It is thus clear that in order to address the question of the extent to which the partial Fourier integrals actually converges back to , we will need to understand the properties of the Hilbert transform, and in particular its boundedness properties.

The formula suggests that behaves “like” multiplication by in some sense; informally, behaves like for positive frequency functions and for negative frequency functions. Now multiplication by is a fairly harmless operation – it is bounded on every normed vector space – so one might expect to similarly be bounded on more normed spaces than just ; in particular, it might be bounded on . This is indeed the case for , and we now turn to the theory which will generate such bounds.

Exercise 6 Demonstrate by example that the Hilbert transform is not bounded on or .

— 2. Calderón–Zygmund theory —

In the Hilbert transform, we have an operator which is bounded on , and we wish to extend this boundedness to other spaces as well. This is of course not automatic – the “enemy” is that the operator might map a broad shallow function into a tall narrow function of comparable norm (which will increase the norms markedly for all ), or conversely map a tall narrow function down to a broad shallow function of comparable norm (which will increase the norms markedly for ). It turns out that these two phenomena are more or less dual to each other, and so when working with an operator which is “comparable” to its adjoint (which is for instance the case with the skew-adjoint Hilbert transform, ) eliminating one of these enemies will automatically remove the other.

For operators such as the Hilbert transform, there are at least two approaches to exclude these scenarios. One is to exploit the phenomenon that if is not too large locally, then also doesn’t fluctuate too much locally; this prevents the “broad shallow to tall narrow” enemy which blocks boundedness for large . The dual approach is to show that if is localized and fluctuating, then is also localized; this prevents the “tall narrow to broad shallow” enemy which blocks boundedness for small . (There are also more algebraic methods, relying on explicit formulae for things like the norm of , or of the size of level sets of for various sets , but we will not discuss those here as they do not extend well to the more general class of singular kernels considered here.)

We shall first discuss the latter approach, which is the traditional approach to the subject, and return to the former approach later. We shall generalize from the Hilbert transform to a useful wider class of operators, which are integral operators “away from the diagonal”. To motivate this concept, observe that if is compactly supported and lies outside of the support of , then

the point being that the integral is now absolutely convergent, Thus while is not, strictly speaking, an integral operator of the type studied in preceding sections, it can be described as such as long as one avoids the diagonal .

Definition 7 (Singular kernels) A singular kernel is a function (defined away from the diagonal ) which obeys the decay estimate for and the Hölder-type regularity estimates and for some Hölder exponent .

Example 8 In one dimension, and are both singular kernels. In higher dimensions, the kernel is a singular kernel whenever is a Hölder-continuous function.

The regularity estimates may seem strange at first sight, although the fundamental theorem of calculus shows that these estimates would follow (with ) from the estimates

given that the gradient of is , these estimates are then consistent with but of course do not follow from it. In most applications one can take (in which case the Hölder type conditions become Lipschitz type conditions). It is more desirable , for large than for small (note in particular that the version of the estimate already follows from ). It is possible to weaken the Hölder type conditions further, but there are few applications that require such a weakening and so we shall stay with the Hölder condition. The factor is not significant; it is not hard to show that the regularity conditions are equivalent to the condition in which is replaced by any other constant strictly between zero and one.

Definition 9 (Calderón–Zygmund operators) A Calderón–Zygmund operator (CZO) is a linear operator which is bounded on , and such that there exists a singular kernel for which whenever is compactly supported, and lies outside of the support of .

It is not hard to see that uniquely determines . However the converse is not true; for instance, given any bounded function , the physical space multiplier is a CZO with kernel . However, this is the only source of ambiguity; see Exercise . A CZO is an example of a singular integral operator; the kernel just barely fails to be locally integrable (Schur’s test, for instance, encounters a logarithmic divergence), and so there is a singularity in the integration which must be understood in order to analyze correctly. In particular, not all singular kernels lead to an CZO, due to possible failure of boundedness. We will return to the question of working out which kernels do give CZOs later.

Remark 10 As with the theory of the Hardy–Littlewood maximal function, one can define CZOs on more general homogeneous spaces than , but we will not do so here for sake of concreteness.

By convention, we allow all implied constants in what follows to depend on the Hölder exponent and on the implied constants in , , and the operator norm bound on . (One could try to formalize things by imposing a norm on CZOs with a given exponent , but this gets a bit confusing and is largely unnecessary since one mostly deals with just a single CZO at a time.)

The Hilbert transform is thus a CZO, with kernel . We will discuss other examples of CZOs, such as the Riesz transforms, Hörmander–Mikhlin multipliers, and pseudodifferential operators of order zero, later on.

The Hilbert transform was skew-adjoint, translation-invariant, and scale-invariant. In general, CZOs are neither. However, the class of CZOs is self-adjoint (and skew-adjoint), translation-invariant, and scale-invariant; if is a CZO then so is , and the translates and dilates of are also CZOs uniformly in and . (What happens to the kernels in each case?)

Now we formalize the key phenomenon alluded to earlier, that CZOs map fluctuating localized functions to nearly localized functions.

Lemma 11 Let be a ball in , let be a CZO, and let have mean zero, thus . Then we have the pointwise bound In particular, we have

Proof: We remark that we could use the translation and scaling symmetries of CZOs to normalize , but we shall not do so here. For we can use the mean zero hypothesis to obtain

and the first claim then follows from and the triangle inequality. The second claim of course follows from the first (recall we allow our implied constants to depend on and ).

Remark 12 Observe that the argument also gives the slightly more general estimate when the mean zero condition is dropped.

This lemma asserts that is sort of bounded on – but only when the input function is localized and mean zero, and as long as one doesn’t look too closely at the region near the support of the function. This extremely conditional boundedness is a far cry from true boundedness; indeed, it is not hard to show that boundedness can fail. However, Lemma , combined with the Calderón–Zygmund decomposition from the previous section (which decomposes any function into a bounded part and a lot of localized, fluctuating parts) is strong enough to obtain the weak-type boundedness instead.

Corollary 13 All CZOs are of weak type , with an operator norm of .

Proof: Let (the condition ensuring that is well-defined), let be a CZO, and let . Our task is to show that

We spend scaling and homogeneity symmetries to normalize and to both be ; this avoids a certain amount of book-keeping regarding these factors later. Applying the Calderón–Zygmund decomposition, we can split , where the “good” function obeys the bounds

each is supported on , has mean zero, and has the bound

and

Now since , we can split

For the first term, we exploit the boundedness of via Chebyshev’s inequality and log-convexity of Hölder norms:

For the second term, we cannot rely on bounds because the have no reason to be bounded in . But the mean zero condition, combined with Lemma , shows that each has bounds outside of a dilate of (for some fixed constant depending only on );

(Actually one could take arbitrarily close to by iterating , to obtain a variant of Lemma , but we will not do so here.).

Restricting to the common domain and summing in using the triangle inequality we obtain

and hence by Chebyshev

This still leaves the question of what happens in the exceptional set . But here is the great advantage of weak-type estimates: we can tolerate arbitrarily bad behavior as long as it is localized to small sets. Indeed we have

Putting it all together, we obtain

as desired.

Combining this with Marcinkiewicz interpolation and the boundedness hypothesis gives that all CZOs are of strong-type for . The self-adjointness of the CZOs as a class then gives the same statement for , thus we have proven

Corollary 14 CZOs are of strong type for all , with an operator norm of .

Remark 15 We thus see that the localizing properties of the singular kernel allow one to automatically leverage boundedness into boundedness for . In fact there is nothing particularly special about here (Exercise ). But in practice boundedness is easier to obtain than other types of boundedness.

In particular, we see that the Hilbert transform is bounded on for (or more precisely, it is bounded on a dense subclass, and thus has a unique bounded extension). This does not by itself directly settle the question of whether the formula makes sense for functions in a norm or pointwise convergence sense (for that, one needs bounds on the truncated Hilbert transforms and the maximal truncated maximal Hilbert transforms), though it does lend confidence that this is the case. However, we can now easily conclude an analogous claim for the inversion formula.

Theorem 16 Let , and for every let be the Dirichlet summation operator, defined on Schwartz functions by Then extends uniquely to a bounded operator on , and converges in norm to as for all .

Proof: From Proposition we obtain the identity

and so the are uniformly bounded on as desired. (Note however that the are not CZOs, since modulation does not preserve the CZO class.) Since converges in norm to for Schwartz functions (by dominated convergence), the claim then follows by the usual limiting arguments.

Remark 17 The corresponding question about pointwise almost-everywhere convergence is the Carleson–Hunt theorem, which is quite difficult. When an example of Kolmogorov shows that both norm convergence and almost-everywhere convergence can fail. In higher dimensions, the norm convergence for is true by Plancherel’s theorem, but a surprising result of Fefferman shows that it fails for . The almost everywhere convergence problem remains open (and is considered very difficult) in higher dimensions.

Exercise 18 Let and be two CZOs with the same kernel . Show that there exists a bounded function such that for all . (Hint: by subtraction we can assume and . Now observe that is an absolutely continuous measure, and use the Radon–Nikodym theorem.)

Exercise 19 Show that if we redefine the notion of a CZO by weakening the hypothesis of boundedness, to merely being of restricted weak-type for some , then Corollary remains unchanged (but with an operator norm of instead of . In particular this apparent expansion of the notion of a CZO does not in fact increase the class of CZOs (up to changes in the implied constants).

— 3. Bounded mean oscillation —

Let us now study what a CZO does on the endpoint space . At first glance things are rather bad; not only is unbounded on in general, is not even known yet to be defined on a dense subset of . Nevertheless, a closer inspection of things will show that actually does have a fairly reasonable definition on , up to constants.

Let us try to define for directly. A naive implementation of the kernel formula

runs into problems both for near , and near infinity, since the decay is not strong enough to give absolute integrability in either case. Localizing to be supported away from helps with the first problem, but not the second. But suppose we instead looked at the difference between at two values. Then we formally have

Now implies that for fixed , the expression decays like as , and so the integral will be absolutely integrable as long as vanishes near and . At the other extreme, if is compactly supported, then it lies in and there is no difficulty defining (up to the usual measure zero ambiguities).

Motivated by this, let us make the following definitions. We say that two functions are equivalent modulo a constant if there exists a complex number such that for almost every ; this is of course an equivalence relation. Given , we define to be the equivalence class associated to the function

where is any ball which contains both and ; it is not hard to see that the right-hand side is well-defined (in particular, the integral is absolutely convergent) and is independent (modulo a constant) of the choice of , and that remains linear. One can also check that this definition is consistent (modulo a constant) with the existing definition of on (or on ) on the common domain of definition (i.e. or ).

Because is only defined modulo a constant, we cannot meaningfully control it in standard norms such as . However, there is a norm which is suitable for this type of function.

Definition 20 (Bounded mean oscillation) Let be a function defined modulo a constant. The (or Bounded Mean Oscillation) norm of is defined as where ranges over all balls. Note that if one shifts by a constant, the norm is unchanged, so this norm is well-defined for functions defined modulo constants. We denote by the space of all functions with finite norm, modulo constants.

It is not hard to verify that this is indeed a norm (but note that one has to quotient out by constants before this is the case!), and that is a Banach space. One can also easily check using the triangle inequality that

where ranges over constants; thus the statement is equivalent (up to changes in the implied constant) to the assertion that for any ball there exists a constant such that , thus deviates from by at most on the average.

It is not hard to see that contains , and that . But it also contains unbounded functions:

Exercise 21 Verify that the function is in .

The primary significance of for Calderón–Zygmund theory is that it can serve as a substitute for , on which CZOs are often ill-behaved. Consider for instance the Hilbert transform applied (formally) to the signum function . The resulting function is not bounded, but lies in :

Exercise 22 Verify that using the previous definitions, modulo constants.

This phenomenon applies more generally to all CZOs:

Proposition 23 (CZOs map to ) Let be a CZO and . Then and

Proof: We can normalize . By definition of the norm, it suffices to show that for every ball there exists a constant such that

By translation and dilation symmetry we may take . We split . Since is bounded on , we have

and hence by Hölder (or Cauchy–Schwarz)

This deals with the “local” part of . For the “global” part, observe that for we have

for some arbitrary constant . But by we have ; since is bounded by , we thus see that , and so

Adding the two facts, we obtain the claim.

In order to exploit this proposition, we need to understand to what extent BMO bounds on a function control its size. The basic insight here is that BMO bounds are “almost as good as” control for many purposes, despite the logarithmically unbounded members of BMO such as .

Let us first work informally. Let be a real-valued BMO function, normalized so that , thus

for all balls . By Markov’s inequality, this means that

for any . Note that this bound is only non-trivial for , otherwise the trivial bound of is superior. It asserts, for instance, that only exceeds its average by (say) on at most of any given ball.

It turns out that one can iterate the above fact to give far better estimates in the limit that initially suggests. This is because of a basic principle in harmonic analysis: bad behavior on a small exceptional set (such as of any given ball) can often be iterated away if we know that the exceptional sets are small at every scale. Roughly speaking, this principle allows us to say that only exceeds its average by on at most of of a given ball (i.e. on at most of the ball), exceeds its average by on at most , and so forth, leading to a bound of the form which tails off exponentially in rather than polynomially. More precisely, we have

Proposition 24 (John–Nirenberg inequality) Let . Then for any ball we have for all and some constant depending only on .

Proof: We can normalize . For each let denote the best constant for which we have

whenever is a ball and is a real-valued function with BMO norm . From and the trivial bound of we have the bounds

and our task is to somehow iterate this to obtain the improved estimate

We shall do this by a variant of the Calderón–Zygmund decomposition. We’ll need a moderately large quantity to be chosen later, and assume that . Let be any ball, and let . Then we have

We consider the set . Let be small, and let be any compact set in such that

Applying the Vitali type covering lemma as in the proof of the Hardy–Littlewood maximal inequality (one can also use the Whitney decomposition or Calderón–Zygmund decomposition here) we can cover

where are a finite collection of balls intersecting such that and

Note that as

we conclude that

Since intersects , which is contained in , we can conclude that if we choose large enough.

Observe that is pointwise bounded almost everywhere by (by the Lebesgue differentiation theorem), and so

On the other hand, on each we see that , and hence

and so by definition of

Summing this we conclude

Since is arbitrary we can remove it. Taking suprema over all and we conclude the recursive inequality

Taking sufficiently large depending on , and iterating starting from the trivial bound , we obtain the claim.

Using the distributional characterization of the norms, we conclude

Corollary 25 (Alternate characterization of BMO norm) For any and all locally integrable we have

Exercise 26 Prove this corollary.

Exercise 27 Show that Corollary is in fact true for all .

One consequence of this corollary, when combined with the Calderón–Zygmund decomposition, is that the norm serves as a substitute for the norm as far as log-convexity of Hölder norms is concerned:

Lemma 28 Let and . Then and

Proof: By rescaling one can normalize . Let us understand the size of the distribution function . From the bound we already have

We can get a better bound for large as follows. Applying the version of the Calderón–Zygmund decomposition at level , we can find a collection of disjoint dyadic cubes with , such that almost everywhere outside of , and such that on each cube . If we let be a ball containing of comparable diameter, we then easily verify that also, and thus . From the John–Nirenberg inequality we thus see that

for large enough; summing over we conclude that

for large . Combining this with the bound for small and using the distributional formula for the norm we obtain the claim.

Because of this, we can also use as a substitute for for real interpolation purposes. For instance, let be a CZO and . It is now easy to check that is restricted type . Indeed, for any sub-step function of height and width , the boundedness on gives

and Proposition gives

and so Lemma gives

for , which is the restricted type. Further interpolation then gives strong type. (Note that these arguments are essentially the dual of those which establish weak-type for ; in particular, note how the Calderón–Zygmund decomposition continues to make an appearance.)

Exercise 29 (Lacunary exponential sums lie in BMO) Let be a sequence of non-zero frequencies in which are lacunary in the sense that for all . Let be a sequence of complex numbers which obeys the bound . To avoid issues of convergence let us assume that only finitely many of the are non-zero. Let be the exponential sum If is an interval in , and is a bump function adapted to , establish the local estimate for some constant . (Hints: split the summation into the high frequencies where , and the low frequencies where . For the high frequencies, forget about and expand out the sum, using some good estimate on the Fourier transform of . For the low frequencies, choose in order to obtain some cancellation and then use the triangle inequality.) Conclude that

Exercise 30 (Dyadic BMO) Given any and any locally integrable , define the dyadic BMO norm of as where ranges over all dyadic cubes inside . Show that we have the John–Nirenberg inequality for all and all , and in particular for all . (As in Exercise , this extends to as well.)

Exercise 31 (BMO and dyadic BMO) For any locally integrable in , show that but that the converse inequality is not true. On the other hand, if we let be the shifted dyadic meshes from the previous week’s notes, and set to be the associated shifted dyadic BMO norms, show that

Exercise 32 (John–Nirenberg via Bellman functions) Let be the function . Use a Bellman function approach to establish the John–Nirenberg type inequality for some absolute constants , all intervals , and all functions with .

Exercise 33 (Sharp function, I) For any locally integrable function , define the function as where the supremum ranges over all balls containing . In particular we see that Show that we have the pointwise bound

Exercise 34 (Sharp function, II) Let be locally integrable, let , and let be any ball on which for at least one , and such that for at least one . Use the Hardy–Littlewood maximal inequality to show that for any sufficiently large , that From this and a Vitali covering lemma or Calderón–Zygmund decomposition argument, conclude the good inequality and then (by taking large, and very small) show that for all and . Use this to give an alternate proof of Lemma .

— 4. Multiplier theory —

An important subclass of CZOs are given by certain types of Fourier multipliers. (We also have spatial multipliers , but their behavior is much more elementary to analyze thanks to Hölder’s inequality; for instance, a spatial multiplier is bounded on if and only if the symbol is in , and indeed the operator norm of the former is the sup norm of the latter.)

Definition 35 If is a tempered distribution we define the Fourier multiplier on Schwartz functions by the formula We refer to the function as the symbol of the multiplier .

Note that if is Schwartz, then so is , and that is enough to make a tempered distribution, and so is well-defined as a tempered distribution. If is bounded, we observe from Plancherel’s identity that

and hence the Fourier multiplier extends continuously to all of in this case. Indeed it is not hard to show that in fact

The partial derivative operator is a Fourier multiplier with symbol . More generally, any constant-coefficient differential operator , where is a polynomial, has symbol ; thus one can formally view in as representing the vector-valued operator . The translation operator is a Fourier multiplier with symbol , thus we formally have . A convolution operator , with a distribution, is a Fourier multiplier with symbol ; conversely, every Fourier multiplier is a convolution operator with the distribution . The Hilbert transform is a Fourier multiplier with symbol , thus .

Let . We say that is an multiplier if extends to a bounded linear operator on , and in that case denote to be the operator norm of on . Thus for instance the norm is the same as the norm. At first glance, it seems one can say quite a lot about these quantities; for instance, since the adjoint of is easily seen to be , which has the same operator norm as , we see that

and thus by Riesz–Thorin interpolation

and so all multipliers are bounded (though the converse turns out to be false, as we shall see later). The space of multipliers turns out to be a Banach algebra, thus the norm is a Banach space norm and furthermore

One can also show (from the symmetries of the Fourier transform) that the norm is invariant under translations, modulations, and dilations:

The dilation invariance can be generalized: given any invertible linear transformation we can check that

Because of the translation invariance, the same is true if is an affine transformation rather than a linear one.

Thus for instance, by modulating the Hilbert transform we see that

for , which by Riemann–Stieltjes integration and Minkowski’s inequality shows that

when has bounded variation, where

Exercise 36 Show that if is a bounded linear operator on which commutes with translations, then it is a Fourier multiplier operator with bounded symbol.

Exercise 37 Let and be multipliers. Show that the tensor product is also an multiplier with (Hint: it may help to first establish the cases when or .)

For or , it is also not hard to show that

whenever the inverse Fourier transform of is absolutely integrable; with a little more work one can also show that is unbounded on and when does not lie in , although this is a little tricky. Thus for we have (by Riesz–Thorin again)

One consequence of this is that if is a bump function adapted to for some fixed bounded domain and some arbitrary affine linear transformation , then is an multiplier with for all .

Suppose a multiplier on does not depend on the final coordinate , thus

for some . Then the -dimensional Fourier multiplier and the -dimensional Fourier multiplier are related by the identity

for all and Schwartz , where is the slice

From this and Fubini’s theorem one easily verifies that if is in , then is in with

Of course, most multipliers on do depend on the final coordinate . Nevertheless we can obtain a partial generalization of for these more general multipliers:

Theorem 38 (De Leeuw’s theorem) If is continuous, then for any .

Proof: By translation invariance we can take . For any , let be the linear transformation

then by we have

Now we send . We observe from the hypothesis that is continuous that pointwise, where is the (non-invertible) projection

On the other hand, from we have

So to conclude the theorem it suffices to show that

This can be achieved by a weak convergence argument. (As usual, weak convergence preserves upper bounds, but does not necessarily preserve lower bounds (cf. Fatou’s lemma).) If are Schwartz functions, we see from Hölder’s inequality that

Taking limit superior as and using Plancherel and dominated convergence on the left-hand side (using the pointwise convergence, as well as the boundedness of ) we conclude

Using the converse to Hölder’s inequality we obtain the claim.

One consequence of de Leeuw’s theorem is that we expect multiplier theory on higher dimensions to be harder than multiplier theory for lower dimensions.

One can view de Leeuw’s theorem as a transference theorem, transferring a bound on one space () to another (), basically by exploiting the fact that there was a projection from to . For a result of similar flavor, see Exercise .

Exercise 39 (Transference theorem for ) If is bounded, define the multiplier on by the formula where we use the Fourier transform on the torus , We then define to be the operator norm of . Let be continuous and an multiplier for some . Show that . (Hint: take functions , extend them periodically to , multiply them by some cutoff to (such as ), apply on , and use some appropriate limiting argument (and possibly a Poisson summation formula) to get down to .)

Despite these encouraging initial results, it appears very difficult to decide in general whether any given multiplier lies in or not for any other than . For instance, the question of determining the precise values of and for which the multiplier is an multiplier is not fully resolved in three and higher dimensions; this problem is known as the Bochner–Riesz conjecture, which we will not discuss in detail here. Broadly speaking, the difficulty with the Bochner–Riesz conjecture is that the symbol is singular on a large, curved set (in this case, the sphere), which necessarily leads to some delicate and still incompletely understood geometric issues (most notably the Kakeya conjecture, which we will also not discuss here).

However, one can do a lot better if the multiplier has no singularities, or is only singular at one point (such as the origin). For instance, if is Schwartz, then is definitely in , and so lies in for all . Note from that any dilate, translate, and/or modulation of a Schwartz function is then also in uniformly in the dilation, translation, and modulation parameters.

Now we address multipliers which only have singularities at the origin.

Theorem 40 (Hörmander–Mikhlin multiplier theorem) Let obey the homogeneous symbol estimates of order for all and . (In particular .) Then is a CZO, and we have for all .

Remark 41 It is possible to use less regularity than derivatives here, but in practice this type of control is sufficient for many applications. The conditions are essentially asserting that is smooth and does not oscillate on each annulus , although it essentially allows different annuli to behave independently. For instance, if is a bump function adapted to the annulus , and are arbitrary signs, then obeys the homogeneous symbol estimates of order uniformly in the choice of .

Proof: The multiplier is already bounded on . Formally, we know that is a convolution operator: , but we run into the formal difficulty that need not be locally integrable. To compute things rigorously it is convenient to use the Littlewood–Paley decomposition. Let be a fixed bump function which equals one on the ball and is supported on the ball , and write

for . Then we have the partition of unity

for almost every , and hence we have the pointwise convergent sum

where . Using Plancherel and dominated convergence we conclude that

for all , where the sum is conditionally convergent in the sense. In particular for any we have

where the sum is at least conditionally convergent (indeed by Parseval one can show that it is absolutely convergent). Now let us assume that are also bounded and compactly supported with disjoint supports. We can write

where

One can justify these formulae via the Fourier inversion formula and Fubini’s theorem using the fact that are bounded with compact support; we omit the details. Now observe that is supported on the annulus , and multiple applications of the Leibnitz rule give us the bounds

for . From the case of this bound and the triangle inequality we obtain the bound

on the other hand, by integrating by parts times (using the identity ) we obtain the bound

for , and hence

Also, we can compute gradients

The additional factor essentially just contributes an extra factor of to the analysis, and we end up with

These estimates are enough to show that away from a neighborhood of , the series is convergent in to a kernel obeying the bounds

and by Fubini’s theorem we have

when are bounded with disjoint support. By a monotone convergence argument (noting that is bounded on ) the same is true when are with compact support.

The bounds (and the fundamental theorem of calculus) ensure that is a singular kernel, and shows that is a CZO with kernel . Thus, by Corollary , is of strong-type with an operator norm of for all , and the claim follows.

Note that the Hörmander–Mikhlin multiplier theorem includes the boundedness of the Hilbert transform for as a special case, since the multiplier certainly obeys the symbol estimates. Another useful instance of the theorem is to conclude that the homogeneous fractional derivative operators , with symbol and real, are also bounded on , with an operator norm of (in particular, it grows at most polynomially in ). This fact turns out to be very useful when one wishes to use Stein interpolation involving fractional differentiation operators . A similar argument also applies to the inhomogeneous fractional derivative operators with symbol . (The factor of is of course irrelevant here and the same results hold if this factor is dropped. Indeed, in PDE it is customary to drop the from the definition of the Fourier transform so as not to see these factors then arise from differential operators.)

Another example of multipliers covered by this theorem are the Riesz transforms for , with symbol . These symbols are easily verified to obey the homogeneous symbol estimates and are thus in with norm for all . The significance of the Riesz transforms is that they connect general second order derivatives to the Laplacian; indeed from Fourier analysis we quickly see that

for all and Schwartz . In particular, since is bounded on , we obtain the elliptic regularity estimate

for all Schwartz and all