After having studied the Hilbert transform in detail we now move to the study of general Calderón-Zygmund operators, that is operators given formally as
for an appropriate kernel . Let us quickly review what we used in order to show that the Hilbert transform is of weak type and strong type . First of all we essentially used the fact that the linear operator is defined on and bounded, that is, that it is of strong type . This information was used in two different ways. First of all, the fact that is defined on means that it is defined on a dense subspace of for every . Furthermore, the boundedness of the Hilbert transform on allowed us to treat the set where is the `good part’ in the Calderón-Zygmund decomposition of a function . Secondly, we used the fact that there is a specific representation of the operator of the form
whenever and has compact support and . For the Hilbert transform we had that the kernel is given as
We used the previous representation and the formula of to prove a sort of restricted boundedness of on functions which are localized and have mean zero, which is the content of Lemma 7 of Notes 6. This, in turn, allowed us to treat the `bad part’ of the Calderón-Zygmund decomposition of . From the proof of that Lemma it is obvious that what we really need for is a Hölder type condition. Note as well that for the Hilbert transform we first proved the bounds for and then the corresponding boundedness for followed by the fact that is essentially self-adjoint.
1. Singular kernels and Calderón-Zygmund operators
We will now define the class of Calderón-Zygmund operators in such a way that we will be able to repeat the schedule used for the Hilbert transform. We begin by defining an appropriate class of kernels , name the singular (or standard) kernels.
Example 1 Let be given as for with . Then is a singular kernel. Observe that is the singular kernel associated with the Hilbert transform.
where is a Hölder-continuous function:
for some . Then is a singular kernel.
Exercise 1 Prove that the kernel of example 2 is a singular kernel.
Example 3 Let satisfy the size estimate
and the regularity estimates
away from the diagonal . Then is a singular kernel. In particular, the kernel given as
We are now ready to define Calderón-Zygmund operators.
Definition 2 (Calderón-Zygmund operators) A Calderón-Zygmund operator (in short CZO) is a linear operator which is bounded on :
and such that there exists a singular kernel for which we have
for all with compact support and .
Remark 2 Note that the integral converges absolutely whenever has compact support and lies outside the support of . Indeed,
by (1), for some . Observe that the integral in the last estimate converges.
Remark 3 For any singular kernel one can define by means of
for with compact support and . It is not necessary however that is a CZO since it might fail to be bounded on .
Remark 4 It is not hard to see that uniquelydetermines the kernel . That is if
for all with compact support, then almost everywhere (why?). The opposite is not true. Indeed, for any bounded function the operator defined as is a Calderón-Zygmund kernel with kernel zero. A more specific example is the identity operator which also falls in the previous class, and is CZO with kernel 0. However, this is the only ambiguity. See Exercise 2.
for all .
If is a CZO, the definition already contains the fact that is defined and bounded on , so we don’t need to worry about that. The next step is to establish the restricted boundedness for functions with mean zero. The following lemma is the analogue of Lemma 7 of Notes 6.
Lemma 3 Let be a Euclidean ball in and denote by the ball with the same center and twice the radius, that is . Let have mean zero, that is . Then we have that
for all . We conclude that
Proof:Using the fact that has zero mean on , for we can estimate
Integrating throughout we also get the second estimate in the lemma.
The only thing missing in order to conclude the proof of the bounds for CZOs is the the fact that they are self adjoint as a class. In particular, we need the following.
Proof: It is immediate from (4) and the fact that is bounded on that is also bounded on with the same norm. Now let have disjoint compact supports. We have
Now let and have support inside with . For , the functions are supported in so, for small enough, the support of is disjoint from the support of . By (5)we conclude that
Letting we get
for almost every . Since the conditions defining singular kernels are symmetric in the variables , the kernel is again a singular kernel so we are done.
The discussion above leads to the main theorem for CZOs:
2. Pointwise convergence and maximal truncations
Let be a CZO. The example of the Hilbert transform suggests that we should have the almost everywhere convergence
at least for nice functions . The truncated operators
certainly make sense for because of (1). However, the limit need not even exist in general or may exist and be different from . Here we can use the trivial example of the operator . As we have already observed this is a CZO operator with kernel . Thus for all but clearly in general.
The following lemma clears out the situation as far as the existence of the limit is concerned:
Lemma 6 The limit
exists almost everywhere for all if and only if the limit
exists almost everywhere.
Proof:First suppose that the limit exists for all and let with on . Then
Observe that by (1)the second integral on the right hands side converges absolutely. Since the limit on the left hand side exists we conclude that the limit on the right hand side exists as well. Conversely, suppose that the limit
exists and let . We have that
By the same considerations are before is a positive number that does not depend on . By the hypothesis we also have that . Finally for observe that we have
by (1). Since
dominated convergence implies that exists as well.
Thus, for specific kernels one has an easy criterion to establish whether the limit exists a.e. for `nice’ functions . For example, for the kernel of the Hilbert transform, the existence of the limit
is obvious. In order to extend the almost everywhere convergence to the class we need to consider the corresponding maximal function.
Definition 7 Let be a CZO and define the truncations of as before
The maximal truncationof is the sublinear operator defined as
The maximal truncation of a CZO has the same continuity properties as itself.
The proof of Theorem 8 depends on the following two results.
The proof of this lemma is a simple application of the representation of the norm in terms of level sets and is left as an exercise.
Exercise 3 Prove Lemma 9 above.
The second result we need is the following lemma that gives a pointwise control of the maximal truncations of the CZO by an expression that involves the maximal function of and the maximal function of .
Proof:Let us fix a function and and consider the balls and its double . We decompose in the form
Since and obviously has compact support we can write
Also every is not contained in the support of thus
by (3), since for in the area of integration above. By this estimate we get that
If then we are done. If then there is such that . Let
Let . Then either or or . In the last case so in every case we conclude that thus . However we have that
Also, by the type of we get
Finally, if then . Otherwise so
Thus in every case we get that
Since the previous estimate is true for any we conclude that
which gives the desired estimate in the case .
For estimate (7)implies that
and integrate in to get
and by Lemma 9the last term is controlled by
since is of weak type . Gathering these estimates we get
as we wanted to show.
We can now give the proof of the fact that maximal truncation of a CZO is of weak type and strong type for .
Thus the proof will be complete if we show that
As we have seen in Corollary 18 of Notes 5 we have that
where is the dyadic maximal function. Furthermore, using the Calderón-Zygmund decomposition it is not hard to see (see Exercise 4) that
Applying the last estimate to we get
For the set has finite measure. Thus by Lemma 9we conclude that
and thus by (8)that
This concludes the proof.
3. Singular integral operators on and .
The theory of Calderón-Zygmund operators developed so far is pretty satisfactory except for one point, the action of a CZO on . Exercise 4 from Notes 6 shows for example that in general a CZO cannot be bounded on . Furthermore, it is at the moment unclear how to define the action of on a general bounded function or even on a dense subset of . With a little effort however this can be achieved.
Let us first fix a function and look at the formula
As we have already mentioned several times, such a formula is not meaningful throughout . Indeed the integral above need not converge, both close to the diagonal , since is singular, as well as at infinity since only decays like , not fast enough to make the integral above absolutely convergent. The first problem we have dealt with so far by considering functions with compact support and requiring the validity of (9)only for . A similar solution could work now but we still have a problem at infinity. Note that we didn’t run into this problem yet since we only considered functions in which necessarily possess decay at infinity. This is not necessarily the case for bounded functions. However, looking at the difference of the values of at two points with , we can formally write
Using the regularity condition (3)we see that
when . This is enough to assure integrability in the previous integral, as long as Motivated by this heuristic discussion we define for :
for some Euclidean ball so that . First of all it is easy to see that the integrals above make sense. Indeed, is well defined since is in . On the other hand, the integral in the second summand converges absolutely since we integrate away from , is bounded and behaves like for . However, (10)only defines up to a constant. Indeed it is easy to see that if are two different balls containing the difference in the two definitions is equal to
which is a constant independent of . Thus we only define modulo constants. This definition of gives a linear operator which extends our previous definitions on or . To deal with the ambiguity in the definition, we have to define the appropriate space.
Definition 11 We say that two functions are equivalent modulo a constant if there exists a constant such that almost everywhere on . This is an equivalence relationship. By abuse of language and notation we will oftentimes identify an equivalence class with a representative of the class, much like we do with measurable functions.
Definition 12 (Bounded Mean Oscillation) Let be a locally integrable function , defined modulo a constant. We set
to be the average of on the Euclidean ball . The norm of is the quantity
where the supremum varies over all Euclidean balls . The space is the set of all locally integrable functions , defined modulo a constant, such that . Thus, an element of is only defined up to a constant.
First of all observe that this is a good definition since replacing a function by for any constant does not affect its BMO norm. Thus, all elements in the equivalence class of have the same BMO norm. The previous quantity actually defines a norm, always keeping in mind that we identify functions that differ by a constant. For example any constant is equivalent to the function in BMO and thus if and only if almost everywhere for some .
It is not hard to give the following alternative description of the BMO norm, which is maybe a bit more revealing:
(ii) For any locally integrable function and a cube set . We set
where the supremum is taken over all cubes Then
as in . Moreover
Proof:For (i) observe that for any ball we have
On the other hand for any we have
which gives the opposite inequality as well by taking the infimum over . The proof of the first claim in is identical. For the second claim in let and be a cube. Consider the smallest ball with the same center as . Then
for any cube . Taking also the supremum over cubes proves the one direction of the inequality. The proof of the opposite inequality is similar.
Thus a function in BMO has the property that for any ball there is a constant such that . That is, the values of oscillate around by at most in average. Locally, and in the mean, the function has bounded oscillation.
The space BMO contains but also contains unbounded functions.
Proposition 14 (i) For every we have that
(ii) The function is in . Thus is a proper subset of .
Our interest in the space BMO mainly lies in the fact that it serves as a substitute endpoint for the boundedness of CZOs, namely a CZO is bounded from to BMO, where should be defined as in (10). Note here that even though (10) only defines `up to constants’, this is the only possible definition of a BMO function.
Theorem 15 Let be a CZO. Then for every we have that
Proof:Let be some ball in . We need to show that
and denote . We set
Since is of strong type we have
Thus by Cauchy-Schwartz we have
On the other hand for , the ball certainly contains both and so
Remembering that (10)only defines up to a constant we get
By Proposition 13 this proves the theorem.
3.1. The John-Nirenberg Inequality
We will now see that although the space BMO contains unbounded functions like , this is in a sense the maximum possible growth for a BMO function. Although such a claim is not precise in a pointwise sense, it can be rigorously proved in the sense of level sets. Indeed, assuming then
for all balls . Using Chebyshev’s inequality this implies
This estimate is interesting for large, and states that on any ball the function exceeds its average by only on a small fraction of the ball . In fact, this can be improved.
Theorem 16 (John-Nirenberg inequality) Let . Then for any Euclidean cube we have that
for all , where the constant depends only on the dimension .
Remark 5 Obviously it doesn’t make any difference to work with balls instead of cubes so the the previous theorem remains valid with balls replacing cubes .
Proof:For let us denote by the best constant in the inequality
valid for any cube and with By Chebyshev’s inequality combined with the trivial bound we get
which is of course quite far from the desired estimate
This will be achieved by iterating a local Calderón-Zygmund decomposition as follows.
Let us fix a cube and consider the family of cubes inside which are formed by bisecting each side of . Then define the second generation by bisecting the sides of each cube in and so on. The family of all cubes in all generation will be denoted by . For a level to be chosen later let be the `bad’ cubes in , that is the cubes such that
Finally let be the family of maximal bad cubes. Since for the original cube , every bad cube is contained in a maximal bad cube. As in the global Calderón-Zygmund decomposition we conclude that
for each cube where the constant depends only on the dimension . We also conclude that
if by the dyadic maximal theorem. Remembering the initial normalization we get
Now consider . We have
However this means that
whenever . Suppose that . Since is non-increasing and the trivial estimate we get
for (say) and . On the other hand, for we have
so the proof is complete.
Exercise 5 Use the John-Nirenberg and the description of norms in terms of level sets to prove Corollary 17
Finally, we show how we can use the space as a different endpoint in the Log-convexity estimates for the norms.
Lemma 18 Let and . Then and
Proof:Obviously it is enough to assume that otherwise there is nothing to prove. Also by homogeneity we can normalize so that . Now form the Calderón-Zygmund decomposition of at level and denote by the family of bad cubes as usual. For each cube we then have
From the John-Nirenberg inequality we conclude that
for all . On the other hand, since we have
Exercise 6 (The sharp Maximal function) For define the sharp maximal function
Observe that if and only if and, in particular,
Show that for every we have
4. Vector valued Calderón-Zygmund Singular integral operators
We close this chapter on CZOs by describing a vector valued setup in which all our results on CZOs go through almost verbatim. We will see an application of these vector valued results in our study of Littlewood-Paley inequalities.
So let be a separable Hilbert space with inner product and norm and consider a function . All the well known facts about spaces of measurable scalar functions have almost obvious generalizations in this setup once we fix some analogies. For example, the function will be called measurable if for every the function is a measurable function of . If is measurable then is also measurable. We then denote the space of all measurable functions such that
and the usual corresponding definition for
It is not hard to check the duality relations for these spaces; for example
for all . Also our interpolations theorems, the Marcinkiewicz interpolation theorem and the Riesz thorin interpolation theorem go through in this setup as well.
Moreover, if a function is absolutely integrable, we can define its integral as an element of by defining the functional
Note here that is uniquely defined as a functional in . Indeed, is obviously linear and by the Cauchy-Schwartz inequality we have
By the Riesz representation theorem on Hilbert spaces, there is a unique element of , which we denote by , such that , that is
Finally, if are separable Hilbert spaces we denote by to be the space of bounded linear operators , equipped with the usual operator norm:
Again, a function will be called measurable if for every the function
is a measurable -valued function.
We are now ready to give the description of vector valued CZOs. We start with the definition of a singular kernel.
Definition 19 (Vector valued singular Kernel) Let be two separable Hilbert spaces and be a function defined away from the diagonal Then will be called a (vector-valued) singular kernel if it obeys the size estimate
Definition 20 Let be separable Hilbert spaces. An linear operator is called a (vector valued) Calderón-Zygmundoperator (vector valued CZO) from to if it is bounded from to
for all , and there exists a vector valued singular kernel such that
whenever has compact support and .
Adjusting the proof of the scalar case to this vector valued setup we get the corresponding statement of Theorem 5.
Theorem 21 Let be separable Hilbert spaces and be a vector valued Calderón-Zygmund operator from to .
(i) The operator is of weak type
for all .
(ii) For all , is of strong type
for all .