Limited range multilinear extrapolation with applications to the bilinear Hilbert transform
We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights for the bilinear Hilbert transform to satisfy weighted norm inequalities of the form
where and . This improves the recent results of Culiuc et al. by increasing the families of weights for which this inequality holds and by pushing the lower bound on from down to , the critical index from the unweighted theory of the bilinear Hilbert transform. Second, as an easy consequence of our method we obtain that the bilinear Hilbert transform satisfies some vector-valued inequalities with Muckenhoupt weights. This reproves and generalizes some of the vector-valued estimates obtained by Benea and Muscalu in the unweighted case. We also generalize recent results of Carando, et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators.
Key words and phrases:Muckenhoupt weights, extrapolation, bilinear Hilbert transform, vector-valued inequalities
2010 Mathematics Subject Classification:42B25, 42B30, 42B35
The Rubio de Francia theory of extrapolation is a powerful tool in harmonic analysis. In its most basic form, it shows that if, for a fixed value , , an operator satisfies a weighted norm inequality of the form
for every weight in the Muckenhoupt class , then for every , ,
whenever . Since its discovery in the early 1980s, extrapolation has been generalized in a variety of ways, yielding weak-type inequalities, vector-valued inequalities, and inequalities in other scales of Banach function spaces. We refer the reader to  for the development of extrapolation; for more recent results we refer the reader to [8, 13, 18].
Extrapolation has been also extended to the multilinear setting. In  it was shown that if a given operator satisfies
for fixed exponents , , and all weights , then the same estimate holds for all possible values of . An extension to the scale of variable Lebesgue spaces was given in .
In this paper we develop a theory of limited range, multilinear extrapolation. In the linear case, limited range extrapolation was developed in  by Auscher and the second author. They proved that if inequality (1.1) holds for a given and for all , then for all and , (1.2) holds. Conditions like this arise naturally in the study of the Riesz transforms and other operators associated to elliptic differential operators.
Our first theorem extends limited range extrapolation to the multilinear setting. To state our results we use the abstract formalism of extrapolation families. Given , hereafter will denote a family of -tuples of non-negative measurable functions. This approach to extrapolation has the advantage that, for instance, vector-valued inequalities are an immediate consequence of our extrapolation results. We will discuss applying this formalism to prove norm inequalities for specific operators below. For complete discussion of this approach to extrapolation in the linear setting, see .
Given , let be a family of extrapolation -tuples. For each , , suppose we have parameters and , and an exponent , , such that given any collection of weights with and , we have the inequality
for all such that , where and depends on . Then for all exponents , , all weights and ,
for all such that , where and depends on . Moreover, for the same family of exponents and weights, and for all exponents , ,
for all such that the left-hand side is finite and where and depends on .
Theorem 1.3 is a consequence of a linear, restricted range, off-diagonal extrapolation theorem, which we believe is of interest in its own right. It generalizes the classical Rubio de Francia extrapolation, the off-diagonal extrapolation theory of Harboure, Macías and Segovia , and the limited range extrapolation theorem proved by Auscher and the second author .
Given and a family of extrapolation pairs , suppose that for some such that , , and all weights such that ,
for all such that , and the constant depends on . Then for every , such that , and , and every weight such that ,
for all such that , and depends on , , , , .
In Theorems 1.3 and 1.8 we make the a priori assumption that the left-hand sides of both our hypothesis and conclusion are finite, and this plays a role in the proof. In certain applications this assumption is reasonable: for instance, when proving Coifman-Fefferman type inequalities (cf. ). However, when using extrapolation to prove norm inequalities for operators we would like to remove this assumption, as the point is to conclude that the left-hand side is finite. But in fact, we can do this by an easy approximation argument. This immediately yields the following corollaries.
Under the same hypotheses as Theorem 1.3, if we assume that (1.4) holds for all (whether or not the left-hand side is finite) then the conclusion (1.5) holds for all (whether or not the left-hand side is finite). Analogously, the vector-valued inequality (1.6) holds for all families (whether or not the left-hand side is finite).
In the statement of Theorem 1.8 there are some restrictions on the allowable exponents and . We make these explicit here; these restrictions will play a role in the proof below.
Because of our assumptions that and it follows that . Moreover, the fact that yields that . Note that if we were to allow that , we could choose very close to and the associated would be negative, which would not make sense.
Moreover, we have that the following hold:
If , then and .
If , then , and .
If , then , and .
When we automatically have that . Further, this implies that all of the weights which appear in both our hypothesis and conclusion (i.e, , , , ) are in . Consequently, they are locally integrable, and so all the Lebesgue spaces that appear in the statement contain the characteristic functions of compact sets. In fact, since , (see Lemma 2.1 below). The same is true for and , since by Remark 1.13, .
When , the condition imposes an upper bound for : . A similar bound holds for . Thus (by Lemma 2.1) and so again all the weights involved are in and thus locally integrable.
Our generalization of off-diagonal extrapolation involves weighted norm inequalities that have already appeared in the literature in the context of fractional powers of second divergence form elliptic operators with complex bounded measurable coefficients. More precisely, in  it was shown that for a certain operator , there exist such that for every and for every . By applying Theorem 1.8 we could prove the same result via extrapolation if we could show that there exists such that for every . Note that the latter condition can be written as with and , and in this case , so the hypotheses of Theorem 1.8 hold.
A restricted range, off-diagonal extrapolation theorem has previously appeared in the literature. Duoandikoetxea [18, Theorem 5.1] proved that if for some and , and all weights (note that unlike in the classical definition of this class he does not require ), if (1.9) holds, then for all and such that , and all weights , (1.10) holds.
This result is contained in Theorem 1.8 in the particular case when if we take and . In this case, (because ) if and only if . Moreover, in this scenario since .
Despite this overlap, our results are different. We eliminate the restriction as we can take . Moreover, for a value of , it is not clear whether our result can be gotten from his by rescaling. On the other hand, we cannot recapture his result for values of .
Finally, in light of Remark 1.15, we note that [18, Theorem 5.1] allows for weights or that may not be locally integrable unless one assumes . For example, if we fix and let , then it is easy to see that and so , but neither nor is locally integrable (and so the characteristic function of the unit ball centered at does not belong to or to ). In light of this, we believe the condition is not unduly restrictive.
To demonstrate the power of our multilinear extrapolation theorem, we use Theorem 1.3 to prove results for the bilinear Hilbert transform and for multilinear Calderón-Zygmund operators. We first consider the bilinear Hilbert transform, which is defined by
The problem of finding bilinear estimates for this operator was first raised by Calderón in connection with the Cauchy integral problem (though it was apparently not published until ). Lacey and Thiele [25, 26] showed that for , ,
The problem of weighted norm inequalities for the bilinear Hilbert transform has been raised by a number of authors: see [15, 16, 20, 29]. The first such results were recently obtained by Culiuc, di Plinio and Ou .
Given , define by and assume that . For , let be such that , and define . Then
If we apply Theorem 1.3, we can extend Theorem 1.16 to a larger collection of weights and exponents. In particular, we can remove the restriction that , replacing it with , the same threshold that appears in the unweighted theory.
Given arbitrary , define and assume that . For every , let . Then, for all —or, equivalently, for — if we write and , we have that
In particular, given arbitrary so that where , there exist values such that , in such a way that if we set , then , and for all weights with (or, equivalently, for ) and ,
We can state Theorem 1.18 in a different but equivalent form. For instance, in the second part of that result, if we let , then our hypothesis becomes , and the conclusion is that
In  the authors actually proved Theorem 1.16 for a more general family of bilinear multiplier operators introduced by Muscalu, Tao and Thiele . Theorem 1.18 immediately extends to these operators. We refer the interested reader to these papers for precise definitions. This extension actually shows that that the bound in Theorem 1.16 and the bound in Theorem 1.18 are natural and in some sense the best possible. In [24, Theorem 2.14], Lacey gave an example of an operator which does not satisfy a bilinear estimate when ; in [14, Remark 1.2] the authors show that Theorem 1.16 applies to this operator. Hence, if Theorem 1.16 could be extended to include the case , we would get weighted estimates for this operator. But by extrapolation, these would yield inequalities below the threshold . Indeed, we could apply the first part of Theorem 1.18 with those fixed exponents and to obtain that this operator maps into for every and . If we fix and let , we would have that and
Given , as part of the proof of Theorem 1.18 we construct the parameters needed to define the weight classes. Thus, while we show that such weights exist, it is not clear from the statement of the theorem what weights are possible. To illustrate the different kinds of weight conditions we get, we give some special classes of weights, and in particular we give a family of power weights.
Given , define by , and assume further that . Then,
holds for all and . In particular,
if or if
As a result, (1.25) holds for all .
By Corollary 1.23 we get weighted estimates for the bilinear Hilbert transform in exactly the same range where the unweighted estimates are known to hold. (Note that when we recover the unweighted case.) Rather than taking equal weights in (1.25), we can also give this inequality for more general power weights of the form ; details are left to the interested reader.
As a consequence of Corollary 1.23 we see that even in the range of exponents covered by Theorem 1.16 from , we get a larger class of weights. Fix and assume that . First, it is easy to show (see Lemma 2.1 below) that if an only if . Hence, if we further assume that this condition becomes or, equivalently, (see Lemma 2.1 below) . Hence, as a corollary of Theorem 1.16 we get that for all . But by Corollary 1.23, again assuming that , we can allow , or equivalently, which is weaker than since .
Further, when , Corollary 1.23 gives the class of weights . To compare this with Theorem 1.16 from  note that their condition is, as explained above, and hence we can weaken to at the cost of assuming that . Alternatively, if , our condition becomes , which removes any reverse Hölder condition for at the cost of assuming that .
We can also prove vector-valued inequalities for the bilinear Hilbert transform for the same weighted Lebesgue spaces as in the scalar inequality. Even in the unweighted case, vector-valued inequalities were an open question until recently. Benea and Muscalu [4, 5] (see also [22, 31] for earlier results) proved that given and such that and , then there exist such that
where , , and, depending on the values of the , there are additional restrictions on the possible values of the . (See [5, Theorem 5] for a precise statement or (5.4) below.) An alternative proof of these estimates when is given in .
By using the formalism of extrapolation pairs, vector-valued inequalities are an immediate consequence of extrapolation. Hence, as a consequence of Theorem 1.18 we get the following generalization of the results in [4, 5, 14]. We note that for some triples our method does not let us recover the full range of spaces gotten in [4, 5] but we do get weighted estimates in our range.
Given arbitrary , define and assume that . For every , let . Then, for all —or, equivalently, for — if we write , and , there holds
In particular, for every such that , and for every such that , if
there are values such that , in such a way that if we set ,