\mathrm{A}_{1}-regularity and Calderón-Zygmund operators

# A1-regularity and boundedness of Calderón-Zygmund operators. II

Dmitry V. Rutsky Steklov Mathematical Institute
St. Petersburg Branch
Fontanka 27
191023 St. Petersburg, Russia
###### Abstract.

Proof is given for the “only if” part of the result stated in the previous paper of the series that a suitably nondegenerate Calderón-Zygmund operator is bounded in a Banach lattice on if and only if the Hardy-Littlewood maximal operator is bounded in both and , under the assumption that has the Fatou property and is -convex and -concave with some . We also get rid of an application of a fixed point theorem in the proof of the main lemma and give an improved version of an earlier result concerning the divisibility of -regularity.

###### Key words and phrases:
-regularity, Calderón-Zygmund operator, Hardy-Littlewood maximal operator
###### 2010 Mathematics Subject Classification:
Primary 46B42, 42B25, 42B20, 46E30, 47B38

This paper is closely related to [7] and contains essentially no new non-technical results, hence for the background and the generalities we refer the reader to [7].

A. Yu. Karlovich and L. Maligranda kindly pointed out to the author that the proof of [7, Theorem 16] has a flaw, namely that the relationship is incorrect (and in fact it is always false). Unfortunately, it is not clear if [7, Theorem 16] is true in the stated form.

Nevertheless, we will see that the main result of [7] is still true with only a slight loss of generality concerning the nondegeneracy assumption imposed on a Calderón-Zygmund operator . Specifically, in place of -nondegeneracy of (which is a condition that the boundedness of in implies with an estimate for the constant) we require that the kernels of both and its conjugate satisfy a standard assumption on growth along a certain singular direction (see [10, Chapter 5, §4.6]).

###### Definition 1.

We say that a singular integral operator on is nondegenerate if there exists a constant and some such that for any ball of radius and any locally summable nonnegative function supported on we have

 (1) |Tf(x)|⩾c1|B|∫Bf

for all .

For example, the Hilbert transform and any of the Riesz transforms are nondegenerate in this sense. A nondegenerate operator is also -nondegenerate. For details see [10, Chapter 5, §4.6].

###### Theorem 2.

Suppose that is a Banach lattice of measurable functions on that satisfies the Fatou property and is -convex and -concave with some . Let be a Calderón-Zygmund operator in such that both and are nondegenerate. The following conditions are equivalent.

1. The Hardy-Littlewood maximal operator acts boundedly in and in the order dual of .

2. All Calderón-Zygmund operators act boundedly in .

3. acts boundedly in .

Thus, concerning the necessity of -regularity we make no claims about the general spaces of homogeneous type, although in many cases a suitable generalization of Definition 1 seems to be possible. Another subtle loss of generality is that in contrast to [10, Theorem 16] in the proof of we take advantage of the assumption that is a Calderón-Zygmund operator as well as a nondegenerate operator, specifically that is bounded in for with norm as .

For the proof of of Theorem 2 see [7]. The proof of essentially follows the scheme of the flawed proof of [7, Theorem 16], but it seems to require a much more delicate approach that we will present throughout the rest of the paper, leading to the proof itself given at the end of Section 5 below. We briefly outline the structure of the argument, the details of which are also of some independent interest.

The following result was established (with some caveats) in [5, Theorem A’]; a complete proof in the stated form can be found in [6, Theorem 4]. Here and elsewhere is a space of homogeneous type and is a -finite measurable space.

###### Theorem 3.

Suppose that is a Banach lattice on with an order continuous norm. If a linear operator is bounded in then for every there exists a majorant , , such that , where depends only on the Grothendieck constant .

This yields almost at once the following version of Theorem 2 that we will need in the proof of Theorem 2, showing that Theorem 2 is also valid for , (and by duality for and ), provided that (respectively, ) has order continuous norm. The proof is given in Section 1 below.

###### Theorem 4.

Suppose that is a -convex Banach lattice of measurable functions on having order continuous norm and the Fatou property. Let be an -nondegenerate linear operator in . If acts boundedly in then the maximal operator is bounded in both  and  with a suitable estimate for the constants.

In contrast to [7], in the present work we use the standard definition of the constant , of a Muckenhoupt weight on based on the Muckenhoupt condition:

 [w]Ap=esssupω∈ΩsupB(1ν(B)∫Bw(⋅,ω))(1ν(B)∫Bw−1p−1(⋅,ω))p−1,

where the supremum is taken over all balls .

Recall that a quasi-normed lattice is called regular with constants if every admits a majorant , , such that and belongs to the Muckenhoupt class with .

In Section 2 we give (Proposition 7) a simplified proof of [7, Proposition 8] that does not use a fixed point theorem. This yields a slightly improved version (Proposition 8) of [7, Proposition 12] stating that -regularity of both and implies -regularity of these lattices, where the assumption that satisfies the Fatou property is replaced by a weaker assumption that is a norming lattice for . Thus it suffices to establish that condition 3 of Theorem 2 implies that is -regular; interchanging with would then show that is also -regular.

Under condition 3 of Theorem 4 we may apply Theorem 3 to lattice with some fixed sufficiently close to and all sufficiently large , since is bounded in by interpolation with some estimate for the norm that grows with . This yields -regularity of , with an estimate on the growth of the constant as . Now the key idea is to show that the -majorants of functions from also satisfy the reverse Hölder inequality with exponent for some sufficiently large , which would yield -regularity of , , and thus the required -regularity of the lattice .

However, as discussed in Section 4 below, in order to get an estimate for with a suitable rate of growth we also need to make sure that the weight appearing in the conclusion of Theorem 3 (applied to ) satisfies some additional assumptions, namely that is a doubling weight with a constant independent of . Theorem 4 allows us to obtain -regularity of from condition 3 of Theorem 2 with a sufficiently large fixed , where an estimate for the constants is independent of . An extension (Theorem 15) of [6, Theorem 2] concerning the divisibility of -regularity, which we introduce in Section 3 below, allows us to prove that admits suitable majorants such that (and hence is a doubling weight) with a constant independent of , and an adaptation (Theorem 18) of the original fixed point argument from [7, §2] makes it possible to impose this condition on the weights appearing in the conclusion of Theorem 3, thus completing the proof.

## 1. Proof of Theorem 4

Suppose that and are normed lattices on a measurable space . Lattice is said to be norming for if for all and and for all . A normed lattice is always norming for its order dual . Conversely, it is well known that is a norming lattice for if satisfies either the Fatou property (implying that ), or if is a Banach lattice having order continuous norm (since then ). The fact that if and only if the maximal operator is bounded in with the appropriate estimates of the constants yields at once the following result; see [6, Proposition 13].

###### Proposition 5.

Suppose that and are normed lattices on such that is a norming space for . If is -regular with some then is -regular with appropriate estimates for the constants.

The following result is a particular case of [8, Proposition 13]; we give a complete proof for clarity.

###### Proposition 6.

Suppose that is an -regular quasi-normed lattice on . Then lattice is -regular.

Indeed, suppose that with norm , so there exist some and with norms at most such that almost everywhere and . Let be a suitable -majorant for in . Then we also have and

 ∥Mf∥Z12L2⩽∥∥∥w12∥∥∥Z12∥∥∥(Mf)w−12∥∥∥L2=∥w∥12Z∥∥∥(Mf)w−12∥∥∥L2⩽c∥∥∥(Mf)w−12∥∥∥L2=c∥(Mf)∥L2(w12)⩽c′∥f∥L2(w12)=c′∥∥∥fw−12∥∥∥L2=c′∥∥∥(gw)12h∥∥∥L2⩽c′∥h∥L2⩽c′′

with some suitable constants , and , so is bounded in the lattice which is thus -regular as claimed.

Now we can prove Theorem 4. Since is -convex, we may apply Theorem 3 to lattice and obtain -regularity of lattice by the assumed -nondegeneracy of operator . By Proposition 5 lattice is then -regular, and by Proposition 6 lattice is also -regular as claimed.

## 2. Main lemma revisited

Recall that a lattice is called -regular if functions from admit majorants with the appropriate control on the norm; see also Definition 10 in Section 3 below. Lattice is -regular if and only if it is -regular with some . -regularity is equivalent to the boundedness of the Hardy-Littlewood maximal operator (see, e. g., [6, Proposition 1]).

The following result was established in [7, Theorem 8] with the help of a fixed point theorem under an additional assumption that is a Banach lattice satisfying the Fatou property. However, we will now see that for the proof it suffices to carry out a slightly modified version of estimate [7, (6)] with the appropriate majorants.

###### Proposition 7.

Suppose that is a quasi-Banach lattice of measurable functions on such that is -regular with some and is -regular with some . Then is -regular with an appropriate estimate for the constants depending only on the corresponding -regularity constants of , -regularity constants of and the value of .

Indeed, let . Then there exists an -majorant for in , and in turn there exists an -majorant for in . We fix some such that and , and let , , , be an arbitrary ball in . Sequential application of the condition satisfied by weight , the Jensen inequality with convex function , , and the condition satisfied by the weight yield

 (2) 1ν(B(x,r))∫B(x,r)|f(⋅,ω)|⩽1ν(B(x,r))∫B(x,r)w(⋅,ω)⩽c[1ν(B(x,r))∫B(x,r)[w(⋅,ω)]−1p−1]−(p−1)=c[1ν(B(x,r))∫B(x,r)[w(⋅,ω)]−1p−1]−δ(p−1)⋅1δ⩽c[1ν(B(x,r))∫B(x,r)[w(⋅,ω)]δ]1δ⩽c[1ν(B(x,r))∫B(x,r)u(⋅,ω)]1δ⩽c′[u(x,ω)]1δ

for almost all with suitable constants and . Since , and are arbitrary, (2) implies that almost everywhere, so with some appropriate constants and . Thus is bounded in with an appropriate estimate of the norm, and so lattice is suitably -regular.

###### Proposition 8.

Let be a normed lattice on such that is norming for . Suppose that both and are -regular. Then both and are -regular.

Indeed, since and are -regular, they are also -regular with some . By Proposition 5 both and are then -regular, and it remains to apply Theorem 7 to and to with .

## 3. Divisibility of Ap-regularity

It is often convenient to think about Muckenhoupt weights in terms of the Jones factorization theorem (see, e. g., [10, Chapter 5, §5.3]: if and only if with some weights with the appropriate estimates on the constants. This makes it intuitive that, for example, division by the weights turns weights into weights, which is the main insight behind the divisibility theorem for -regularity [6, Theorem 2]: under certain assumptions on Banach lattices and , if lattice is -regular and lattice is -regular then lattice is -regular.

However, in the present work a somewhat more general problem arises: we need to make sure that a lattice admits majorants such that based on the assumption that lattice is -regular with an -regular lattice and some . With that in mind we introduce the following notions; see also [8, §1].

###### Definition 9.

Let . We say that a weight on belongs to class with a constant if there exist two weights with constant such that .

###### Definition 10.

Let , and suppose that is a quasi-normed lattice on . We say that is -regular with constants if for any there exists a majorant , such that and with constant .

” in the notation stands for “factorizable weight”, and the properties of the weights imply that at least in the local terms roughly represents the “poles” of the weight where the weight takes relatively large values, whereas represents the “zeroes” of where the weight is relatively small. The corresponding factorization is generally not unique.

Since implies for , we see that for and . Likewise, -regularity of a lattice implies its -regularity.

It is easy to see that these properties are closely related to -regularity.

###### Proposition 11.

Suppose that , and is a weight on . Then if and only if , and if and only if with the appropriate estimates on the constants.

Indeed, it suffices to observe that for .

Proposition 11 yields at once the corresponding result for -regularity.

###### Proposition 12.

Let be a quasi-normed lattice on , and suppose that , . Lattice is -regular if and only if lattice is -regular.

Incidentally, as a corollary we get yet another characterization of the property and the corresponding -regularity in terms of with some and and, respectively, -regularity of lattice .

Notation allows convenient computations for exponents and products of weights. The following property is immediate from the definitions.

###### Proposition 13.

Suppose that and . Then with the same constants. If almost everywhere then with the same constants. If a lattice is -regular and then lattice is -regular with the same constants.

###### Proposition 14.

Suppose that , and . then with the appropriate estimates on the constants. Likewise, if and are some lattices on such that is -regular and is -regular then lattice is -regular with the appropriate estimates on the constants.

Indeed, since the sets of weights with constant at most are logarithmically convex (see (8) below), it is easy to see that if and with some appropriate then

 w0w1=(ωα0α0+α100ωα1α0+α110)α0+α1⎛⎝ωβ0β0+β101ωβ1β0+β111⎞⎠β0+β1∈Fα0+α1β0+β1

with an appropriate estimate for the constant.

It is remarkable that the statement of Proposition 14 can be reversed not only for weights but also for lattices. The following result is a generalization of [6, Theorem 2]; in the proof of Theorem 2 in Section 5 below it is applied with , , ,

###### Theorem 15.

Suppose that and are quasi-Banach lattices on satisfying the Fatou property, is -regular and is -regular. Then lattice is -regular.

Examining the case of weighted lattices with suitable weights shows that the conclusion of Theorem 15 is sharp in the sense that the indexes of regularity cannot be replaced by smaller values.

A complete proof of theorem 15 is given in Section 6 below. A weaker statement can be obtained directly from [6, Theorem 2]; however, the resulting indexes of regularity are too crude for our purposes. However, we may deduce the case needed in the present work from the following recently obtained result, which seems to be somewhat less involved technically than the proof of Theorem 15 in full generality that, among other things, uses a fixed point theorem.

###### Theorem 16 ([8, Theorem 14]).

Suppose that is a Banach lattice of measurable functions on satisfying the Fatou property and , . Then is -regular if and only if is -regular.

Indeed, suppose that under the conditions of Theorem 15 both lattices and are -convex with some such that and ; these conditions are satisfied in the application to the proof of Theorem 2 in Section 5 below with some sufficiently close to value of . Then lattice is -regular and lattice is -regular by Proposition 13, so lattice is -regular by Theorem 16, thus lattice is -regular by Proposition 13. By the Lozanovsky factorization theorem [4] we have , and lattice is -regular by Proposition 14, which by Theorem 16 implies that lattice is -regular, and thus lattice is -regular by Proposition 13. Applying Theorem 16 to lattice yields -regularity of lattice , which by Proposition 13 implies the required -regularity of lattice .

## 4. An estimate for nondegenerate operators

It is well known that if is a nondegenerate operator in the sense of Definition 1 then the boundedness of in implies that . However, in quantitative terms the standard argument establishing this (see, e. g., [10, Chapter 5, §4.6]) only yields an estimate , which is too rough for the proof of Theorem 2 in Section 5 below to work in full generality. The value  cannot be estimated in terms of ; see [2, §8.B].

Nevertheless, securing an additional restriction on the doubling constant of either the weight or the weight leads to a suitable estimate. We denote by the Legbesgue measure on .

###### Proposition 17.

Suppose that is a nondegenerate operator that is bounded in with a weight on such that either or satisfies the doubling condition with a constant . Then

 (3) [w]A2⩽cTCw∥T∥2L2(w−12)→L2(w−12)

with a constant independent of the weight and a constant depending only on .

Indeed, let under the assupmtions of Proposition 17. The argument in [6, Proposition 19] shows that

 (4) ∫|Tf(⋅)|2w(⋅,ω)⩽2m2∫|f(⋅)|2w(⋅,ω)

for almost all and all .

Suppose that is a ball in and let with and taken from the definition of a nondegenerate operator (Definition 1) as applied to . It is easy to see that the boundedness of implies that both and are locally summable for almost all . Substituting the condition (1) from the definition of a nondegenerate operator into (4), we see that

 (5) 2m2∫Bf2(⋅)w(⋅,ω)⩾∫|Tf(⋅)|2w(⋅,ω)⩾∫B′|Tf(⋅)|2w(⋅,ω)⩾c2(1|B|∫Bf)2∫B′w(⋅,ω)

for almost all and all such that and . Putting into (5) yields

 (6) (1|B|∫B′w(⋅,ω))(1|B|∫Bw−1(⋅,ω))⩽2c−2m2.

Since the balls and are mutually comparable in the sense that and , the doubling condition of either the weight or the weight implies that one of the factors on the left-hand side of (6) is suitably comparable to a similar factor with either replaced by or vice versa. This observation yields (3), since both and are arbitrary balls of .

We apply Proposition 17 to the situation arising in Theorem 3.

###### Proposition 18.

Suppose that is a Banach lattice on with an order continuous norm, and let be a nondegenerate operator acting boundedly in . Suppose also that lattice is -regular with some . Then for every there exists a majorant , , such that

 (7) [w]A2⩽C2∥T∥2Y12→Y12

with some constants independent of and .

To prove Proposition 18 we need to show that it is possible to take weights in the conclusion of Theorem 3 that also satisfy with a suitable control on the norm. To do this we adapt the fixed point argument from the proof of [7, Theorem 8]. This requires a few preparations.

We introduce the following sets of Muckenhoupt weights for :

 BA(MC)p(C)={w∈Ap∣[w]Ap⩽C},
 (8) BA1(C)={w∈A1∣esssupMww⩽C}.

Here “” denotes “the ball of ”, and “” indicates that these sets are defined by the Muckenhoupt condition to avoid confusion with earlier work (e. g. [6, Section 3]), where different (for ) sets were used. The latter have the advantage of being convex and they can also be used to establish the results of the present work; however, we do not need the convexity, and the basic facts about sets seem to be simpler. Such a definition is more in line with the rest of the arguments.

###### Proposition 19.

Sets are logarithmically convex and closed with respect to the convergence in measure.

Indeed, the logarithmic convexity follows at once from the Hölder inequality, and the closedness with respect to the convergence in measure is obtained by twice applying the Fatou lemma: if and almost everywhere then

 1ν(B)∫Bw(⋅,ω)⩽liminfn1ν(B)∫Bwn(⋅,ω)⩽Climinfn(1ν(B)∫Bw−1p−1n(⋅,ω))−(p−1)=C(limsupn1ν(B)∫Bw−1p−1n(⋅,ω))−(p−1)⩽C(liminfn1ν(B)∫Bw−1p−1n(⋅,ω))−(p−1)⩽C(1ν(B)∫Bw−1p−1(⋅,ω))−(p−1)

for all balls and almost all , so .

According to Proposition 11, we can define for , the corresponding sets of weights with a control on the constant by

 BFαβ(C)={wα∣w∈BA(MC)βα+1(C)},
 BF0β(C)={w−β∣w∈BA1(C), w>0 almost everywhere}.

Consequently, these sets are also logarithmically convex and closed with respect to the convergence in measure.

###### Proposition 20.

Suppose that is a Banach111It is easy to see that Proposition 20 also holds true for quasi-normed lattices . lattice on a -finite measurable space, , almost everywhere, is a bounded set in such that for all . Then there exists some weight , almost everywhere, such that is a bounded set in .

To prove Proposition 20, take any such that and almost everywhere, any such that and almost everywhere, and define a weight . Then implies

 ∫w⩾1|logw|2ω⩽∫w⩾1|logw|2a=∫w⩾14∣∣log(w12)∣∣2a⩽4∫wa⩽4∥w∥Z∥a∥Z′⩽4∥w∥Z

and

 ∫w<1|logw|2ω=∫w<1(−logw)2ω⩽∫(−logω1)2ω=∫([logω1]−)2ω⩽∫([logw]−)2σ(1−[logω1]−)−2⩽∫σ=1,

so indeed is a bounded set in .

We now begin the proof of Proposition 18. For convenience, let ; lattice always has the Fatou property. Let be the constant from Theorem 3. We introduce a set

 BT={w∈X∣w⩾0,∫|Tg|2w⩽(C∥T∥Y12→Y12)2∫|g|2w for all g∈L2(w−12)}.

Theorem 3 shows that this set is nonempty. By the complex interpolation is logarithmically convex. The closedness of the set with respect to the convergence in measure is verified routinely (see, e. g., the proof of [6, Proposition 16]): if and almost everywhere then we put and see that by the Fatou lemma and the Lebesgue dominated convergence theorem

 (9)

for all , so extending (9) to all by density yields .

Suppose that . We may assume that and almost everywhere. By the assumptions lattice is -regular with some constants . Let be a sufficiently small number to be determined later. We introduce a set and a set-valued map by

for all with a sufficiently large constant to be determined in a moment.

Let . Then with . Applying Theorem 3 to function yields a majorant , , such that . On the other hand, by the -regularity of there exists some majorant , , such that