Weighted norm inequalities for singular integral operators

WEIGHTED NORM INEQUALITIES FOR SINGULAR INTEGRAL OPERATORS C. P´erez Departmento de Matem´aticas Universidad Aut´onoma de Madrid 28049 Madrid, Spain Abstract For a Calder´on–Zygmund singular integral operator T, we show that the following weighted inequality holds ZRn |Tf(y)|pw(y)dy ≤CZRn |f(y)|pM[p]+1w(y)dy, where Mkis the Hardy–Littlewood maximal operator Miterated ktimes, and [p]is the integer part of p. Moreover, the result is sharp since it does not hold for M[p]. We also give the following endpoint result: w({y∈Rn:|Tf(y)|> λ})≤C λZRn |f(y)|M2w(y)dy. 1 1 Introduction and statements of the results 2 1 Introduction and statements of the results A classical result due to C. Fefferman and E. Stein [4] states that the Hardy– Littlewood maximal operator Msatisfies the following inequality for arbitrary 1 < p < ∞, and weight w ZRn |Mf(y)|pw(y)dy ≤CZRn |f(y)|pMw(y)dy, (1) where Cis independent of f. A weight win Rnwill always be a nonnegative locally integrable function. The study of weighted inequalities like the above, for other operators has played a central rˆole in modern of Harmonic Analysis since they appear in duality arguments. We refer the reader to [5] Chapters 5 and 6 for a very nice exposition. Although we could work with any Calder´on–Zygmund operator (cf. §3), we shall only consider singular integral operators of convolution type defined by: Tf(x) = p.v. ZRn k(x−y)f(y)dy, where the kernel kis C1away from the origin, has mean value on the unit sphere centered at the origin and satisfies for y6= 0 |k(y)| ≤ C |y|nand |∇k(y)| ≤ C |y|n+1 . It is well known that the analogous version of inequality (1) fails for the Hilbert transform for all p. In [3] A. C´ordoba and C. Fefferman have shown that there is a similar inequality for any T, but with Mw replaced by the pointwise larger operator Mrw=M(wr)1/r,r > 1, that is, for 1 <p<∞ ZRn |Tf(y)|pw(y)dy ≤CZRn |f(y)|pMrw(y)dy, (2) with Cindependent of f. The purpose of this paper is to prove weighted norm inequalities of the form (2), where Mrw,r > 1, will be replaced by appropiate smaller maximal–type operators w→Nw satisfying Mw(x)≤Nw(x)≤C Mrw(x),(3) 1 Introduction and statements of the results 3 for each x∈Rn. We shall also be concern with corresponding endpoints results such as weak type (1,1) and H1–L1estimates. Before stating our main results, we shall make the following observation. Let Mkbe the Hardy–Littlewood maximal operator Miterated ktimes, where k= 1,2,· · · . We claim that for k= 2,· · · , and r > 1, there exists a positive constant C independent of wsuch that Mw(x)≤Mkw(x)≤C Mrw(x),(4) for each x∈Rn. The left inequality follows from the Lebesgue differentiation theorem; for the other, we let Bbe the best constant in Coifman’s estimate M(Mrw)≤ B Mrw, where Bis independent of w. Then, it follows easily that Mkw≤Bk−1Mrw, k= 1,2,· · · . In view of this observation, it is natural to consider whether or not (2) holds for some Mk, with k= 2,3,· · · . In a very interesting paper [8], M. Wilson has recently obtained the following partial answer to this question: Let 1 <p<2, then ZRn |Tf(y)|pw(y)dy ≤CZRn |f(y)|pM2w(y)dy. (5) Moreover, he shows that this estimate does not hold for p≥2, and also that when p= 2, M2wcan be replaced by M3w. However, his method does not yield corresponding estimates for p > 2 (cf. §3 of that paper), and M2wmust be replaced by a much more complicated expression. M. Wilson’s approach to this problem is based on certain (difficult) estimates for square functions that he obtained in the same paper, together with a couple of related estimates for the area function, obtained essentially by S. Chanillo and R. Wheeden in [1]. In this paper we give a complete answer to Wilson’s problem by means of a different method. Our main result is the following. Theorem 1.1: Let 1 <p<∞, and let Tbe a singular integral operator. Then, there exists a constant Csuch that for each weight w ZRn |Tf(y)|pw(y)dy ≤CZRn |f(y)|pM[p]+1w(y)dy, (6) where [p] is the integer part of p. Furthermore, the result is sharp since it does not hold for M[p]. The corresponding weak–type (1,1) version of this result is the following. 1 Introduction and statements of the results 4 Theorem 1.2: Let Tbe a singular integral operator. Then, there exists a constant Csuch that for each weight wand for all λ > 0 w({y∈Rn:|Tf(y)|> λ})≤C λZRn |f(y)|M2w(y)dy. (7) Remark 1.3: Let 1 < p < ∞, a natural question is whether (7) can be extended to the case (p, p), that is whether w({y∈Rn:|Tf(y)|> λ})≤C λpZRn |f(y)|pM[p]w(y)dy, holds for some constant Cand for all λ > 0. At the end of section 2 we give an example showing that this inequality is false when pis not an integer; however, we do not know what happens when pis an integer. Although we do not know whether (7) holds for Mw (cf. remark 1.7) we can give the following estimate. For a measure µwe shall denote by H1(µ) the subspace of L1(µ) of functions fwhich can be written as f=Pjλjaj, where ajare µ–atoms and λjare complex numbers with Pj|λj|<∞. A function ais a µ–atom if there is a cube Qfor which supp(a)⊂Q, so that |a(x)| ≤ 1 µ(Q), and ZQ a(y)dy = 0. Theorem 1.4: Let Tbe a singular integral operator. Then, there exists a constant Csuch that for each weight w ZRn |Tf(y)|w(y)dy ≤CkfkH1(Mw).(8) Theorem 1.1 is in fact a consequence of a more precise estimate than (6). The idea is to replace the operator M[p]+1 by an optimal class of maximal operators. We explain now what “optimal” means. We want to define a scale of maximal–type operators w→MAwsuch that Mw(x)≤MAw(x)≤Mrw(x) 1 Introduction and statements of the results 5 for each x∈Rn, where r > 1. Astands for a Young function; i.e. A: [0,∞)→[0,∞) is continuous, convex and increasing satisfying A(0) = 0. To define MAwe introduce for each cube Qthe A–average of a function fover Qby means of the following Luxemburg norm kfkA,Q = inf{λ > 0 : 1 |Q|ZQ A|f(y)| λdy ≤1}. We define the maximal operator MAby MAf(x) = sup x∈Q kfkA,Q , where fis a locally integrable functions, and where the supremum is taken over all the cubes containing x. When A(t) = trwe get MA=Mr, but more interesting examples are provided by Young functions like A(t) = tlog(1 + t),  > 0. The optimal class of Young functions Ais characterized by the following theorem. Theorem 1.5: Let 1 < p < ∞, and let Tbe a singular integral operator. Suppose that Ais a Young function satisfying the condition Z∞ ct A(t)p0−1dt t<∞,(9) for some c > 0. Then, there exists a constant Csuch that for each weight w ZRn |Tf(y)|pw(y)dy ≤CZRn |f(y)|pMAw(y)dy. (10) Furthermore, condition (9) is also necessary for (10) to hold for all the Riesz transforms: T=R1, R2,· · · , Rn. We recall that the j–th Riesz transform Rj,j= 1,2,· · · , n, is the singular integral operator defined by Rjf(x) = p.v. ZRn xj−yj |x−y|n+1 f(y)dy. The proof of this theorem is given in §2, and it is based on the following inequality of E.M. Stein [7] ZQ w(y) logk(1 + w(y)) dy ≤CZQ Mw(y) logk−1(1 + Mw(y)) dy, (11) 2 Proof of the Theorems 6 with k= 1,2,3,· · · . As for the strong case, there is an estimate sharper than (7). Theorem 1.6: Let Tbe a singular integral operator. For arbitrary  > 0, consider the Young function A(t) = tlog(1 + t).(12) Then, there exists a constant Csuch that for each weight wand for all λ > 0 w({y∈Rn:|Tf(y)|> λ})≤C λZRn |f(y)|MAw(y)dy. (13) Remark 1.7: For 1 <p<∞let us denote by Bpthe callection of all Young functions Asatisfying condition (9): Z∞ ct A(t)p0−1dt t<∞, for some c > 0. Observe that Bp⊂Bq, 1 < p < q < ∞. Then it follows easily from the proof of last theorem that we may replace Aby any Young function belonging to the smallest class ∩p>1Bp. We could consider for instance A(t) = tlog(1 + t)[log log(1 + t)].(14) If we let = 0 in (12) MA0=Mis the Hardy–Littlewood maximal operator. Since A0does not belong to ∩p>1Bpwe think that the estimate: w({y∈Rn:|Tf(y)|> λ})≤C λZRn |f(y)|Mw(y)dy, (15) for some constant C, and for all λ > 0, does not hold. 2 Proof of the Theorems Proof of Theorem 1.5: We prove first that condition (9) is sufficient for (10) to hold for any singular integral operator T. We may assume that MAwis finite almost everywhere, and we let T∗be the adjoint operator of T.T∗is also a singular integral operator with kernel k∗(x) = k(−x). Then, by duality (10) is equivalent to 2 Proof of the Theorems 7 ZRn |T∗f(y)|p0MAw(y)1−p0dy ≤CZRn |f(y)|p0w(y)1−p0dy. (16) We shall be using some well known facts about the Aptheory of weights for which we remit the reader to [5] Chapter 4. To prove (16) we shall use the following fundamental estimate due to Coifman ([2]): Let Tbe any singular integral operator; then for each 0 <p<∞, and each u∈A∞, there exists C=Cu,p >0 such that for each f∈C∞ 0(Rn) ZRn |Tf(y)|pu(y)dy ≤CZRn Mf(y)pu(y)dy. (17) Therefore, to apply this estimate to T∗we need to show that (MAw)1−p0satisfies the A∞condition. To check this, we claim first that (MAw)δsatisfies the A1condition for 0 < δ < 1. However, this is an straightforward generalization of the well known fact that (Mw)δ∈A1, 0 < δ < 1, also due to Coifman (cf. [5] p. 158), and we shall omit its proof. Now, since w1−r∈Ar, for any w∈A1and r > 1, we have that (MAw)1−p0=h(MAw)p0−1 r−1i1−r ∈ ∩r>p0Ar⊂A∞. After these observations, we have reduced the problem to showing that ZRn Mf(y)p0MAw(y)1−p0dy ≤CZRn |f(y)|p0w(y)1−p0dy. (18) But this is a particular instance of the following characterization which can be found in [6] Theorem 4.4. Theorem 2.1: Let 1 < p < ∞. Let Abe a Young function, and denote B=A(tp0). Then the following are equivalent. i) Z∞ ct A(t)p−1dt t<∞; (19) ii) there is a constant csuch that ZRn MBf(y)pdy ≤cZRn f(y)pdy (20) 2 Proof of the Theorems 8 for all nonnegative, locally integrable functions f; iii) there is a constant csuch that ZRn MBf(y)pu(y)dy ≤cZRn f(y)pMu(y)dy (21) for all nonnegative, locally integrable functions fand u; iv) there is a constant csuch that ZRn Mf(y)pu(y) [MA(w)(y)]p−1dy ≤cZRn f(y)pMu(y) w(y)p−1dy, (22) for all nonnegative, locally integrable functions f,wand u. Observe that (18) follows from (22) by taking u= 1, and by replacing pby p0. Now we shall prove that condition (9) is also necessary for (10) to hold for all the Riesz transforms. That is, suppose that the Young function Ais fixed, and that the inequality ZRn |Tf(x)|pw(x)dx ≤CZRn |f(x)|pMAw(x)dx, (23) is verified for each Riesz transform T=Rj,j= 1,2,· · · , n. Fix one of these j. As above, by duality (23) is equivalent to ZRn |Rjf(x)|p0MAw(x)1−p0dx ≤CZRn |f(x)|p0w(x)1−p0dx, (24) We shall adapt an argument from [5] p. 561. We define the cone Ej={x∈Rn: max{|x1|,|x2|,· · · ,|xn|} =xj}, so that Rn=∪n j=1 (Ej∪(−Ej)). Let Bbe the unit ball, and consider the function f=w=χB∩(−Ej). Then, (24) implies ∞> C ZRn |f(x)|p0w(x)1−p0dx =C|B∩(−Ej)| ≥ ≥ZEj∩{|x|>2} |Rjf(x)|p0MAf(x)1−p0dx. 2 Proof of the Theorems 9 Observe that for |x|>2, MAf(x)≈A−1(|x|n)−1. Also, for every x∈Ej Rjf(x) = CZB∩(−Ej) xj−yj |x−y|n+1 dy ≥CZB∩(−Ej) 1 |x−y|ndy ≥C |x|n. Therefore ZEj∩{|x|>2} 1 |x|np0A−1(|x|n)p0−1dx ≤C|B∩(−Ej)|. A corresponding estimate can be proved for Ej, and for each j= 1,2,· · · , n, by using in each case the corresponding Riesz transform. Since the family of cones {+ −Ej}j=1,2,··· ,n is disjoint, we finally have that Z|x|>2 1 |x|np0A−1(|x|n)p0−1dx ≈Z∞ c 1 tp0A−1(t)p0−1tdt t≈ Z∞ ct A(t)p0−1dt t<∞, since tA0(t)≈A(t). This concludes the proof of the theorem. 2 Proof of Theorem 1.6: We shall assume that MAwis finite almost everywhere, since otherwise there is nothing to be proved. For f∈C∞ 0(Rn) we consider the standard Calder´on–Zygmund decomposition of fat level λ(cf. [5] p. 414). Let {Qj}be the Calder´on–Zygmund nonoverlapping dyadic cubes satisfying λ < 1 |Qj|ZQj |f(x)|dx ≤2nλ. (25) If we let Ω = ∪jQj, we also have that |f(x)| ≤ λa.e. x∈Rn\Ω. Using the notation fQj=1 |Qj|RQjf(x)dx, we write f=g+bwhere g, the “good part”, is given by g(x) = f(x)x∈Rn\Ω fQjx∈Qj Observe that |g(x)| ≤ 2nλa.e. 3 Calder´on–Zygmund operators 16 3 Calder´on–Zygmund operators In this section we shall state our main results for the more general Calder´on– Zygmund operators. We recall the definition of a Calder´on–Zygmund operator in Rn. A kernel on Rn×Rnwill be a locally integrable complex–valued fuction K, defined on Ω = Rn×Rn\diagonal. A kernel Kon Rnsatisfies the standard estimates, if there exist δ > 0 and C < ∞such that for all distinct x, y ∈Rnand all zsuch that |x−z|<|x−y|/2: (i) |K(x, y)| ≤ C|x−y|−n; (ii) |K(x, y)−K(z, y)| ≤ C|x−z| |x−y|δ |x−y|−n; (iii) |K(y, x)−K(y, z)| ≤ C|x−z| |x−y|δ |x−y|−n. We say that a linear and continuous operator T:C∞ 0(Rn)→ D0(Rn) is associated with a kernel K, if hTf, gi=ZRnZRn K(x, y)g(x)f(y)dxdy, whenever f, g ∈C∞ 0(Rn) with supp(f)∩supp(g) = ∅. We say that Tis a Calder´on–Zygmund operator if the associated kernel Ksatisfies the standard estimates, and if it extends to a bounded linear operator in L2(Rn). Theorem 3.1: Let 1 <p<∞, and let Tbe a Calder´on–Zygmund operator. Then, there exists a constant Csuch that for each weight w ZRn |Tf(y)|pw(y)dy ≤CZRn |f(y)|pM[p]+1w(y)dy, (32) and there exists another constant Csuch that for all λ > 0 w({y∈Rn:|Tf(y)|> λ})≤C λZRn |f(y)|M2w(y)dy. (33) The proof of Theorem 3.1 is essentially the same as Theorems 1.1 and 1.2, after observing that the adjoint T∗of any Calder´on–Zygmund operator Tis also a Calder´on–Zygmund operator with kernel K∗(x, y) = K(y, x). 3 Calder´on–Zygmund operators 17 There are corresponding results to Theorems 1.2, 1.4, 1.5, and for 1.6 for any Calder´on– Zygmund operator. We shall omit the obvious statements. Acknowledgements. I am very grateful to Prof. M. Wilson for sending me a preprint of his work [8], and also to Prof. S. Hofmann for interesting conversations concerning Wilson’s problem. References [1] S. Chanillo, R. L. Wheeden, Some weighted norm inequalities for the area integral, Indiana Univ. Math. J. 36 (1987), 277–294. [2] R. Coifman, Distribution function inequalities for singular integrals, Proc. Acad. Sci. U.S.A. 69 (1972), 2838–2839. [3] A. C´ordoba and C. Fefferman, A weighted norm inequality for singular integrals , Studia Math. 57 (1976), 97–101. [4] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. [5] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North Holland Math. Studies 116, North Holland, Amsterdam, (1985). [6] C. P´erez, On sufficient conditions for the boundedness of the Hardy– Littlewood maximal operator between weighted Lp–spaces with different weights, preprint (1990). [7] E. M. Stein, Note on the class Llog L, Studia Math. 32 (1969), 305–310. [8] J. M. Wilson, Weighted norm inequalities for the continuos square functions, Trans. Amer. Math. Soc. 314 (1989), 661–692.