# On some sharp weighted norm inequalities On some sharp weighted norm inequalities

A.K. Lerner / Journal of Functional Analysis 232 (2006) 477 – 494 481and (2.3) such that∫Tf (x) = K(x,y)f(y)dyR nfor any f ∈ C ∞ 0 (Rn ) and x/∈ supp(f ).2.3. Littlewood–Paley operatorsLet φ ∈ C0 ∞(Rn ) with ∫ φ = 0. Write φ t (y) = t −n φ(y/t). The area integral S(f)and the Littlewood–Paley function gμ ∗ (f ) are defined byand(∫ ∞ ∫S(f )(x) =g ∗ μ (f )(x) = ( ∫ ∫R n+1+where R n+1+ = Rn × R + .0{y:|y−x| 1),Given a weight ω, set ω(E) = ∫ Eω(x) dx. The non-increasing rearrangement of ameasurable function f with respect to a weight ω is defined by (cf. [5, p. 32])f ∗ ω (t) =supω(E)=tinf |f(x)| (0x∈E

482 A.K. Lerner / Journal of Functional Analysis 232 (2006) 477–494Roughly speaking, f # is the Hardy–Littlewood maximal operator of M λ # f (see [13,17]):c 1 MM # λ f(x)f # (x)c 2 MM # λ f(x).In [13,16], the following version of the Fefferman–Stein inequality was established:‖f ‖ Lpωc‖M # λ f ‖ L p ω , (3.1)where 0

A.K. Lerner / Journal of Functional Analysis 232 (2006) 477 – 494 483Given a measurable function f, define the local maximal function m λ f bym λ f(x)= sup(f χ Q ) ∗ (λ|Q|) (0 < λ < 1).Q∋xThe following propositions are proved in  (see Propositions 2.1 and 4.2, andTheorem 5.4 there).Proposition 3.2. Let ω be any weight such that ω(R n ) =+∞. Then f ∗ ω (+∞) = 0iff the distribution function μ f,ω (α) = ω{x :|f(x)| > α} is finite for any α > 0.Proposition 3.3. For any locally integrable f and for all x ∈ R n ,Mf (x)3f # (x) + m 1/2 f(x), (3.4)f # (x)8MM # λ nf(x) (3.5)andM # λ (m 1/2f )(x)4M # λ/2·9 n f(x). (3.6)Proposition 3.4. For any weight ω such that ω(R n ) =+∞and for any measurablefunction f with fω ∗ (+∞) = 0,‖f ‖ L r ωc r,n ‖(M # λ nf )(P λ′nω)‖ L r ω(0

A.K. Lerner / Journal of Functional Analysis 232 (2006) 477 – 494 4854. Pointwise estimates for Calderón–Zygmund and Littlewood–Paley operatorsGiven 0 < δ < 1, consider the maximal function f # δdefined byf # δ( ∫ 11/δ(x) = sup inf |f(y)− c| dy) δ .Q∋xc |Q| QBy Chebyshev’s inequality, it is easy to see thatM λ # f(x)(1/λ)1/δ f δ # (x). (4.1)In [2, Theorem 2.1], Alvarez and Pérez proved that for any Calderón–Zygmundoperator T and for any f ∈ C ∞ 0 (Rn ),(Tf ) # δ (x)c δ,nMf (x) (0 < δ < 1).From this and from (4.1) we have the following.Proposition 4.1. For any Calderón–Zygmund operator T and for any f ∈ C ∞ 0 (Rn ),M # λ (Tf )(x)c λ,nMf (x).We note that for specific classes of Calderón–Zygmund operators this proposition iscontained in [13,16].For the Littlewood–Paley function gμ ∗ (f ), it was proved by Cruz-Uribe and Pérez that for any f ∈ C0 ∞(Rn ),(g ∗ μ (f ))# δ (x)c δ,nMf (x) (0 < δ < 1, μ > 2).Therefore, a full analogue of Proposition 4.1 holds for gμ ∗ (f ). However, we will showthat a more precise result holds, although our proof works in the case μ > 3. Notealso that our approach is different from the one of .Proposition 4.2. Let μ > 3. Then for any f ∈ C ∞ 0and for all x ∈ R n ,M # λ (g∗ μ (f )2 )(x)cMf (x) 2 , (4.2)where c depends on λ, μ and n.

A.K. Lerner / Journal of Functional Analysis 232 (2006) 477 – 494 487c∞∑( ∫1 12 k |2 k+1 Q|k=16·2 k Q|f |) 2cMf (x) 2 .Combining this estimate with (4.4) yields() ∗inf (g ∗ cμ (f ( ) ∗ )2 − c)χ Q (λ|Q|) (I1 + I 2 − I 2 (z 0 ))χ Q (λ|Q|) (I 1 ) ∗ (λ|Q|) + cMf (x) 2 cMf (x) 2 ,which proves the desired result.□5. Proof of main resultsFirst, we show how to deduce the estimates contained in the statements of Theorems1.1 and 1.2, and then we shall discuss the sharpness of exponents α p and β p .Observe that the L p ω boundedness of the Calderón–Zygmund and Littlewood–Paleyoperators when ω ∈ A p is well known (see, e.g., [15,22]). Therefore, assuming that‖f ‖ p MLωis finite, we clearly obtain that (Tf ) ∗ ω (+∞) = 0, where T is any one of theoperators appearing in Theorems 1.1 and 1.2.Suppose now that T is a Calderón–Zygmund operator. Letting Tf instead of f in (3.3)with q = 1 and applying Proposition 4.1, we immediately obtain‖Tf‖ MLpωc p,n ‖ω‖ α pA p‖f ‖ MLpω(1

488 A.K. Lerner / Journal of Functional Analysis 232 (2006) 477–494the exponents 1 and2 1 are best possible. This follows easily from general observationsby an argument of Fefferman and Pipher .Suppose, for example, that for some p 0 2 we have (5.1) but with ‖ω‖ Ap0 replacedby φ(‖ω‖ Ap0 ), where φ is a non-decreasing function such that φ(t)/t → 0 ast →+∞. Then we clearly obtain‖Tf‖ Lp 0ω cφ(‖ω‖ A 1)‖f ‖ MLp 0ω . (5.3)From this, arguing exactly as in [10, pp. 356–357], i.e., using the Rubio de Franciaalgorithm and the duality, we get‖T ‖ L p c 1 φ(c 2 p) as p →∞. (5.4)But it is well known that ‖T ‖ L p = O(p), and this is sharp, in general. Thus, we haveobtained a contradiction which shows that the exponent 1 in (5.1) is sharp.For the sake of completeness, we outline briefly how (5.4) follows from (5.3). Letp>p 0 , and ψ0, ‖ψ‖ L (p/p 0 )′ = 1. Form the operatorRψ = ψ +∞∑k=1Then ‖Rψ‖ L (p/p 0 ) ′ 2 and ‖Rψ‖ A 12‖M‖ L (p/p 0 ) ′(5.3) and Hölder’s inequality,M k ψ(2‖M‖ L (p/p 0 ) ′ )k .= O(p) as p →∞. Therefore, by∫∫∫|Tf|p 0ψ |Tf|p 0Rψcφ(‖Rψ‖ A1 ) p 0R n R n R n(Mf )p 0Rψ 2cφ(c ′ p) p 0‖Mf ‖ p 0L p c ′′ φ(c ′ p) p 0‖f ‖ p 0L p .Taking the supremum over all ψ with ‖ψ‖ L (p/p 0 )′ = 1 yields (5.4).Exactly the same observations show that in (5.2), ‖ω‖ 1/2A p0,p 0 3, cannot be replacedby φ(‖ω‖ Ap0 ) with φ(t)/ √ t → 0ast →+∞, because it is well known that ‖S‖ L p =O( √ p), and this is sharp, in general (see, e.g., ).It remains to show that the exponents α p and β p are sharp in the cases 1

A.K. Lerner / Journal of Functional Analysis 232 (2006) 477 – 494 489Then ‖ω‖ Ap ∼ δ 1−p ,Mf (x) ∼ 1 )(|x| δ−1 χδ{|x| 2).Next, simple calculations show thatandTherefore,‖Mf ‖ Lpωcδ 1+1/p‖M(Hf)‖ p Lω c (∫ ∞(log x/x) p x (p−1)(1−δ) dxδ1(∫ ∞) 1/p= c (log x/x) p 1dx1δ 2+1/p .) 1/p‖Hf ‖ MLpω‖f ‖ MLpω c δ c‖ω‖1/(p−1) A p.This shows that the exponent α p is sharp for 1

490 A.K. Lerner / Journal of Functional Analysis 232 (2006) 477–494Remark 5.1. In , pointwise estimates by means of f δ # for a large class of operatorshave been obtained. As a result, a full analogue of Proposition 4.1 holds for weaklystrongly singular integral operators by Fefferman, for pseudo-differential operators inthe Hörmander class, for oscillatory integral operators introduced by Phong and Stein.The proof of Theorem 1.1 shows that for all these operators, their ML p ω operator normsare bounded by a multiple of ‖ω‖ α pA p, 1

A.K. Lerner / Journal of Functional Analysis 232 (2006) 477 – 494 491The third estimate of Proposition 3.3 is based on a quite standard technique. Namely,using the fact that for two intersecting cubes Q 1 and Q 2 we have either Q 1 ⊂ 3Q 2or Q 2 ⊂ 3Q 1 , one can show that for all x,y ∈ Q,m 1/2 f(y)m 1/2((f − mf (3Q) ) χ 3Q)(y) + 2M#1/2·3 nf(x)+ infQ m 1/2f.Next, (3.9) in the unweighted case gives(m 1/2 f) ∗ (t)f ∗ (t/2 · 3 n ). (A.2)Combining the last two inequalities yields easily (3.6).□Proof of Proposition 3.4 (Sketch). The proof of Proposition 3.4 is more complicated.We define the weighted centered versions of M # λ f and m λf by˜M # λ,ωf(x)= supQ∋x( ) ∗inf (f − c)χQcω (λω(Q))and( ) ∗˜m λ,ω f(x)= sup f χQ (λω(Q)) (0 < λ < 1),ωQ∋xwhere the supremum is taken over all cubes centered at x. The main tool used inProposition 3.4 is the following rearrangement inequality which holds for arbitraryweights ω (see [19, Theorem 1.2]):f ∗ ω (t)2( ˜M # λ n ,ω f)∗ ω (t/2) + f ∗ ω (2t) (0

492 A.K. Lerner / Journal of Functional Analysis 232 (2006) 477–494Next, the most technical part of the proof is the following pointwise estimate:)˜M λ,ω # f(x)c n˜m λ/4,ω(M λ # nfP λ/2 ω (x).(A.5)Observe that (A.5) along with (A.4) implies almost immediately the desired result.Indeed, exactly as in proving (A.2), one can show (using the weighted weak type(1, 1) property of the weighted centered Hardy–Littlewood maximal operator) that(˜m λ,ω f) ∗ ω (t)f ∗ ω (λt/c n).Therefore, the operator ˜m λ,ω is bounded on L r ω for any r>0 with the operator normsnot depending on ω. Combining this fact with (A.5) and (A.4) gives the statement ofProposition 3.4.The proof of (A.5) is based on several ingredients. The first one [19, Lemma 5.2]says that for any closed set F ⊂ Q with |F | > 0 there is a function g such that f = gon F and‖M 1/2;Q # g‖ ∞ sup M λ # n ;Q f(x)x∈F(A.6)(one can take λ n = 1/100 n ). Here, Mλ,Q # f denotes the local sharp maximal functionrestricted to a cube Q. Note that this result is close in spirit to characterizing theE-functional for (L 0 , BMO) [13, Theorem 3.2]. The second ingredient is the John–Strömberg theorem [14,27] saying that for any cube Q ⊂ Q 0 ,((f − mf (Q) ) χ Q) ∗ (t)cn ‖M # 1/2;Q 0f ‖ ∞ log 2|Q|t(0

A.K. Lerner / Journal of Functional Analysis 232 (2006) 477 – 494 493( ) ∗ (g − mg (Q))χ Q (|Eλ |). Applying to this term (A.7) and then (A.6), we get( )( ) ∗inf (f − c)χQcω (λω(Q)) c n sup M λ # n ;Q f log 2|Q|F|E λ |) ∗ c n(M λ # n ;Q f (λω(Q)/4) inf P λ/2ωω Q c n((M # λ nfP λ/2 ω)χ Q) ∗ω (λω(Q)/4).This implies (A.5), and therefore the proof is complete.□References J. Alvarez, J. Hounie, C. Pérez, A pointwise estimate for the kernel of a pseudo-differential operator,with applications, Rev. Un. Mat. Argentina 37 (3–4) (1991) 184–199. J. Alvarez, C. Pérez, Estimates with A ∞ weights for various singular integral operators, Boll. Un.Mat. Ital. A (7) 8 (1) (1994) 123–133. S.M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities,Trans. Amer. Math. Soc. 340 (1) (1993) 253–272. D.L. Burkholder, Martingale theory and harmonic analysis in Euclidean spaces, Proc. Sympos. PureMath. 35 (1979) 283–301. K.M. Chong, N.M. Rice, Equimeasurable rearrangements of functions, Queen’s Papers in Pure andApplied Mathematics, vol. 28, Queen’s University, Kingston, Ont., 1971. R.R. Coifman, C. Fefferman, Weighted norm inequalities for maximal functions and singularintegrals, Studia Math. 15 (1974) 241–250. D. Cruz-Uribe, C. Pérez, On the two-weight problem for singular integral operators, Ann. ScuolaNorm. Sup. Pisa Cl. Sci. (5) 1 (4) (2002) 821–849. O. Dragičević, L. Grafakos, M.C. Pereyra, S. Petermichl, Extrapolation and sharp norm estimatesfor classical operators on weighted Lebesgue spaces, Publ. Math. 49 (1) (2005) 73–91. O. Dragičević, A. Volberg, Sharp estimate of the Ahlfors–Beurling operator via averaging martingaletransforms, Michigan Math. J. 51 (2) (2003) 415–435. R. Fefferman, J. Pipher, Multiparameter operators and sharp weighted inequalities, Amer. J. Math.119 (2) (1997) 337–369. C. Fefferman, E.M. Stein, H p spaces of several variables, Acta Math. 129 (1972) 137–193. S. Hukovic, S. Treil, A. Volberg, The Bellman functions and sharp weighted inequalities for squarefunctions, in: Complex Analysis, Operators, and Related Topics, Operator Theory Advances andApplications, vol. 113, Birkhäuser, Basel, 2000, pp. 97–113. B. Jawerth, A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985) 231–270. F. John, Quasi-isometric mappings, Seminari 1962–1963 di Analisi, Algebra, Geometria e Topologia,Rome, 1965. J.-L. Journé, Calderón–Zygmund Operators, Pseudo-differential Operators and the Cauchy Integralof Calderón, Lecture Notes in Mathematics, vol. 994, 1983. A.K. Lerner, On weighted estimates of non-increasing rearrangements, East J. Approx. 4 (1998)277–290. A.K. Lerner, On the John-Strömberg characterization of BMO for nondoubling measures, Real.Anal. Exchange 28 (2) (2003) 649–660. A.K. Lerner, Weighted norm inequalities for the local sharp maximal function, J. Fourier Anal.Appl. 10 (5) (2004) 465–474. A.K. Lerner, Weighted rearrangement inequalities for local sharp maximal functions, Trans. Amer.Math. Soc. 357 (6) (2005) 2445–2465.

494 A.K. Lerner / Journal of Functional Analysis 232 (2006) 477–494 B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math.Soc. 165 (1972) 207–226. B. Muckenhoupt, Weighted norm inequalities for classical operators, Proc. Sympos. Pure Math. 35(1) (1979) 69–83. B. Muckenhoupt, R.L. Wheeden, Norm inequalities for the Littlewood–Paley function g ∗ λ , Trans.Amer. Math. Soc. 191 (1974) 95–111. S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in termsof the classical A p -characteristic, 2002, preprint. S. Petermichl, S. Pott, An estimate for weighted Hilbert transform via square functions, Trans.Amer. Math. Soc. 354 (4) (2002) 1699–1703. S. Petermichl, A. Volberg, Heating of the Ahlfors–Beurling operator: weakly quasiregular maps onthe plane are quasiregular, Duke Math. J. 112 (2) (2002) 281–305. S. Petermichl, J. Wittwer, A sharp estimate for the weighted Hilbert transform via Bellman functions,Michigan Math. J. 50 (1) (2002) 71–87. J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, IndianaUniv. Math. J. 28 (1979) 511–544. J.M. Wilson, Weighted inequalities for the dyadic square function without dyadic A ∞ , Duke Math.J. 55 (1987) 19–49. J.M. Wilson, Weighted norm inequalities for the continuous square functions, Trans. Amer. Math.Soc. 314 (1989) 661–692. J. Wittwer, A sharp estimate on the norm of the martingale transform, Math. Res. Lett. 7 (1)(2000) 1–12. J. Wittwer, A sharp estimate on the norm of the continuous square function, Proc. Amer. Math.Soc. 130 (8) (2002) 2335–2342.