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 [19] (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[7] 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 [7].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 **sharp**ness 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 [10].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., [4]).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 [1], 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 **norm**sare 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 un**weighted** 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 **norm**snot 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

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