Improved Muckenhoupt-Wheeden inequality and weighted inequalities for potential operators

Publicacions Matem`atiques, Vol 39 (1995), 23–41. IMPROVED MUCKENHOUPT-WHEEDEN INEQUALITY AND WEIGHTED INEQUALITIES FOR POTENTIAL OPERATORS Y. Rakotondratsimba Abstract By a variant of the standard good λinequality, we prove the Muckenhoupt-Wheeden inequality for measures which are not necessarily in the Muckenhoupt class. Moreover we can deal with a general potential operator, and consequently we obtain a suitable approach to the two weight inequality for such an operator when one of the weight functions satisfies a reverse doubling condition. 0. Introduction In this paper dµ,dω are locally finite positive Borel measures of Rn, n≥1. For a nonnegative locally-dµ integrable function K(x, y) (a.e. continuous in the first variable) we define the potential operator (Tfµ)(x)=y∈Rn K(x, y)f(y)dµ(y). For each C1>0, we assume the existence of a C2>0 so that (H)K(x, y)≤C2K(z,y),for each x, y, z with 0 <|z−y|<C 1|x−y|. The dual operator T∗is the operator defined by the kernel K∗(x, y)= K(y,z). The usual fractional integral operator Is, with 0 <s<n,is given by K(x, y)=|x−y|s−n. Other examples of operators Tare those introduced by Chanillo-Stromberg-Wheeden [Ch-St-Wh] and given by kernels K(x, y)=a(y,|x−y|) |x−y|n. Here ais considered as a function defined on balls of Rnand which satisfies some growth conditions we precise below. 24 Y. Rakotondratsimba We are interested in finding a constant C>0 for which (PT)TfµLq ω≤CfLp µfor all nonnegative functions f with 1 <p,q<∞. The constant Cdepends only on n,p,q,ω,µand K; and when it is necessary we denote this dependance by writing C= C(n, p, q, K, ω, µ). Here gLr ν=Rn |g|rdν1 r. The inequality (PT) includes the usual two weight norm inequality TfLq u≤CfLp v since it is sufficient to replace fby fv 1 p−1, and to take dω =udx,dµ = v−1 p−1dx, where dx is the usual Lebesgue measure on Rn. Inequality (PT) with T=Ishas been studied extensively by many authors (see for instance [Ke-Sa], [Sa-Wh] and [Pe] and the reference given by them). Kerman and Sawyer [Ke-Sa] solved the problem (PIs) with dω =dx. This particular case is first interesting since it is the usual form which appears in many mathematic and physic areas. It also appears that the case dω =dx is naturally suitable to be treated. In fact using a good λ-inequality, they proved that the left member of (PT) is majorized by the Lqnorm of the fractional maximal function. So (PT) is reduced to a weighted inequality for maximal operator, whose study was done by the first author [Sa]. Problem (PT) with general measures dµ and dω was solved by Sawyer and Wheeden [Sa-Wh]. Let us consider the operator T=Iswith 0 <s<n. We have the pointwise inequality Msg≤c(s, n)Isg where Msis the fractional maximal operator defined by (Msf)(x) = sup{|Q|s n−1f1I QL1(dy);Qa cube with Qx}. We generally use the letter Qto denote a cube of Rn, and by which we mean a product of nintervals [ai,a i+t](0<t<∞). The MuckenhouptWheeden inequality [Mu-Wh] yields a sort of converse (in norm) of the above inequality, and asserts that for 0 <q<∞: IsfLq ω≤c(s, n, q, ω)MsfLq ωfor all functions f whenever the measure dω satisfies the Muckenhoupt condition A∞, i.e. there are c=c(ω),δ >0 such as |E|ω |Q|ω ≤c|E| |Q|δ for all cubes Qand all measurables sets E⊂Q. Muckenhoupt-Wheeden inequality 25 In his thesis Perez [Pe] gave a weaker condition than the A∞’s. He proved the above Muckenhoupt-Wheeden inequality for measures dω satisfying D∞and Bρconditions with ρ>1−s n, and which can be noted as dω ∈D∞∩Bρ. These conditions respectively mean: |2Q|ω=2Q ω≤C(ω)|Q|ωfor all cubes Q, (here 2Qis the cube having the same center as Qand the length expanded twice) |Q|ω |Q|ω ≤C(ω)|Q| |Q|ρ for all cubes Q, Qwith Q⊂Q. Contrary to the Muckenhoupt-Wheeden technique [Mu-Wh], the Perez’s analysis [Pe] is not based on the standard good-λinequalities. This last author used some estimates obtained by Frazier and Jawerth [Fr-Ja] for local maximal operator, and moreover he was able to treat the problem with a general convolution operator. In this paper we also prove the Muckenhoupt-Wheeden inequality for measures which are not necessarily in the Muckenhoupt class (see Corollary 4), and with the general potential operator Tdescribed above. We do this, with a sort of a variant of the standard good λinequality and by introducing a suitable maximal operator MT,ω (see Theorem 1). The additional conditions on the measure dω arise only in order to relate this “exotic” maximal operator to a more standard one like Ms(see Theorem 3). Consequently we obtain a suitable approach to the two weight inequality for such an operator when one of the weight functions satisfies a reverse doubling condition (see Theorem 5). 1. Statements of results Let us define the dyadic maximal operator (Md T,ω,µf)(x) = sup{|Q|ω−1f(T∗1I Qω)1I3QL1(dµ) ; Qa dyadic cube with Qx}. A dyadic cube Qis a product of nintervals of the form [2kai,2k(ai+1)], where kand aiare integers. Fix q≥1. By using the Holder inequality we can observe that (Md T,ω,µf)(x) is a.e. finite for all bounded functions with compact supports whenever measures dω and dµ satisfy the condition (ST)T1I |x|<RµLq ω≤c(R)<∞for all R>0. Our first result is as follow: 26 Y. Rakotondratsimba Theorem 1. Let 0<q<∞and Kbe a nonnegative kernel satisfying the hypothesis H. Assume the measures dω and dµ satisfy the condition (ST). Then there is C=C(n, q, K)>0so that TfµLq ω≤CMd T,ω,µfLq ω. If Md ωis the dyadic maximal operator defined by (Md ωf)(x) = sup{|Q|ω−1f1I QL1(dω) ;Qa dyadic cube with Qx}, then (see Lemma 1 below) (Md T,ω,µg)(.)≤(Md ω(Tgµ))(.) and consequently we get Proposition 2. Let K,dω,dµ be as above Then for q>1we have TfµLq ω≈Md T,ω,µfµLq ω. Moreover this equivalence also holds for the range of q∈]0,1] whenever Md ω(Tg)(.)≤c(n, K, ω)(Tg)(.) for all f nonnegative functions g. The above equivalence means C1(n, q, K)TfµLq ω≤Md T,ω,µfµLq ω≤C2(n, q, ω)TfµLq ω. The extra assumption in this result is satisfied for instance for the kernel K(x, y)=|x−y|s−n, with dω =dx the Lebesgue measure, and more generally for measures dω ∈D∞∩Bρwith 1 −s n<ρ. Thus in view of Theorem 1, the inequality (PT) is reduced to the following one, related for Md T,ω,µ (˜ PT)Md T,ω,µfµLq ω≤CfLp µ. In order to get this last one, we impose more hypothesis on the kernel K. So as to simplify, we only deal with kernels K(x, y)=Ka(x, y)=a(y,|x−y|) |x−y|n=a(B(y,|x−y|)) |x−y|n Muckenhoupt-Wheeden inequality 27 where ais a function defined on balls satisfying the following hypotheses H: (i) a(B1)≤c1(n, a)a(B2) for all balls B1,B 2with B1⊂B2; there are λ, σ > 0 so that (ii) c 1(n, a)tnλ a(B)≤a(tB)≤c 2(n, a)tnσ a(B) for all balls Band t≥1. We also define the function aon cubes by a(Q)=a(B), where Bis the smallest ball which contains the cube Q. A suitable dyadic maximal operator related to the potential operator T=Ta(with kernel K=Ka) is (Md Φf)(x) = sup{a(Q)|Q|−1f1I QL1(dy);Qa dyadic cube with Qx}. The nondyadic version of Md Φis merely denoted by MΦ. The measure dω satisfies the condition RDρwith ρ>0 (and we write as dω ∈RDω) when there c=c(ω,n)>0 for which tnρ |B|ω≤c|tB|ωfor all balls Band t>1. Our second result ensures the link between the two maximal operators we have defined above. Theorem 3. Let K=Kabe a kernel satisfying Hi)-ii) with 0<λ,σ≤1. Suppose dω ∈RDρwith 1−λ<ρ. Then C1(Md Φfµ)(.)≤(Md Ta,ω,µf)(.)≤C2(MΦfµ)(.)for all functions f here C1=C1(n, a)>0and C2=C2(n, a, ω). In fact C2does not depend on the individual measure dω but only on the RDρconstant of dω. The claim we announced in the introduction can be stated as Corollary 4. Let 0<s<nand 0<q<∞. Suppose dω ∈RDωwith (1 −s n)<ρ. Then IsgLq ω≈MsgLq ωfor all nonnegative functions g whenever R<|x| |x|(s−n)qdω(x)<c(R)<∞for all R>0. 28 Y. Rakotondratsimba This is an immediate consequence of Theorems 1 and 2. Indeed since for all R>0: (Is1I B(0,R))(.)1IB(0,2R)(.)≈Rs1I B(0,2R)(.) and (Is1I B(0,R))(x)1I|x|>2R(x)≈ |x|s−n1I |x|>2R(x) so the condition Is1I B(0,R)Lq ω<∞is reduced to the one written in this corollary. Note also in studying the two weight inequality IsfLq ω≤cfLq νit is necessary that Is1I B(0,R)Lq ω≤c1I B(0,R)Lp ν<∞. By Theorems 1 and 3, the problem (PT) is then reduced to the following maximal inequality MΦfµLq ω≤cfLp µ. By the study of this last case (see [Ra1], or adapt the proof given in [Sa]) then we get Theorem 5. Let 1<p,q<∞, and K=Kabe a kernel satisfying Hi)-ii) with 0<λ,σ<1. Suppose dω ∈RDωwith (1 −λ)<ρ. Then the inequality (PT)TfµLq ω≤cfLp µ holds if and only if (1) |x|>Ra(x, |x|) |x|nq dω(x)<c(R)<∞for all R>0, and (2) (T k εk1I Qkµ)1IQkLq ω ≤C k εk1I QkLp µ , where C>0is a constant which does not depend of each sequence (Qk)k of cubes and (εk)kof nonnegative reals εk. Moreover in the case 1<p≤qthe condition (2) can be replaced by (2’) (T1I Qµ)1IQLq ω≤C1I QLp µfor all cubes Q Sawyer and Wheeden [Sa-Wh] proved that for 1 <p≤qand for all general measures dω and dµ, then (PT) is equivalent to (2) and (2”) (T∗1I Qω)1IQLp µ≤c1I QLq ωp=p p−1,q=q q−1. With an additional hypothesis on the measure dµ we can simplify the conditions in Theorem 5. We first consider the case p≤q. Muckenhoupt-Wheeden inequality 29 Proposition 6. Let 1<p≤q<∞, and K=Kabe a kernel satisfying Hi)-ii) with 0<λ,σ<1. Suppose dω ∈RDρ,dµ ∈RDρwith (1 −λ)<ρand ρ>0. Then the inequality (PT)holds if and only if (T1I Qµ)1IQLq ω≤C1I QLp µ. If moreover dµ ∈RDρwith (1 −λ)<ρ or dµ ∈A∞then, an easy necessary and sufficient condition for (PT)is a(Q) |Q||Q|1−1 p µ|Q| 1 q ω≤Cfor all cubes Q. This second part is already known [Sa-Wh], and here we deduce it by using results on maximal functions (see [Ra2] and [Pe]). To deal with the range of q<p, we introduce the two conditions dµ ∈ RD(p), dω ∈Dε,q with ε∈[1,∞[ (see [Ch-St-Wh]) and which mean respectively  j≥0 k εk|Qk|µ |2jQk|µ1I 2jQkLp µ ≤c(µ) k εk1I QkLp µ  k εk1I tQkLq ω ≤c(ω)tnε 1 q k εk1I QkLq ω for all t≥1, εk>0 and all cubes Qand Qk. Thus we can state Proposition 7. Let 1<q<p<∞, and K=Kabe a kernel satisfying Hi)-ii) with 0<λ,σ<1, and dω ∈RDρ, with (1 −λ)<ρ. Suppose dµ ∈ RD(p). Then the inequality (PT)holds if and only if for some m≥4and C>0  k εk(T1I Qk)1I(mQk)Lq ω ≤C k εk1I QkLp µ for all cubes Qkand all εk>0. For dµ ∈RDρ∩Dε,p with max(1 −λ, 1 pε)<ρ , a necessary and sufficient condition for (PT)is  k εka(Qk) |Qk||Qk|µ1I QkLq ω ≤C k εk1I QkLp µ . 30 Y. Rakotondratsimba This equivalence is also true when dµ ∈RDρ∩D∞,dω ∈Dε,q with 1−λ<ρ and ε<q(1 −σ). Remark. Now we show that the use of the sharp maximal M#(see [Ya] for a definition) is not well adapted to weaken the weight condition in the Muckenhoupt-Wheeden inequality (1) IsfLq ω≤cMsfLq ω. Indeed such a purpose is based on the two inequalities: (2) (Isf)#≤c(Msf); (3) gLq ω≤cg#Lq ω. Inequality (2) is valid for all functions fwith (Isf)∈L1 loc and was proved in [Ad]. Although (3) is well known to be true for w∈A∞, Yabuta [Ya] had obtained such an inequality with a weak condition he denoted as w∈Cr(with r>q). Thus we think get (1) with this last condition. But since Msf≤cIsf=hthen (4) h#Lq ω≤chLq ω It was proved in [Ya] that condition like (4) implies necessarily w∈A∞. 2. Some Lemmas We first state two Lemmas we need and then we give their proofs. Lemma 1. Let fbe a nonnegative (dµ-locally integrable) function. Then (Md T,ω,µf)(.)≤(Md ω(Tfµ))(.). Lemma 2. Let T=Tabe an operator with the kernel K=Kasatisfying Hi)-ii), and let dν be a positive Borel measure. A) If 0<σ≤1then there is C=C(a, n)>0so that for all cubes Q a(Q) |Q||Q|ν1I Q(.)≤C(T1I Qν)(.)1IQ(.). B) Let m≥1. There is C=C(a, n, m)>0so that (T1I Qν)(.)1ImQ(.)≤C[S1(.)+S2(.)] Muckenhoupt-Wheeden inequality 31 where S1(.)=a(Q) |Q||Q|ν1I mQ(.) and S2(.)=a(Q) |Q| j≥0 2−jn[λ−1] Q∩{|y−.|∼2−j|Q| 1 n} ν1I mQ(.) C) Let m≥4. There is C=C(a, n, m)>0so that (T1I Qν)(.)1I(mQ)c(.)≤C|Q|ν j≥0 a(2jQ) |2jQ|1I 2jQ. Proof of Lemma 1: Let Qbe a dyadic cube. Then we have Q (Tfµ)dω =Rn [T∗1I Qω]fdµ ≥3Q [T∗1I Qω]fdµ. Diving by |Q|ωthis inequality and taking the supremum, we obtain the conclusion. Proof of Lemma 2: A) Let Qbe a cube with center x0and length 2R>0. Then |x−y|≤ c2Rfor all x, y ∈Qwith c=c(n). We obtain a(Q) |Q||Q|ν1I Q(y)≤c1(a, n)x∈Q a(y,c2R) Rndν1I Q(y) ≤c2(a, n)x∈Q|x−y| c2Rn1 |x−y|na y, c2R |x−y||x−y|dν(x)1I Q(y) ≤c3(a, n)x∈Q|x−y| c2Rn[1−σ]a(y,|x−y|) |x−y|ndν(x) 1I Q(y). Since 0 <σ≤1 and |x−y|<2cR we get a(Q) |Q||Q|ν1I Q(y)≤c(a, n)(Ta1I Qν)(y)1IQ(y). 38 Y. Rakotondratsimba ≤c2|Q|µ j≥0 |2jQ|−1 µ(T1I 2jQµ)1I(2jQ)Lq ω ≤c2A|Q| 1 p µ j≥0|Q|µ |2jQ|µ1−1 p ≤c3A|Q| 1 p µ. For the second part of this proposition, the point is to note that MΦµ: Lp µ→Lq ωis equivalent to a(Q) |Q||Q|1−1 p µ|Q| 1 q ω<A<∞ whenever dµ ∈A∞(see [Pe]) or dµ ∈RD∞with 1 −λ<ρ (see [Ra2]). Proof of Proposition 7: It is clear that a necessary condition for (PT)is (*)  k εk(T1I Qkµ)1I(mQk)Lq ω ≤A k εk1I QkLp µ with m≥4 and for all cubes Q,Qkand all εk>0. Conversely we suppose this condition be satisfied and dµ ∈ RD(p). Once we have (**)  k εk(T1I Qkµ)1I(mQk)cLq ω ≤cA k εk1I QkLp µ then (i) and (ii) hold as in proof of Theorem 5, and consequently the inequality (PT) is satisfied. Now using Part C) of Lemma 2, the above condition (∗) and the hypothesis dω ∈ RD(p)wehave S= k εk(T1I Qkµ)1I(mQk)cLq ω ≤c1 k εk j≥0 a(2jQk) |2jQk||Qk|µ1I (2jQk)Lq ω by part C of Lemma 2 ≤c2 k εk j≥0|Qk|µ |2jQk|µ(T1I 2jQkµ)1I(2jQk)Lq ω Muckenhoupt-Wheeden inequality 39 ≤c2A j≥0 k εk|Qk|µ |2jQk|µ1I (2jQk)Lp µ by the condition (∗) ≤c3A k εk1I QkLp µ since dµ ∈ RD(p). It is also clear that a necessary condition for (PT)is (**’)  k εka(Qk) |Qk||Qk|µ1I QkLq ω ≤A k εk1I QkLp µ . Conversely we assume this condition be satified and dµ ∈Dε,p ∩RDρ with 1 −λ<ρ and ε<pρ . It is sufficient to get the conditions in the first part of the present Proposition. As in the proof of Theorem 3 by using part A) of Lemma 2) and since dµ ∈D∞then (T1I Qkµ)1I(mQk)≤ca(Qk) |Qk||Qk|µ1I (mQk) and consequently  k εk(T1I Qkµ)1I(mQk)Lq ω ≤c k εka(Qk) |Qk||Qk|µ1I (mQk)Lq ω ≤c k εk1I (mQk)Lp µ . Now using dµ ∈Dε,p ∩RDρwith ε<pρ we can get the condition dµ ∈ RD(p) as follow: S= k εk j≥0|Qk|µ |2jQk|µ |Qk|µ1I (2jQk)Lp µ ≤c1 j≥0 2−jnρ k εk1I (2jQk)Lp µ ≤c2 j≥0 2−jn[ρ−1 pε] k εk1I QkLp µ =c3 k εk1I QkLp µ . Finally we suppose dµ ∈D∞∩RDρand dω ∈Dε,q ∩RDρwith 1−λ<ρ and ε<q(1 −σ). It remains to get the above condition (∗∗). Thus we have S= k εk(T1I Qkµ)1I(mQk)cLq ω 40 Y. Rakotondratsimba ≤c1 k εk j≥0a(2jQk) |2jQk||Qk|µ1I (2jQk)Lq ω ≤c2 j≥0 2−jn[1−σ] k εka(Qk) |Qk||Qk|µ1I (2jQk)Lq ω ≤c3 j≥0 2−jn[1−σ−1 qε] k εka(Qk) |Qk||Qk|µ1I QkLq ω ≤c3A j≥0 2−jn[1−σ−1 qε] k εk1I QkLp µ ≤c4A k εk1I QkLp µ . Acknowledgement. I would like to thank the referee for his helpful comments and suggestions. References [Ad] D. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), 765–778. [Fr-Ja] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93(1) (1990), 34–170. [Ke-Sa] R. Kerman and E. Sawyer, Weighted norm inequalities for potentials with applications to Schrodinger operators, Fourier transforms and Carleson measures, Ann. Inst. Fourier 36 (1986), 207–228. [Mu-Wh] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. [Pe] C. P´ erez, Weighted norm inequalities for potential and maximal operators, Ph. D. Thesis, Washington University (1989). [Ra1] Y. Rakotondratsimba,In´egalit´es `a poids pour des op´erateurs maximaux et des op´erateurs de type potentiel., Th`ese de Doctorat, Universit´e d’Orl´eans France (1991). [Ra2] Y. Rakotondratsimba, On Muckenhoupt and Sawyer conditions for maximal operators, Pub. Mat. 37 (1993), 57–93. Muckenhoupt-Wheeden inequality 41 [Sa] E. Sawyer, A characterization of a two weight norm inequality for maximal operators, Studia Math. 75 (1982), 1–11. [Sa-Wh] E. Sawyer and R. L. Wheeden, Weighted norm inequalities for fractional integral on euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813–874. [Ya] K. Yabuta, Sharp maximal functions and Cpcondition, Archiv. Math. 55 (1990), 151–155. Keywords. Weighted inequalities, Potential operators, Maximal operators 1980 Mathematics subject classifications: 42B25 Universit´e d’Orl´eans D´epartement de Math´ematiques U.F.R. Facult´e des Sciences B.P. 6759 45067 Orl´eans Cedex 2 FRANCE Primera versi´o rebuda el 16 de Mar¸c de 1993, darrera versi´o rebuda el 15 de Gener de 1995