Composition of maximal operators

Publicacions Matem`atiques, Vol 40 (1996), 397–409. COMPOSITION OF MAXIMAL OPERATORS Menita Carozza and Antonia Passarelli di Napoli Abstract Consider the Hardy-Littlewood maximal operator Mf(x) = sup Qx 1 |Q|Q |f(y)|dy. It is known that Mapplied to ftwice is pointwise comparable to the maximal operator MLlog Lf, defined by replacing the mean value of |f|over the cube Qby the Llog L-mean, namely MLlog Lf(x) = sup x∈Q 1 |Q|Q |f(y)|log e+|f| |f|Q(y)dy, where |f|Q=1 |Q|Q|f|(see [L], [LN], [P]). In this paper we prove that, more generally, if Φ(t) and Ψ(t) are two Young functions, there exists a third function Θ(t), whose explicit form is given as a function of Φ(t) and Ψ(t), such that the composition MΨ◦MΦis pointwise comparable to MΘ. Through the paper, given an Orlicz function A(t), by MAfwe mean MAf(x) = sup Qx ||f||A,Q where ||f||A,Q = inf λ>0: 1 |Q|QA|f| λ(x)dx ≤1. 1. Introduction. Let f∈L1 loc(Rn), the Hardy-Littlewood maximal operator Mf of fis defined by Mf(x) = sup Qx 1 |Q|Q |f(y)|dy. This work has been supported by M.U.R.S.T. (40%). 398 M. Carozza, A. Passarelli di Napoli A well-known result of Coifman and Rochberg, (see [CR], [T]), states that if Mf < ∞a.e. and if δ∈(0,1), then (Mf)δ∈A1, where A1is the Muckenhoupt class of the non negative weights wsuch that A1(w) = sup Q Q w ess infQw<∞, where Qwstands for the average of wover Qand the supremum being taken over all cubes Qof Rn. Setting Mrf= sup QxQ |f|r1 r r>1, from the mentioned result, with δ=1 r, it follows that M◦Mr∼Mr. This means that there exists a constant c, such that Mrf(x)≤M(Mrf(x)) ≤cMrf(x) a.e. in Rn. For r= 1 the situation is different, namely we have that M◦M∼ MLlog L, i.e. c1MLlog Lf(x)≤M(Mf(x)) ≤c2MLlog Lf(x) a.e. in Rn (see [L], [LN], [P]), and this corresponds to Stein’s result, i.e. for f supported in a cube Q f∈Llog L(Q)⇐⇒ Mf ∈L1(Q) (see [S]). The maximal operator MLlog Lfis defined by replacing the mean value of |f|over the cube Qby the Llog L-mean, namely (1.1) MLlog Lf(x) = sup x∈Q 1 |Q|Q |f(y)|log e+|f| |f|Q(y)dy, where |f|Q=1 |Q|Q|f|. The previous results justify the introduction of a maximal operator in an Orlicz space such as Llog L. More precisely, let Ω be a cube of Rn. A continuosly increasing function on [0,∞], say Ψ : [0,∞]→[0,∞] such that Ψ(0) = 0, Ψ(1) = 1 and Ψ(∞)=∞, will be referred to as an Orlicz function. Composition of maximal operators 399 The generalized Orlicz space denoted by LΨ(Ω) consists of all functions g:Ω⊆Rn→Rsuch that Ω Ψ|g| λ(x)dx < ∞ for some λ>0. Let us define the Ψ-average of gover a cube Qcontained in Ω by (1.2) ||g||Ψ,Q = inf λ>0:Q Ψ|g| λ(x)dx ≤1. When Ψ(t) is a Young function, i.e. a convex Orlicz function, the quantity ||g||Ψ= inf λ>0:Ω Ψ|g| λ(x)dx ≤1 is the well known Luxemburg norm in the space LΨ(Ω) (see [KR], [RR]). If f∈LΨ(Rn), the maximal function of fwith respect to Ψ is defined by setting (1.3) MΨf(x) = sup x∈Q ||f||Ψ,Q where the supremum is taken over all cubes Qof Rncontaining xwith sides parallel to the coordinate axes. Let us remark that if we choose Ψ(t)=tlog(e+t), the maximal operator MΨfdefined by (1.3) is equivalent to the MLlog Loperator defined by (1.1) (see [IS]). In this paper we generalize the mentioned results: namely, given two Young functions Φ(t) and Ψ(t), we get a third Young function Θ(t), such that the composition, MΨ◦MΦ, between MΦand MΨis equivalent to the operator MΘ. As an application, we reobtain, in a simple way, the Herz type inequality for the nonincreasing rearrangement of the maximal operator in Llog L(see [B]). Moreover, we obtain a pointwise estimate for the maximal function of the jacobian of a function fsuch that |Df|nbelongs to L1. 2. The main result. LetΩbeacubeofRnand set MΦf(x) = sup x∈Q⊆Ω ||f||Φ,Q. First, let us prove a result which will be useful in the following. 400 M. Carozza, A. Passarelli di Napoli Theorem 1. Let Ψ(t)be an Orlicz function and Φ(t)be a Young one. For Θ(t)=t 0 Ψ(s)Φ t sds, there exist two positive constants c1,c2such that (2.1) c1||MΦf||Ψ,Ω≤||f||Θ,Ω≤c2||MΦf||Ψ,Ω for every f∈LΘ(Ω). Proof: In order to prove that (2.2) ||MΦf||Ψ,Ω≤c||f||Θ,Ω, we use the following equality: Ω ΨMΦf(x) λdx =∞ 0 Ψ(t)|{x∈Ω:MΦf(x)>tλ}| dt. Let us set Etλ ={x∈Ω:MΦf(x)>tλ}. Thanks to Proposition 4.1 in [BP], we can consider a sequence of cubes {Qk}such that Etλ =∪kQkand Qk Φ|f| λt (x)dx > |Qk|. Now, we observe that |Qk|<Qk Φ|f| λt (x)dx ={x∈Ω:|f|>λt 2}∩Qk Φ|f| λt (x)dx +{x∈Ω:|f|≤ λt 2}∩Qk Φ|f| λt (x)dx ≤{x∈Ω:|f|>λt 2}∩Qk Φ|f| λt (x)dx +Φ1 2|Qk|. Without loss of generality, we may assume Φ 1 2<1, then we have |Qk|<c{x∈Ω:|f|>λt 2}∩Qk Φ|f| λt (x)dx <c{x∈Ω:|f|>λt 2}∩Qk Φ2|f| λt (x)dx Composition of maximal operators 401 by monotonicity of Φ. We get |Etλ|≤c{x∈Ω:|f|>λt 2}∩Etλ Φ2|f| λt (x)dx. After that, we obtain Ω ΨMΦf(x) λdx ≤c∞ 0 Ψ(t){x∈Ω:|f|>λt 2}Φ2|f| λt (x)dx dt =cΩ2|f| λ 0 Ψ(t)Φ 2|f| λt (x)dt dx(2.3) =cΩ Θ2|f| λ(x)dx. By estimate above, we have (2.2). Now, we have to prove that (2.4) ||f||Θ,Ω≤c||MΦf||Ψ,Ω. By Calderon-Zygmund lemma, we may cover Etλ ={x∈Ω:MΦf(x)> tλ}by a sequence of nonoverlapping cubes Qk, each having the property 2−n|Qk|≤|Qk∩Etλ|<|Qk| and such that 2n|Etλ|≥|Qk|≥Qk Φ|f| λt dx ≥Etλ Φ|f| λt dx. We have that (2.5) Ω ΨMΦf(x) λdx ≥˜cΩ Θ|f| λ(x)dx. In fact Ω ΨMΦf(x) λdx =∞ 0 Ψ(t)|{x∈Ω:MΦf(x)>tλ}| dt ≥c∞ 0 Ψ(t)Etλ Φ|f| tλ dx dt =cΩMΦ(f) λ 0 Ψ(t)Φ |f| tλ dt dx ≥cΩf(x) λ 0 Ψ(t)Φ |f| tλ dt dx 402 M. Carozza, A. Passarelli di Napoli since Φ(t) is convex. Finally, we get Ω ΨMΦf(x) λdx ≥cΩ Θ|f| λdx, which implies (2.4), then the theorem is proved . Remark 1. Theorem 1 with Φ and Ψ both Young functions, is proved in [BP]. Moreover, in the particular case Φ(t)=tand Ψ(t) any Orlicz function, Theorem 1 gives Proposition 3.1 of [GIM]. Using the previous result, we develop a useful estimate for the composition MΨ◦MΦ, where Φ and Ψ are Young functions. Theorem 2. Let Ψ(t)and Φ(t)be two Young functions. For Θ(t)=t 0 Ψ(s)Φ t sds, there exist two positive constants, c1and c2, such that for every f∈ LΘ loc(Rn)we have (2.6) c1MΘf(x)≤MΨ(MΦf(x)) ≤c2MΘf(x) almost everywhere. Proof: Let us fix x∈Rnand a cube Qcontaining x. Put f=f1+f2 with f1=fχ3Q, we have, by triangle inequality of the Luxemburg norm || ||Ψ, (2.7) ||MΦf||Ψ,Q ≤||MΦf1||Ψ,Q +||MΦf2||Ψ,Q =I+II. In order to estimate I, consider MΦf(x) = sup{||f||Φ,¯ Q:x∈¯ Q, ¯ Q⊆3Q} and we observe that there exists a constant c(n) such that (2.8) MΦf1(x)≤c(n)MΦf1(x). Composition of maximal operators 403 Namely, for every cube ˜ Q⊆Rn,˜ Qx,˜ Q∩C(3Q)=∅the following inequality holds ˜ Q Φ(|f1|)= 1 |˜ Q|˜ Q∩3Q Φ(|f1|)≤3n3Q Φ(|f1|) and then, if λ>0 is such that 3Q Φ|f1| λ≤1 we have 1 3n˜ Q Φ|f1| λ≤1. By convexity of Φ, we get ˜ Q Φ|f1| 3nλ≤1 and this implies (2.9) ||f1||Φ,˜ Q≤3n||f1||Φ,3Q. Note that (2.9) is trivial if ˜ Q⊆3Q. Observing that ||f1||Φ,3Q≤ MΦf1(x) and taking the supremum over all cubes ˜ Qof Rncontaining xon the left hand side of (2.9), we have (2.8). By formulas (2.8) and (2.2), applied with Mand Ω = 3Q,we deduce I=||MΦf1||Ψ,Q ≤C||f||Θ,Q. To estimate II it suffices to observe that (2.10) MΦf2(y)≤Cinf QMΦf2∀y∈Q. In fact, let us fix a point y∈Qand a cube ¯ Qysuch that ¯ Q∩C(3Q)=∅; the cube 3 ¯ Qcontains every point x∈Q. Reasoning as before, we obtain ||f2||Φ,¯ Q≤3n||f2||Φ,3¯ Q≤CMΦf2(x) and so (2.10). Now we observe that there are positive constants c1,c 2,t 0, depending on Φ and Ψ, such that Φ(c1t)≤c2Θ(t), for t≥t0. Namely, Θ(t)=t 0 Ψ(s)Φ t sds ≥t 0 Ψ(s)Φt st s2ds ≥c0t t0 Φt st s2ds =c0t t0 1 Φ(σ)dσ =c−1 2Φt t0 404 M. Carozza, A. Passarelli di Napoli and then there exists a positive constant c3such that MΦ f(x)≤c3MΘf(x) a.e. . By (2.7), (2.9) and (2.10), we conclude that (2.11) ||MΦf||Ψ,Q ≤C1||f||Θ,Q +C2MΘf(x). Taking the supremum over the cubes Qcontaining xin (2.11), we get (2.12) MΨ(MΦf(x)) ≤c2MΘf(x) a.e. . On the other hand, formula (2.4) implies that (2.13) MΨ(MΦf(x)) ≥c1MΘf(x) a.e. . Formulas (2.12) and (2.13) give the thesis. Let us give some example of such compositions. Example 1. Let us consider Ψ(t)=tp Φ(t)=1t≤1 tqt>1,p, q ≥1 recalling that Mrf(x) = sup x∈QQ fr1 r we have Mpf(x)=MΨf(x) Mqf(x)=MΦf(x). Theorem 2 implies that Mp◦Mq∼Mrr= max{p, q},p=q MLqlog Lp=q. Let us note that in some very special cases one can determine the constants c1,c2. Example 2. For r>1, if fis a nonincreasing function f:(0,∞)→ (0,∞) then Mrf(x)≤M(Mrf(x)) ≤r r−1Mrf(x), Composition of maximal operators 405 by Kolmogorov inequality (see [BDS]). Moreover x 0 f(t)1 + log xf(t) x 0f(s)dsdt ≤M(Mf(x)) ≤e e−1x 0 f(t)1 + log xf(t) x 0f(s)dsdt. Let us consider Bagby’s formula, (see [B]), for such f.IfΦ α(t)=t[1 + (log+t)α], α>0, then c1MΦαf(t)≤1 tt 0log t sα f(s)ds ≤c2MΦαf(t), and observe that MΦα◦MΦβ∼MΦα+β+1 . Remark 2. If in Theorem 2 Φ(t)=t, we get Θ(t) t=t 0 Ψ(s) sds. It is easy to verify that c1MΨf(x)≤MΨ(Mf)(x)≤c2MΨf(x) if and only if Ψ(t)=A(t)tp, where A(t) is a continuous increasing function and p>1. Moreover, we have that c1MΦf(x)≤M(MΦf)(x)≤c2MΦf(x) if and only if Φ(t)=tp, where p>1. So, we reobtain the mentioned result of [CR]. 3. Some consequences. Corollary 1. Let A(t)be a Young function in (0,∞). The n-composition, MA◦MA◦···◦MA, of the maximal operator MA, is equivalent to the maximal operator MAn, where (3.1) An(t)=t 0 A(s)An−1t sds.