COMPOSITION OF MAXIMAL OPERATORS

Consider the Hardy-Littlewood maximal operator Mf(x) = sup Q x 1 |Q| ∫

It is known that M applied to f twice is pointwise comparable to the maximal operator M L log L f , defined by replacing the mean value of |f | over the cube Q by the L log L-mean, namely [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 M Ψ • M Φ is pointwise comparable to M Θ .Through the paper, given an Orlicz function A(t), by M A f we mean

Introduction.
Let f ∈ L 1 loc (R n ), the Hardy-Littlewood maximal operator Mf of f is defined by 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 ∈ A 1 , where A 1 is the Muckenhoupt class of the non negative weights w such that where Q w stands for the average of w over Q and the supremum being taken over all cubes Q of R n .Setting from the mentioned result, with δ = 1 r , it follows that M • M r ∼ M r .This means that there exists a constant c, such that (see [L], [LN], [P]), and this corresponds to Stein's result, i.e. for f supported in a cube [S]).The maximal operator M L log L f is defined by replacing the mean value of |f | over the cube Q by the L log L-mean, namely The previous results justify the introduction of a maximal operator in an Orlicz space such as L log L.
The generalized Orlicz space denoted by L Ψ (Ω) consists of all functions for some λ > 0.
Let us define the Ψ-average of g over a cube Q contained in Ω by When Ψ(t) is a Young function, i.e. a convex Orlicz function, the quantity is the well known Luxemburg norm in the space L Ψ (Ω) (see [KR], [RR]).
If f ∈ L Ψ (R n ), the maximal function of f with respect to Ψ is defined by setting where the supremum is taken over all cubes Q of R n containing x with sides parallel to the coordinate axes.
Let us remark that if we choose Ψ(t) = t log(e + t), the maximal operator M Ψ f defined by (1.3) is equivalent to the M L log L operator 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, M Ψ • M Φ , between M Φ and M Ψ is equivalent to the operator M Θ .
As an application, we reobtain, in a simple way, the Herz type inequality for the nonincreasing rearrangement of the maximal operator in L log L (see [B]).
Moreover, we obtain a pointwise estimate for the maximal function of the jacobian of a function f such that |Df | n belongs to L 1 .

The main result.
Let Ω be a cube of R n and set First, let us prove a result which will be useful in the following.
Theorem 1.Let Ψ(t) be an Orlicz function and Φ(t) be a Young one.For there exist two positive constants c 1 , c 2 such that for every f ∈ L Θ (Ω).
Proof: In order to prove that we use the following equality: Let us set Thanks to Proposition 4.1 in [BP], we can consider a sequence of cubes {Q k } such that Now, we observe that Without loss of generality, we may assume Φ 1 2 < 1, then we have by monotonicity of Φ.We get After that, we obtain By estimate above, we have (2.2).Now, we have to prove that By Calderon-Zygmund lemma, we may cover and such that We have that (2.5) In fact which implies (2.4), then the theorem is proved .
Remark 1. Theorem 1 with Φ and Ψ both Young functions, is proved in [BP].
Using the previous result, we develop a useful estimate for the composition M Ψ • M Φ , where Φ and Ψ are Young functions.
Theorem 2. Let Ψ(t) and Φ(t) be two Young functions.For there exist two positive constants, c 1 and c 2 , such that for every almost everywhere.
Proof: Let us fix x ∈ R n and a cube Q containing x.Put f = f 1 + f 2 with f 1 = fχ 3Q , we have, by triangle inequality of the Luxemburg norm In order to estimate I, consider and we observe that there exists a constant c(n) such that Namely, for every cube Q By convexity of Φ, we get and this implies (2.9) and taking the supremum over all cubes Q of R n containing x on the left hand side of (2.9), we have (2.8).By formulas (2.8) and (2.2), applied with M and Ω = 3Q, we deduce To estimate II it suffices to observe that (2.10) In fact, let us fix a point y ∈ Q and a cube Q y such that Q∩C(3Q) = ∅; the cube 3 Q contains every point x ∈ Q. Reasoning as before, we obtain and so (2.10).Now we observe that there are positive constants c 1 , c 2 , t 0 , depending on Φ and Ψ, such that Φ(c 1 t) ≤ c 2 Θ(t), for t ≥ t 0 .Namely, and then there exists a positive constant c 3 such that a.e. .By (2.7), (2.9) and (2.10), we conclude that Taking the supremum over the cubes Q containing x in (2.11), we get (2.12) On the other hand, formula (2.4) implies that Formulas (2.12) and (2.13) give the thesis.
Let us give some example of such compositions.
Example 1.Let us consider we have Theorem 2 implies that Let us note that in some very special cases one can determine the constants c 1 , c 2 .
by Kolmogorov inequality (see [BDS]).Moreover Let us consider Bagby's formula, (see [B]), for such and observe that It is easy to verify that if and only if Ψ(t) = A(t)t p , where A(t) is a continuous increasing function and p > 1.Moreover, we have that if and only if Φ(t) = t p , where p > 1.So, we reobtain the mentioned result of [CR].
Corollary 1.Let A(t) be a Young function in (0, ∞).The n-composition, of the maximal operator M A , is equivalent to the maximal operator M An , where Proof: Put A 1 (t) = A(t).For n = 2, (3.1) follows by Theorem 1.By an induction argument, we have the assertion for every positive integer n.
In particular, the n-iterate of the Hardy-Littlewood maximal operator is equivalent to the operator (see [L], [LN], [P]).
Let f be a measurable function defined on R n .We denote by µ the distribution function of f , namely, for t > 0 we set Then we define the decreasing rearrangement f * of f : The following theorem states the equivalence between (Mf) * and M (f * ).
Another corollary of Theorem 2 is the following Herz type inequality for the L log n L-maximal operator.

Corollary 2. There exist c
Proof: Let us prove the thesis for k = 1.Herz inequality states that Formula (3.2), with g replaced by Mf, becomes and using (3.2) again in the right hand side of (3.3), we get Applying Theorem 1 in (3.4), with Φ(t) = Ψ(t) = t, we get The thesis follows arguing by induction.
As another application of Theorem 2, we obtain a pointwise estimate about the maximal function of the jacobian of a function f (see [IS], [M]).Namely we have the following

Theorem 4 .
If |Df | n ∈ L 1 loc (R n ), then we have (3.6)M L log L J(x) ≤ c(n)M (|Df | n )(x) a.e.x ∈ R n where J = J f (x) ≥ 0 is the jacobian of f .Proof: In [IS] is proved that if |Df | n ∈ L 1 loc (R n ) for any cube Q ⊂ R n , 0 < σ < 1,then we have σQ J dy ≤ c(n) to Theorem 2, applying the maximal function M to both sides in (3.7), we get (3.8)M L log L (J)(x) ≤ cM [(M (|Df | Let us observe that, in general, the following estimateM Θ J(x) ≤ c(n)M Ψ (|Df | n )(x) a.e.x ∈ R nholds, where Ψ(t) is a Young function, |Df | n ∈ L Ψ loc (R n