TWO WEIGHT NORM INEQUALITY FOR THE FRACTIONAL MAXIMAL OPERATOR AND THE FRACTIONAL INTEGRAL OPERATOR

New sufficient conditions on the weight functions u(.) and v(.) are given in order that the fractional maximal [resp. integral] operator Ms [resp. Is], 0 ≤ s < n, [resp. 0 < s < n] sends the weighted Lebesgue space Lp(v(x) dx) into Lp(u(x) dx), 1 < p < ∞. As a consequence a characterization for this estimate is obtained whenever the weight functions are radial monotone.


Introduction
The fractional maximal operator M s of order s, with 0 ≤ s < n, acts on locally integrable function f (.) of R n as These cubes have sides parallel to the coordinate-axes.Here M 0 is the well known Hardy-Littlewood maximal operator.The purpose of this paper is to determine weight functions u(.) and v(.) for which M s is bounded from L p v = L p (R n , v dx) into L p u with 1 < p < ∞.This means there is C > 0 for which For convenience such a estimate will be denoted by M s : L p v → L p u .
Sawyer [Sa1] proved that M s : L p v → L p u if and only for a constant S > 0: Here 1I Q (.) denotes the characteristic function of the cube Q. Unfortunately for given weight functions, this condition is not easy to check since it is expressed in term of the maximal operator M s itself.According to Pérez [Pe] the above estimate holds whenever for some t > 1 and A > 0 ≤ A for all cubes Q.
It is an "almost necessary condition" in the sense that M s : L p v → L p u implies this last inequality with t = 1.Although this Pérez's condition is not expressed in term of M s , it can be non-satisfactory because of integrations on arbitrary cubes.Take, for instance, w(x) = |x| 1 2 ln −1 (e + |x|)[ln(e + |x|) − |x|(e + |x|) −1 ].For cubes Q noncentered at the origin, a direct computation of Q w(x) dx seems to be extremely hard to do.This is not the case in evaluating |x|<R w(x) dx, R > 0. Such an observation leads to consider and study again the estimate M s : L p v → L p u , which necessarily [see Section 2] implies for all R > 0, and for all R > 0.
So the main question, answered in Section 2, is to find a third condition so that the three conditions together are sufficient to derive M s : L p v → L p u [Theorem 2.1].As a consequence it will be proved [Corollary 2.5] that (1.1) and (1.2) together are necessary and sufficient for this estimate to hold whenever the weight functions are radial monotone.
The corresponding problem and study for I s : L p v → L p u , 0 < s < n, are performed in Section 3, where I s is the fractional integral operator given by Ideas for the proofs are inspired from a former paper due to B. Muckenhoupt and R. L. Wheeden [Mu-Wh].
Throughout this paper, it will be always assumed that: are nonnegative locally integrable functions.

Results for the Fractional Maximal Operator
A variant of M s is the restricted maximal operator M s defined by The first main result, which is also the high point of the present paper, asserts that the two weight problem for M s can be essentially reduced to the corresponding weighted inequality for the restricted operator M s .Precisely, we have So the remainder of this paragraph will be devoted first to derive sufficient conditions for the estimate of M s , and then to give applications showing the gain over past results.

Proposition 2.2. The estimate M
The sufficient condition involved in this result does not require In applications, the restricted operator M 0 in (2.1) is not a brake for computations, since trivially ( M 0 w)(x) ≤ sup 1 4 |x|<|y|<4|x| w(y).Then M s : L p v → L p u whenever for a constant C > 0: (2.2) |x| s σ(x) a.e. and with σ(x) = sup For a weight function v(.) constant on annuli then σ(x As a first application of the above results is a sort of "improvement" of a Cordoba-Fefferman's inequality [Co-Fe] which states that ), with the constant C > 0 depending only on n and p.Here T is a Calderon-Zygmund operator, i.e. a linear operator taking And K(x, y) is a continuous function defined on {(x, y); x = y} and satisfying the standard estimates: do not depend on x, y and x .Proposition 2.3.Suppose u(.) ∈ RD ρ ∩ H for a ρ > 0. Then for a constant C > 0: (2.3) Here C depends on n, p, and on the constants in the conditions RD ρ and H. Inequality (2.3) is better than the preceding one since (M 0 w)(x) ≤ (M 0 w t ) This results yields a complete solution of the two weight inequalities for monotone weight functions as announced in the introduction.
We will end this paragraph by a second example of applications of Theorem 1, which seems difficult to treat by a direct use of the Pérez's result [Pe] quoted above.

Results for the Fractional Integral Operator
In fact the above results for M s are inspired from those for I s , whose proofs seem rather simplistic, but appear significant.
The estimate I s : L p v → L p u was first characterized by Sawyer [Sa2] by using both two test conditions expressed in term of I s and arbitrary (dyadic) cubes.Latter Sawyer and Wheeden [Sa-Wh], [see also [Pe]] introduced the sufficient condition Here A > 0 and t > 1 are fixed constants.This test condition requires t < n s and u(. ).Since M s is pointwise dominated by I s , then each of (1.1) and (1.2) is a necessary condition for I s : L p v → L p u to be held.The dual of the Hardy inequality (1.2): is also a necessary condition for the above estimate.As in the case of the fractional maximal operator, the corresponding restricted operator will be useful for the sequel.
The main result also states that the two weight problem for I s is essentially reduced to the corresponding weighted inequality for the restricted operator I s .
Note that Verbitsky and Wheeden [Ve-Wh] got and by assuming ( Compared to this last inequality, condition (3.1) is easier to check at least for usual weight functions.Indeed, clearly (3.1) is satisfied if for some constant C > 0: where σ(x) = sup 1 4 |x|<|y|<4|x| v − 1 p−1 (y) and u(x) = sup 1 4 |x|<|y|<4|x| u(y).As a first application of the above results is a sort of "improvement" of a Adam's inequality [Ad] which states that, for 1 < p < n s : R n with the constant C > 0 depending only on s, n and p.
Proposition 3.3.Let 1 < p < n s .Suppose u(.) ∈ RD ρ ∩ H for a ρ > 0. Then for a constant C > 0: Here C depends on n, p and on the constants in the conditions RD ρ and H.As for the case of the fractional maximal operator, we have Corollary 3.6.Let ρ, β, δ, ν, u(.) and v(.) as in Corollary 2.6, and where instead of (2.4): Then Observe that for fixed constants C, c > 0 then So we are attempted to state that the Muckenhoupt condition (1.1) implies I s : L p v → L p u whenever both u(.) and v − 1 p−1 (.) satisfy the reverse doubling condition RD ρ with 1 − s n < ρ.Unfortunately this is not the case, since Lemma 3.7.There is no nontrivial weight functions u(.) and v(.) so that both u(.) and v − 1 p−1 (.) satisfy the reverse doubling condition RD ρ with 1 − s n < ρ and for which the Muckenhoupt type condition (1.1) is satisfied.
Theorems 3.1 and 2.1 will be proved in Section 5. Proofs for all propositions and corollaries are presented in the next paragrah.

Proofs of Propositions and Corollaries
Proof of Propositions 3.2 and 2.2: To derive I s : L p v → L p u from condition (3.1), we set C(x) = {y : 1 2 |x| < |y| ≤ 2|x|}.By the Hölder inequality Next our purpose is to get M s : L p v → L p u from condition (2.1).Also by the Hölder inequality, for each t > 1: With this last inequality we can conclude as follows This last inequality is a consequence of the well-known maximal theorem which asserts that M 0 : L 1 whenever 1 < p (tp ) (which is true since p < tp and t > 1).
Proof of Propositions 2.3 and 3.3: Since these results are proved in details in [Ra], we only outline the main points.
Proof of Lemma 3.7: The Muckenhoupt condition (1.1) can be writ- Thus the contradiction will appear once we get The proof can be limited for the second identity.Since u(.) is not a trivial weight function, then 0 < |y|<c u(y) dy < ∞ for a constant c > 0. Without loss of generality it can be assumed that c = 1 and Q1 u(y) dy for some fixed constants c 0 and c 1 > 0, here Q 0 Proof of Corollaries 2.6 and 3.6: These results are just based on the following lemmas.
Conversely, using (2.7) and these above computations, then A(R) < C and H(R) < C for all R > 0 and for a fixed constant C > 0. Thus by Corollary 2.5 then For the case of I s then (4.5) and (4.6) [with q = p ] and condition (3.4) [which is stronger than (2.4)] are used to get and an estimate of Thus I s : L p v → L p u if and only condition (2.7) is satisfied.Now the above two lemmas remain to be proved.
Inequality (4.5) appears after using (4.2).Indeed for 0 For R > e then (4.6) can be also deduced from (4.2) and by using the fact that t → ln −µ t is a nonincreasing function and (s − n)q + η < 0.

Proof of Theorems 3.1 and 2.1
The main key for these results is the well-known weighted inequalities for the Hardy operator [Mu], which in our context is stated as follows Lemma.There is C > 0 such that if and only for a constant A > 0 (5.1) R<|y| w(y) dy We first begin with the proof for the fractional integral operator I s , 0 < s < n, which is curiously easier to handle than the case of M s .
The Fractional Integral Operator.First suppose . So the Hardy condition (1.2) appears in virtue of the above lemma [with w , so condition (1.2 * ) also holds by using (5.1 * ) in the lemma.
For the converse, it is assumed that I s : L p v → L p u and both the conditions (1.2) and (1.2 * ) are satisfied.Our purpose is to get with Now we give the proof for M s , 0 ≤ s < n.
The Fractional Maximal Operator.

First suppose M
. Thus the assumed estimate implies both (1.1) and (1.2).

≤
) dx ≤ C p R n g p (x)w(x) dx for all g(.)A for all R > 0.The constants A and C are related by the relation c 1 A ≤ C ≤ c 2 A with c 1 , c 2 > 0 depending on n and p.

Theorem 3.1. The
estimate I s : L p v → L p u holds if and only if I s : L p v → L p u and both the Hardy condition (1.2) and its dual version (1.2 * ) are satisfied.A characterizing condition for I s : L p v → L p u is not known, however a sufficient condition [easily verifiable for a large weight functions] can be obtained by elementary arguments.Proposition 3.2.The estimate I s : L p v → L p u holds whenever ).Let us consider again the weighted inequality for the restricted operator I s .