Weighted Inequalities
for the Fractional Maximal Operator
on radial and nonincreasing functions
Yves Rakotondratsimba
(Received August 23, 1999)
Abstract. We find a sufficient condition on weights u(.) and v(.) for which the
fractional maximal operator Mα, 0 ≤ α < n, is bounded from Lpdec(v(x)dx) to Lq(u(x)dx). Here 1 < p ≤ q < ∞, and Lp
dec(v(x)dx) denotes the set of all radial and nonincreasing functions which belong to the weighted Lebesgue space Lp(v(x)dx). Actually a characterization of this boundedness is obtained whenever the weight v(.) satisfies some reverse doubling condition.
AMS 1991 Mathematics Subject Classification. 42B25.
Key words and phrases. Fractional maximal operators, weighted inequalities.
§1. Introduction and Result
The fractional maximal operator Mα of order α, with 0≤ α < n, acts on
locally integrable functions f (.) ofRn, n≥ 1, as (Mαf )(x) = sup t>0 { tα−n ∫ B(x,t) |f(y)|dy}, x∈ Rn.
As usual the ball B(x, t) is the set{y ∈ Rn; |x − y| < t}.
Our purpose in this paper is to determine weights u(.) and v(.) for which
Mαis bounded from Lpdec(v(x)dx) to Lq(u(x)dx), 1 < p≤ q < ∞. This means
that for some C > 0
(1.1) (∫ Rn(Mαf ) q(x)u(x)dx )1 q ≤ C (∫ Rnf p(y)v(y)dy )1 p
for all functions f (.)≥ 0 given by f(·) = ϕ(| · |), where ϕ(.) ≥ 0 is defined and nonincreasing on ]0,∞[. For convenience such a boundedness will be denoted by Mα: Lpdec(v)→ Lq(u). Inequalities for radial and nonincreasing functions
as (1.1) can arise naturally in study of some rough maximal and singular integral operators.
The inequality (1.1) with no restriction on functions f (.) has been investi-gated by many authors (see the references in [Ra2]). No result seems available
through the literature about Mα: L p
dec(v)→ L
q(u) except for the case α = 0
which was investigated by the author in [Ra3]. However, for some linear op-erators T , more are now known about T : Lpdec(v)→ Lq(u), [Ke-Sa], [Sa], [St]
and [Ra1].
Throughout this paper it will be assumed that 0≤ α < n,
1 < p≤ q < ∞, p0= p
p− 1
and
u(.) and v(.) are weights e.g. nonnegative and locally integrable functions.
The main result of this paper reads as follows.
Theorem. A necessary condition for the boundedness Mα: Lpdec(v)→ Lq(u)
is that for some constant A > 0
(1.2) Rα (∫ |x|<Ru(x)dx )1 q ≤ A (∫ |y|<Rv(y)dy )1 p for all R > 0 and (1.3) (∫ R<|x| |x|(α−n)qu(x)dx )1 q × (∫ |y|<R [∫ |z|<|y|v(z)dz ]−p0 |y|np0 v(y)dy )1 p0 ≤ A for all R > 0. Conversely the boundedness Mα : Lpdec(v) → Lq(u) holds whenever (1.3)
and (1.4) Rα (∫ |x|<R u(x)dx )1 q ≤ A (∫ 2−1R<|y|<R v(y)dy )1 p for all R > 0
are both satisfied.
Remarks. 1) Condition (1.4) is slightly stronger than (1.2). Indeed if for some
constant C > 0 (1.5) ∫ |y|<Rv(y)dy≤ C ∫ 2−1R<|y|<R
then the boundedness Mα: L p
dec(v)→ L
q(u) is just characterized by (1.2) and
(1.3) together. Property (1.5) arises when v(.) satisfies the reverse doubling condition
∫
|y|<2−1R
v(y)dy ≤ ρ
∫
|y|<Rv(y)dy for all R > 0
where ρ is a fixed constant such that 0 < ρ < 1. The characterization claimed in the abstract is therefore justified.
2) For α = 0, conditions (1.2) and (1.3) are seen in [Ra3] to be necessary and sufficient for Mα : Lpdec(v) → Lq(u). It is an open question whether an
analogue result remains true when α > 0 and for general weights v(.).
3) Here (1.2), (1.3) and (1.4) are just expressed in terms of balls centered at the origin. Therefore these conditions should be easy to verify than those expressed in terms of cubes which are largely used by many authors to deal with (1.1) for general functions f (.).
To tackle the problem related to (1.1), for nonincreasing functions, we ex-ploit some ideas already used in [Ra2] and [Ra3]. And the main key to realize our purpose is contained in the following result.
Proposition. The boundedness Mα: L p dec(v)→ L q(u) is equivalent to (1.6) H : Lpdec(v)→ Lq(|.|(α−n)qu(.)) and (1.7) Mα: Lpdec(v)→ Lq(u) where (Hf )(x) = ∫ |y|<|x|f (y)dy and (Mαf )(x) = sup j≥1;j integers { (2j|x|)α−n ∫ 2j|x|≤|y|<2j+1|x||f(y)|dy } .
Here (1.6) can be viewed as a weighted Hardy inequality for nonincreasing functions.
Lemma. The boundedness H : Lpdec(v) → Lq(|.|(α−n)qu(.)) is equivalent to the existence of a constant C > 0 such that
(1.8) (∫ ∞ 0 [ t−1 ∫ t 0 ψ(r)dr ]q µ(t)dt )1 q ≤ C(∫ ∞ 0 ψp(r)ν(r)dr )1 p
for all functions ψ(.)≥ 0 nonincreasing on ]0, ∞[. The weights µ(t) and ν(t) are given by ν(t) = tn1[1−n]ev(t 1 n), ev(r) = rn−1 ∫ Sn−1 v(rω)dω and µ(t) = tn1[αq+1−n]eu(t 1 n), eu(r) = rn−1 ∫ Sn−1 u(rω)dω, where dω is the area-measure on the unit sphere Sn−1 ofRn.
Such a lemma can be easily obtained by making use of some variables changes and polar coordinates as in Proposition 1 of [Ra3].
Inequality (1.8) was first characterized by E. Sawyer [Sa] by means of
(1.9) (∫ R 0 µ(r)dr )1 q ≤ A (∫ R 0 ν(r)dr )1 p for all R > 0 and (1.10) (∫ ∞ R r−qµ(r)dr )1 q(∫ R 0 [∫ r 0 ν(t)dt ]−p0 rp0ν(r)dr )1 p0 ≤ A for all R > 0. The difficulty concerning the application of our Proposition is therefore about a realization of (1.7). In fact we will see below that this boundedness is implied by condition (1.4).
§2. Proofs of the Results
The proof of Theorem is first given. And next the Proposition will be justified.
Proof of Theorem.
The Necessary Part
Assume that Mα: L p
dec(v)→ L
q(u) and take R > 0. Observe that for each
f (.)≥ 0 supported by the ball B(0, R) Rα−n ∫ |y|<Rf (y)dy≤ 2 n−α (2R)α−n ∫ B(x,2R) f (y)dy ≤ 2n−α (Mαf )(x) for|x| < R.
Consequently if f (.) satisfies (1.1) then
Rα−n (∫ |z|<Rf (z)dz )(∫ |x|<Ru(x)dx )1 q ≤ (2n−α C) (∫ |y|<Rf p (y)v(y)dy )1 p .
Condition (1.2) follows from this last inequality by taking f (y) = ϕ(|y|), where
ϕ(.) is the nonincreasing function defined on ]0,∞[ by ϕ(r) = 1 for 0 < r < R
and ϕ(r) = 0 otherwise.
Conditions (1.9) and (1.10), with µ(t) and ν(t) defined as in lemma, are satisfied because of our Proposition, Lemma and E. Sawyer’s result [Sa] quoted above. These conditions, after standard computations, can be written as (2.1) (∫ |x|<R1/n|x| αqu(x)dx )1 q ≤ A (∫ |y|<R1/n v(y)dy )1 p for all R > 0 and (2.2) (∫ R1/n<|x||x| (α−n)qu(x)dx )1 q × (∫ |y|<R1/n [∫ |z|<|y|v(z)dz ]−p0 |y|np0 v(y)dy )1 p0 ≤ A for all R > 0
Condition (1.3) follows directly from (2.2). The constant A in (2.1) and (2.2) is different from that in (1.9) and (1.10). Such a standard notation abuse might occur in many places of the text.
The Sufficient Part
To derive Mα : L p
dec(v) → L
q(u), by our Proposition, the task remains to
check (1.6) and (1.7).
Boundedness in (1.6) holds under conditions (2.2) and (2.1) due to the above arguments. Here (2.2) is satisfied because it is the same as condition (1.3). And (2.1) also arises since it is just implied by condition (1.2).
Now let us prove that (1.7) holds under condition (1.4). To this end, an idea used in [Ra2] (see p.p. 99-100) is exploited. Consider f (.) ≥ 0 with
f (y) = ϕ(|y|) for some ϕ(.) ≥ 0 nonincreasing on ]0, ∞[. The conclusion arises
since ∫ Rn(Mαf ) q (x)u(x)dx = ∞ ∑ k=−∞ ∫ 2k≤|x|<2k+1 [ sup j≥1;j integers { (2j|x|)α−n ∫ 2j|x|≤|y|<2j+1|x| f (y)dy }]q u(x)dx ≤c1 ∞ ∑ k=−∞ [ sup j≥1;j integers { (2j+k)α−n ∫ 2j+k≤|y|<2j+k+2 f (y)dy }q](∫ 2k≤|x|<2k+1 u(x)dx ) ≤c1 ∞ ∑ k=−∞ ∞ ∑ j=1 [ (2j+k)α−n ∫ 2j+k≤|y|<2j+k+2 f (y)dy ]q(∫ 2k≤|x|<2k+1 u(x)dx ) =c1 ∞ ∑ m=−∞ [ 2m(α−n) ∫ 2m≤|y|<2m+2 f (y)dy ]q( m∑−1 k=−∞ ∫ 2k≤|x|<2k+1 u(x)dx )
≤c2 ∞ ∑ m=−∞ [ ϕp(2m) ]q p 2mαq (∫ |x|<2m u(x)dx ) since ϕ(.) is nonincreasing ≤c2Aq ∞ ∑ m=−∞ ( ϕp(2m) ∫ 2m−1≤|y|<2m v(y)dy )q p by condition (1.4) ≤c2Aq ∞ ∑ m=−∞ (∫ 2m−1≤|y|<2m fp(y)v(y)dy )q p
again by the decrease of ϕ(.)
≤c2Aq ( ∑∞ m=−∞ ∫ 2m−1≤|y|<2m fp(y)v(y)dy )q p because q p ≥ 1 =c2Aq (∫ Rnf p(y)v(y)dy )q p .
Proof of the Proposition.
The Necessary Part
Assume Mα: Lpdec(v)→ L
q(u) holds. The boundedness in (1.6) is satisfied
since for each function f (.)≥ 0
|x|α−n
∫
|y|<|x|
f (y)dy≤ 2n−α(Mαf )(x) for x6= 0.
Similarly the boundedness in (1.7) is true because for all integers j≥ 1
(2j|x|)α−n ∫
2j|x|≤|y|<2j+1|x|
f (y)dy≤ 22(n−α)(Mαf )(x) for x6= 0.
The Sufficient Part
To derive Mα: L p
dec(v)→ L
q(u) the main point is that for a fixed constant
c3> 0 and for each nonnegative function f (.) (Mαf )(x)≤ c3 ( F1(x) +F2(x) +F3(x) +F4(x) ) , where F1(x) = sup 0<t { tα−n ∫ B(x,t)∩{|y|<2−1|x|} f (y)dy } F2(x) = sup 0<t<2−1|x| { tα−n ∫ B(x,t)∩{2−1|x|≤|y|<2|x|} f (y)dy } F3(x) = sup 2−1|x|≤t { tα−n ∫ B(x,t)∩{2−1|x|≤|y|<2|x|} f (y)dy }
and F4(x) = sup |x|≤t { tα−n ∫ B(x,t)∩{2|x|≤|y|} f (y)dy } .
This cut out of the operator Mα was proved in [Ra2] (see p.p. 97-100). It is
also seen in this previous paper that a fixed constant c4> 0 exists such that
(2.3) F4(x)≤ c4(Mαf )(x).
Therefore the problem remains to get the estimates
(2.4) ∫ RnF q i(x)u(x)dx≤ C q i (∫ Rnf p(y)v(y)dy) q p for all i∈ {1, 2, 3, 4}.
Once again Ci > 0 is a constant which does not depend on the function f (.).
And from now it is assumed that f (y) = ϕ(|y|) for some function ϕ(.) ≥ 0 nonincreasing on ]0,∞[.
The estimate (2.4) for i = 1 is satisfied because of the boundedness (1.6) and since F1(x)≤ 2n−α|x|α−n
∫
|y|<|x|f (y)dy.
The estimate (2.4) for i = 2 arises once F2(x) ≤ c5|x|α−n ∫
|y|<|x|f (y)dy.
And this last inequality is true by the nonincreasing property of ϕ(.) since
F2(x)≤ c6|x|αϕ(2−1|x|) ≤ c7|x|α−n ∫
|y|<2−1|x|f (y)dy.
Similarly the estimate (2.4) for i = 3 follows from F3(x) ≤
c8|x|α−n ∫
|y|<|x|f (y)dy which is satisfied because
F3(x)≤ c9|x|α−n ∫ 2−1|x|<|y|<2|x| f (y)dy ≤ c10|x|αϕ(2−1|x|) ≤ c11|x|α−n ∫ |y|<2−1|x|f (y)dy.
Finally the estimate (2.4) for i = 4 appears from (2.3) and the boundedness in (1.7).
References
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norm, Studia Math. 108 (1994), 103-126.
[Ra1] Y. Rakotondratsimba, Weighted inequalities for the fractional integral operators on
monotone functions, Zeit. Anal. Anw. 15 (1996), 75-93.
[Ra2] Y. Rakotondratsimba, Two-weight norm inequality for the fractional maximal
[Ra3] Y. Rakotondratsimba, Weighted norm inequalities for the Hardy-Littlewood
maxi-mal operator on radial and nonincreasing functions, Rendiconti Mat. Serie VII, 18
(1998), 487-496.
[Sa] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158.
[St] V. Stepanov, Integral operators on the cone monotone functions, J. London Math. Soc. 48 (1993), 465-487.
Yves Rakotondratsimba
Institut polytechnique St Louis, EPMI
13 bd de l’Hautil 95 092 Cergy Pontoise France
