BULLETIN Bull. Malaysian Math. Soc. (Second Series) 21 (1998) 101-105 of the
MALAYSIAN MATHEMATICAL SOCIETY
Some Weighted Estimates for Stein’s Maximal Function
HENDRA GUNAWAN
Department of Mathematics, Institut Teknologi Bandung, Jalan Ganesa 10, Bandung 40132, Indonesia e-mail: [email protected]
Abstract. In this brief article, we prove some weighted estimates for Stein’s maximal function by using interpolation techniques. In some cases, our results agree with those previously obtained in [2]
and [4].
Suppose f is a Schwartz function on Rn (n ≥ 3). For Re(α)> 0 and r> 0, define the operators Mα,r by
) ( )
( ,
, f x m f x
Mαr = αr ∗ where
( )
⎪⎪
⎩
⎪⎪⎨
⎧ Γ <
−
=
−
otherwise
, 0
1 if ) ,
( 1 ) (
2 1
x x x
m α
α α
and mα,r(x) = r−nmα(x/r)[5]. As is known,
) 2 ( )
(
ˆα ξ =π−α+1 ξ −n/2−α+1Jn/2+α−1 π ξ m
(see [6], p. 171). Thus, for α ∈C in general, we can define Mα,rf by the relation
(
Mα,rf)
∧(ξ) = mˆα(rξ)fˆ(ξ).(One may also define Mα,rf by analytic continuation; see [1] for how it works.) Observe that
f M f
M r r
2 1
,
0 =
where Mrf(x) denotes the average of f on the sphere of radius r centered at x.
Now define
. ) ( sup
)
( ,
0
x f M x
f
M r
r α
α = >
Here
{ }
Mα forms an analytic family of operators. Particularly we have fM f
M S
2 1
0 =
where MS f(x) = supr>0 Mrf(x) denotes Stein’s maximal function. Also note that f
cM f
M1 = HL
for some constant c. (Here MHLf denotes the well-known Hardy-Littlewood maximal function.) Stein [5] shows that
p p
p C f
f
Mα ≤ α,
if (a) Re(α) >1− pn′ for 1< p ≤ 2 or (b) Re(α)> 2p−n for 2 ≤ p ≤ ∞. As a consequence of this, one can derive the estimate
p p
Sf p C f
M ≤
provided that nn−1 < p ≤ ∞.
In this paper, we are concerned with the weighted estimate for Stein’s maximal function, namely
w w p w p
Sf p C f
M , ≤ , ,
for any possible values of p >1 and weights w∈Ap. (Here f pp,w p
Rn f x
∫
= ( )
. ) (x dx
w For definition of Ap weights, see [3].) With the above estimates for Mαf and the fact that Mαf is majorized by MHLf when Re(α) ≥1,we prove by using the Stein’s analytic interpolation theorem [6] (applied to the analytic family of operators
}
{Mα ) that the weighted estimate for Stein’s maximal function holds for some w∈Ap
where p > nn−1. Precisely, we have the following theorem:
Theorem. The weighted estimate
w w p w p
Sf p C f
M , ≤ , ,
holds for
(a) 1 2, 1 n .
p
n p
n p w A
− ′
− < ≤ ∈
(b) 2 , 2.
2− +−
∈
∞
≤
≤ p w Apnnp (c) 1 , 1( n2).
p
p n
n
Ap
w n
n p
n −
− −
∈ −
<
<
Remark. w∈Apq means that w can be written as w = vθ for some v∈Ap and .
0≤θ < q For power weights p
a w A
x x
w( )= , ∈ if and only if −n<a<n(p−1), and so w∈Apq means that −nq<a<n(p−1)q.
Proof.
(a) For 2,
1 < ≤
− p
n
n we have
p f n
C f
Mα p ≤ α,p p, Re(α) > 1− ′ . ,
1 ) Re(
, ,
,
,w ,pw pw p
p C f w A
f
Mα ≤ α α ≥ ∈
By the Stein’s analytic interpolation theorem,
, ,
, ,
, θ α, θ θ
αf pw C pw f pw
M ≤
Re( ) 1 , w Ap.
p n p
n ⎟ ∈
⎠
⎜ ⎞
⎝
⎛ − ′
′ +
> θ α
In particular, when α = 0, we have
, 1 0
,
, ,
,
, , n
A p w f
C f
MS pw pw pw p ′
−
<
≤
∈
≤ θ θ
θ θ
or equivalently
.
, 1
, , ,
n p
w p w p w p
S f p C f w A
M
− ′
∈
≤
(b) For 2≤ p ≤ ∞,we have
p f n
C f
M p p p
> −
≤ 2
) Re(
, , α
α α
. ,
1 ) Re(
, ,
,
,w ,pw pw p
p C f w A
f
Mα ≤ α α ≥ ∈
The Stein’s analytic interpolation theorem gives
, ,
, ,
, θ α, θ θ
αf pw C pw f pw
M ≤
2 2 , .
)
Re( w Ap
p n p
p
n ⎟ ∈
⎠
⎜ ⎞
⎝
− ⎛ −
⎟⎠
⎜ ⎞
⎝
⎛ + −
>θ α
When α =0, we have
2, 0 2
,
, ,
,
, ≤ , ∈ ≤ < +− −
p n A n
w f
C f
M p
w w p
w p
S p θ θ θ θ
or equivalently
.
, 2
2
, , ,
− +−
∈
≤ pnnp
w w p w p
Sf p C f w A
M
(c) For nn−1 < p < n, let q = p(nn−−p2). Then clearly q < p ≤ 2 or 2≤ p < q (depending on the value of p). Now, we have
1 2 ) Re(
2,
2
f n C f
Mα ≤ α α > −
. ,
1 ) Re(
, ,
,
,w ,qw qw q
q C f w A
f
Mα ≤ α α ≥ ∈
Interpolation will give
, ,
, ,
, θ α, θ θ
αf rw C rw f rw
M ≤
, , .
2 1 1 2 , 2 1
)
Re( q
A rt q w t t r n
tn ⎟ = − + ∈ q =
⎠
⎜ ⎞
⎝⎛ − +
> θ
α
When α = 0,we have
, ,
,
,wθ r,wθ rwθ
Sf r C f
M ≤
2.
0 , ,
2 , 1 1
n t n q A rt
q w t t
r q
< −
≤
=
∈
− +
= θ
Taking t arbitrarily close to nn−2, we get r close to p and θ close to 1− np. Hence we conclude
.
, 1
, , ,
n p
w q w p
w p
Sf p C f w A
M ≤ ∈ −
Remark. Estimate (a) is sharp in θ, in the sense that θ cannot be greater than 1− np′. This result has been previously obtained in [2] and [4]. Estimate (b) is certainly not sharp in θ, except for p = 2. Estimate (c) is better than (a) and (b), particularly for negative power weights. (For power weights w(x) = xa, (c) says that the estimate holds provided that p −n < a < np−n − p.)
Acknowledgement. This research was supported by The Young Academics Program, URGE Project, Directorate General of Higher Education, Ministry of Education and Culture. The author would also like to thank Professor Michael Cowling of UNSW, Sydney, Australia, for his helpful ideas about using interpolation techniques.
References
1. M. Cowling and G. Mauceri, Inequalities for some maximal functions II, Trans. Amer. Math.
Soc. 296 (1986), 341-365.
2. J. Duoandikoetxea and L. Vega, Spherical means and weighted estimates, J. London Math.
Soc. 53 (1996), 343-353.
3. J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.
4. H. Gunawan, On weighted estimates for Stein’s maximal function, Bull. Austral. Math. Soc.
54 (1996), 35-39.
5. E.M. Stein, Maximal functions: spherical means, Proc. Nat. Acad. Sci. USA 73 (1976), 2174-2175.
6. E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Uni. Press, 1971.
AMS Mathematics Subject Classification: 42B25
