A maximal inequality for the tail of the bilinear Hardy-Littlewood function

0.12 MB·English

by  I. Assani

A MAXIMAL INEQUALITY FOR THE TAIL OF THE BILINEAR
HARDY-LITTLEWOOD FUNCTION
8
I. ASSANI(*) AND Z. BUCZOLICH(**)
0
0
2
Abstract. Let(X, ,µ,T)beanergodicdynamicalsystemonanon-atomicfinite
n measure space. We aBssume without loss of generality that µ(X)=1. Consider the
a f(Tnx)g(T2nx)
J maximal function R∗ : (f,g) Lp Lq R∗(f,g)(x) = sup . We
9 ∈ × → n≥1 n
2 obtain the following maximal inequality. For each 1 < p there exists a finite
constantC suchthatforeachλ>0,andnonnegativefun≤cti∞onsf Lp andg L1
p ∈ ∈
]
DS µ{x:R∗(f,g)(x)>λ}≤Cp(cid:18)kfkpλkgk1(cid:19)1/2.
h. We also show that for each α>2 the maximalfunction R∗(f,g) is a.e. finite for
t pairs of functions (f,g) (L(logL)2α,L1).
a
∈
m
[

2
1. Introduction
v
9
Let (X, ,µ,T) be an ergodic dynamical system on a non-atomic finite measure
8
B
9 space. We assume without loss of generality that µ(X) = 1.
0
In [1] we proved the following maximal inequality about the maximal function
.
2 f(Tnx)g(T2nx)
1 R∗(f,g)(x) = sup . For each 1 < p , there exists a finite constant
7 n ≤ ∞
n≥1
0 C′ such that for each λ > 0, for every f Lp,f > 1 and g L1,g > 1
: p ∈ ∈
v
i
X f p g 1/2
r (1) µ x : R∗(f,g)(x) > λ C′ k kpk k1 .
a { } ≤ p(cid:18) λ (cid:19)
Furthermore the constant C′ behaves like 1 when p tends to 1. To be more precise,
p p−1
we will use that there exists C′ such that for any 1 < p < 2 we have
e C′
(2) C′ .
p ≤ p 1
−e
ThefirstauthoracknowledgessupportbyNSFgrantDMS0456627. Thesecondlistedauthorwas
partially supported by NKTH and by the Hungarian National Foundation for Scientific Research
T049727.
2000 Mathematics Subject Classification: Primary 37A05; Secondary 37A45.
Keywords: Maximal inequality, maximal function .
1
2 I.ASSANI(*)ANDZ. BUCZOLICH(**)
f(Tnx)g(T2nx)
Inequality(1)wasenoughtoprovethea.e. convergencetozeroofthetail
n
n
1
of the double recurrence averages f(Tkx)g(T2kx) for pairs of functions (f,g) in
n
X
k=1
Lp L1 (or L1 Lp) as soon as p > 1. On the other hand, in [2] the tail is used to
× ×
show that these averages do not converge a.e. for pairs of (L1,L1) functions.
During the 2007 Ergodic Theory workshop at UNC-Chapel Hill, J.P. Conze asked
if this inequality could be made homogeneous with respect to f and g. In this paper
first we derive from (1) the following homogeneous version.
Theorem 1. For each 1 < p < there exists a finite constant C such that for each
p
∞
f,g 0 and for all λ > 0 we have
≥
f(Tnx)g(T2nx) f g 1/2
p 1
(3) µ x : sup > λ C k k k k ,
p
n ≤ (cid:18) λ (cid:19)
(cid:8) n≥1 (cid:9)
and there exists C such that for any 1 < p < 2 we have
e C
(4) C .
p
≤ p 1
−e
At the same meeting a question was raised about the a.e. finiteness of R∗(f,g)
for pairs of functions in (LlogL,L1). Our second result is based on an adaptation of
Zygmund’s extrapolation method [4] (vol. II, ch. XII, pp. 119-120) to R∗(f,g). With
somewhat crude estimates we prove the following theorem.
Theorem 2. If α > 2 and the pair of nonnegative functions (f,g) belongs to
(L(logL)2α,L1) then R∗(f,g) = sup f(Tnx)g(T2nx) is a.e. finite.
n≥1 n
2. Proofs
Proof of Theorem 1. First we can notice that the original inequality (1) is homoge-
neous with respect to the L1 function g. Indeed, a simple change of variables shows
that the case g > t can easily be obtained from the case g > 1 with the same constant
C′. So by approximating g with g (x) = max g(x),1/n we can see that (1) holds
p n { }
if the assumption g > 1 is replaced by g 0. Without loss of generality we can also
≥
suppose in the sequel that g = 1.
1
k k
If f = 0 we have nothing to prove. Otherwise, if we can show that (3) holds for
p
k k
f = f/ f for all λ > 0, then this implies that it is true for f as well for all λ > 0.
p
k k
Thus, we just need to prove (3) for f Lp with f = 1.
p
e ∈ k k
Set
f(Tnx)g(T2nx)
M = µ x : sup > λ
n
(cid:8) n≥1 (cid:9)
THE TAIL OF THE BILINEAR HARDY-LITTLEWOOD FUNCTION 3
and h = max f,1 . By our remark about the assumption g 0 the maximal in-
equality (1) is{appl}icable and we obtain that M C′ khkpp 1/2, ≥and (2) also holds for
≤ p λ
1 < p < 2. As h 11 + f = 2 we have the es(cid:0)tima(cid:1)te
p p p
k k ≤ k k k k
1/2 1/2
1 f g
M 2p/2C′ = 2p/2C′ k kpk k1 ,
≤ p(cid:18)λ(cid:19) p(cid:18) λ (cid:19)
with C′ satisfying (2) for 1 < p < 2. Therefore, we obtain
p
f(Tnx)g(T2nx) f g 1/2 f g 1/2
µ x : sup > λ 2p/2C′ k kpk k1 C k kpk k1
n ≤ p(cid:18) λ (cid:19) ≤ p(cid:18) λ (cid:19)
n
(cid:8) (cid:9)
with C = 2p/2C′ and from (2) it follows that there exists C such that (4) holds for
p p
1 < p < 2. (cid:3)
e
Proof of Theorem 2. The starting point is (3) and (4).
There exists a finite constant C such that for every 1 < p < 2, for each f,g 0
≥
and for all λ > 0 we have
e
f(Tnx)g(T2nx) C f g 1/2
p 1
(5) µ x : sup > λ k k k k .
n ≤ p 1(cid:18) λ (cid:19)
(cid:8) n (cid:9) −e
Wecan againassume without loss of generality that g = 1.We fix the function g
1
k k
and denote by R∗(f)(x) the maximal function sup f(Tnx)g(T2nx). Now we can rewrite
n n
(5) as
1/2
C f
(6) µ x : R∗(f)(x) > λ k kp .
≤ p 1(cid:18) λ (cid:19)
(cid:8) (cid:9) −e
The important element for the extrapolation is the factor 1 in the above inequal-
p−1
ity.
Our goal is to prove that for α > 2 there is C such that for any f L(logL)2α we
α
∈
have for each λ > 0
1+ f (log+ f )2α 1/2
(7) µ x : R∗(f)(x) > λ C | | | | .
≤ α (cid:0)R λ1/2 (cid:1)
(cid:8) (cid:9)
∞
Let γ be a positive sequence of numbers such that γ = 1.
j j
Xj=0
∞
The function f being in L(logL)2α we have j2α2jµ x : 2j f < 2j+1 < .
≤ ∞
Xj=1 (cid:8) (cid:9)
We denote by t the quantity µ 2j f < 2j+1 , by f the function 2j11
j j
≤ x:2j≤f<2j+1
(cid:8) (cid:9)
(cid:8) (cid:9)
4 I.ASSANI(*)ANDZ. BUCZOLICH(**)
and by p the number 1+ 1. We set f (x) = f(x) if 0 f(x) < 2, otherwise we put
j j 0 ≤
f (x) = 0. Then
0
∞
(8) f 2 f .
j
≤
Xj=0
We also have
λγ
(9) µ x : R∗(f )(x) > 0
0
2 ≤
(cid:8) (cid:9)
λγ 4 g 4
µ x : R∗(2 11 )(x) > 0 k k1 =
X
· 2 ≤ λγ λγ
(cid:8) (cid:9) 0 0
by the standard maximal inequality for the ergodic averages (see [3] for instance).
For j 1 by (6) used with p = 1+ 1 we obtain
≥ j j
λγ
(10) µ x : R∗(f )(x) > j
j
2 ≤
(cid:8) (cid:9)
1 2j/2[t ]1/2pj j2j/2[t ]1/2pj
C j √2C j .
(1+(1/j)) 1 (cid:18) (λγ /2)1/2 (cid:19) ≤ (λγ )1/2
j j
−
e e
∞
We choose γ = 1/2 and γ = Cγ with β > 1 and C such that γ = 1.
0 j j(log(j+1))β γ j
Xj=0
√2C
Set C = .
1/2
C
γ e
Usinbg (8) and adding (9) and (10) for all j we obtain
∞ λγ 8 ∞ j2j/2 t 1/2pj
(11) µ x : R∗(f)(x) > λ µ R∗(f ) > j +√2C j
≤ j 2 ≤ λ (λγ(cid:2) )(cid:3)1/2 ≤
(cid:8) (cid:9) Xj=0 (cid:8) (cid:9) Xj=1 j
e
8 ∞ j3/2[log(j +1)]β/22j/2 t 1/2pj 8 A
j=1 j 1
+C = +C .
λ P λ1/2 (cid:2) (cid:3) λ λ1/2
To estimate A denbote by J the set of those j for which t1/2pj b3−j. Then
1 1 j ≤
∞
(12) j3/2[log(j +1)]β/22j/2 t 1/2pj j3/2[log(j +1)]β/22j/23−jd=efC .
j s
≤
jX∈J1 (cid:2) (cid:3) Xj=1
If j J then t1/2pj > 3−j, that is,
6∈ 1 j
−1j 1−(1+1j) 1 −1
3 > t2pj = t 2pj = t2pj 2,
j j j
THE TAIL OF THE BILINEAR HARDY-LITTLEWOOD FUNCTION 5
which implies t1/2pj < 3t1/2. Hence
j j
∞
(13) j3/2[log(j +1)]β/22j/2 t 1/2pj 3 j3/2[log(j +1)]β/22j/2 t 1/2d=efB .
j j 1
≤
jX6∈J1 (cid:2) (cid:3) Xj=1 (cid:2) (cid:3)
Suppose that α > δ > 2. By rewriting and applying the Cauchy–Schwartz inequality
we obtain with a suitable constant C that
δ
∞
B = 3 j3/2j−δ jδ log(j +1)]β/22j/2 t 1/2
1 j
≤
Xj=1 (cid:2) (cid:3) (cid:2) (cid:2) (cid:3)
∞ ∞
3 j3−2δ 1/2 j2δ log(j +1) β2jt 1/2 =
j
(cid:2)Xj=1 (cid:3) (cid:2)Xj=1 (cid:2) (cid:3) (cid:3)
∞
C j2δ log(j +1) β2jt 1/2d=efB .
δ j 2
(cid:2)Xj=1 (cid:2) (cid:3) (cid:3)
There exists C such that for all j = 1,2,...
δ,α,β
log(j +1) β C j2(α−δ).
δ,α,β
≤
(cid:2) (cid:3)
Hence,
1/2
(14) B B C C f (log+ f )2αdµ .
1 2 δ δ,α,β
≤ ≤ (cid:18)Z | | | | (cid:19)
By (11-14) we have
C +C C ( f (log+ f )2αdµ)1/2
µ x : R∗(f)(x) > λ C s δ δ,α,β | | | |
{ } ≤ R λ1/2
this implies (7) with a suitable C .b
α
(cid:3)
Remark 1. Inequality (7) implies also that for the pair of nonnegative functions
(f,g) in (L(logL)2α,L1) we have
f(Tnx)g(T2nx)
(15) lim = 0.
n n
Indeed, consider asequence ofboundedfunctions0 f f converging monotone
M
≤ ≤
increasingly to f L(logL)2α. Then we have
∈
f (Tnx)g(T2nx)
M
(16) lim = 0.
n n
Given ε (0,1) choose M so large that
∈
1/2
2 2
(17) I(M,ε,1/2)d=ef f f (log+ f f )2αdµ < 1.
(cid:18)Z ε2| − M| ε2| − M| (cid:19)
6 I.ASSANI(*)ANDZ. BUCZOLICH(**)
Then
f(Tnx)g(T2nx)
µ x : limsup > ε
{ n } ≤
n→∞
(f f )(Tnx)g(T2nx) ε f (Tnx)g(T2nx) ε
M M
µ x : limsup − > +µ x : limsup >
{ n 2} { n 2} ≤
n→∞ n→∞
(by using (16))
ε 2 1
µ x : R∗((f f ),g)(x) > = µ x : R∗( (f f ),g)(x) >
{ − M 2} { ε2 − M ε} ≤
(by using (7) and (17))
C √ε(1+I(M,ε,1/2)) 2C √ε.
α α
≤
Since this holds for any ε (0,1) we obtained (15).
∈
References
[1] I. Assani and Z. Buczolich: “The (Lp,Lq) BilinearHardy–Littlewoodmaximalfunction for
the tail”, Preprint 2007.
[2] I. Assani and Z. Buczolich: “The (L1,L1) Bilinear Hardy–Littlewoodmaximalfunction for
the tail”, in preparation.
[3] U. Krengel: “Ergodic theorems”, de Gruyter Studies in Mathematics, 6. Walter de Gruyter
& Co., Berlin, 1985.
[4] A. Zygmund:“TrigonometricSeries”,vol.I-II correctedsecondedition,CambridgeUniversity
Press, 1968.
(*) Idris Assani - Department of Mathematics- University of North Carolina at
Chapel Hill-email: [email protected]
(**)Zolt´anBuczolich-DepartmentofAnalysis, E¨otv¨osLor´andUniversity, P´azm´any
P´eter S´et´any 1/c, 1117 Budapest, Hungary -email: [email protected]