M ath. Res. Lett.
c International Press 2007
14 (2007), no. 2, 227–237
WEAK TYPE (1,1) ESTIMATES FOR A CLASS OF DISCRETE ROUGH MAXIMAL FUNCTIONS Roman Urban and Jacek Zienkiewicz Abstract. We prove weak type (1, 1) estimate for the maximal function associated with 1 . As a consequence, the sequence [mα ] is universally the sequence [mα ], 1 < α < 1 + 1000 L1 -good.
1. Introduction and statement of the result Let
1 M >0 M
M∗ f (x) = sup
X
f (x − [mα ]), x ∈ Z.
0 0. Then we have (2.18)
x − 1 ≤ (m + s)α − mα ≤ x + 1
and hence (2.19)
cM α−1 ≤ csM α−1 ≤ x ≤ CsM α−1 .
Observe that for the increasing function g(m) = (m + s)α − mα we have g(m + 1) − g(m) ≈ sM α−2 . 1, for |x| ≤ CM, and thus for a fixed s there are at most 1 + s−1 M 2−α ≈ M x−1 different consecutive values of m for wchich (2.18) can hold. Since moreover, by (2.19), s has to satisfy s ≈ xM 1−α , the total number of solutions of (2.18) is bounded from above by CM x−1 (xM 1−α ) = CM 2−α . Hence we easily obtain ρM (x) ≤ CM −α for 0 < |x| < CM and the lemma follows.
3. Proof of Theorem 1.1 Let M ∗ f (x) = sup |µ2n ∗ f (x)|. n>0
With no loss of generality it suffices to show the weak type (1, 1) of M ∗ . We will use the argument of [5] and [11] adapted to our setting. Let λ > 0 and f ∈ `1 (Z). Let N = 2n , n ∈ N. We consider the Calder´ on-Zygmund decomposition X X f =g+ fs,j = bs + g, where |g| < λ and bs contains terms fs,j supported by Qs,j withR|Qs,j | ' 2αs , P −1 kf k`1 and kfs,j k`1 ' λ|Qs,j |. We do not assume fs,j = 0, ins,j |Qs,j | ≤ λ (N )
(N )
(N )
stead we decompose further each bs writing bs (x) = bs (x) + Bs (x) + gs (x), (N ) (N ) (N ) where bs (x) = χ{|bs |>λN } (x)bs (x), and for hs (x) = bs (x) − bs (x) we have P χQ (x) R (N ) (N ) (N ) (N ) (N ) Bs (x) = hs (x) − gs (x) and gs (x) = j 2s,j h . Consequently, αs Qs,j s X X X (3.1) f =g+ gs(N ) + Bs(N ) + bs(N ) . s
s
s
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R. URBAN AND J. ZIENKIEWICZ
Observe that
1 2αs
(N )
R Qs,j
|hs
(3.2)
| ≤ Cλ and, since Qs,j are mutually disjoint, we get X |g(x)| + |gs(N ) (x)| ≤ Cλ. s
We have ) {x : µN ∗ |b(N s |(x) > 0} =
[
([mα ] + {x : |bs(N ) |(x) > 0}).
m≈N
Thus, X
) |{x : µN ∗ |b(N s |(x) > 0}| =
) |{x : |b(N s |(x) > 0}|
m≈N ) =N |{x : |b(N s |(x) > λN }|.
Consequently (remember that N = 2n ), X X X ) |{x : µN ∗ |b(N s |(x) > 0}| ≤ (3.3)
s
s
N - dyadic
≤
X
X1 1 kbs k`1 ≤ kf k`1 . λ λ s (N )
Moreover, since for the fixed dyadic N the supports of Bs it is easy to see, for a fixed x ∈ Qs0 ,j0 , X X X (N ) N −1 Bn−s (x)2 ≤ N dyadic
s
N |{x : |bs |(x) > λN }|
N - dyadic
(x) are mutually disjoint, N −1 |bs0 (x)|2
{N - dyadic:N λ≥|bs0 (x)|}
X
+ λ2
(3.4)
N −1 χ{supp bs0 } (x))
N - dyadic
≤Cλ|bs0 (x)| + λ2 χ{supp bs0 } (x) X ≤Cλ (|bs (x)| + Cλχ{supp bs } (x). s
Hence by (3.4), we have (3.5)
λ−2
X
X
s
N - dyadic
(N )
N −1 kBn−s k2`2 ≤
C kf k`1 . λ
We will use the following lemma, Lemma 3.6. Let N = 2n , n ∈ N. For sufficiently small δ > 0 we have the following estimates, see [5], (3.7)
(N )
(N )
(N )
kµN ∗ Bn−s k2`2 ≤ CλkBn−s k`1 2−δs + 2−n kBn−s k2`2
and for s1 > s2 , (3.8)
(N )
(N )
(N )
|hµN ∗ Bn−s1 , µN ∗ Bn−s2 i| ≤ CλkBn−s2 k`1 2−δs1 .
DISCRETE ROUGH MAXIMAL FUNCTIONS
235
Proof. By Lemma 2.5, 1
µN ∗ µ ˜N (x) = CρN (x) + 2−n δ0 (x) + O(2−n(α+ 1000 ) ), where ρN satisfies C C = nα and ρN (0) = 0. Nα 2 Moreover, for |x| > CM and |x + h| > CM, |ρN (x)| ≤
|ρN (x + h) − ρN (x)| ≤
(3.9)
C |h| . 2nα 2nα
(N )
Let for s ≤ n, supp Bs ⊂ [0, 4 · 2nα ] and supp ρn ⊂ [0, 4 · 2nα ]. Denote by Fs,j the (N ) (N ) restriction of Bs to Qs,j By the estimate kBs k`1 ≤ λ2sα we have, (N )
(N )
(N )
(N )
A :=|hµN ∗ Bn−s1 , µN ∗ Bn−s2 i| = |hµN ∗ µ ˜N ∗ Bn−s1 , Bn−s2 i| * + X n (N ) (N ) (N ) −nα − 1000 kBn−s1 k`1 kBn−s2 k`1 + |ρn ∗ Fn−s1 ,j1 |, |Bn−s2 | ≤2 2
(3.10)
j1
+N
−1
(N ) (N ) |hBn−s1 , Bn−s2 i|. (N )
(N )
Observe that for s1 6= s2 the supports of Bn−s1 , Bn−s2 are disjoint and consequently the third summand is equal to zero. Consider R the second summand in (3.10). By the regularity estimate (3.9) and the fact that Fn−s,j = 0 we get in a standard way |ρN ∗ Fn−s1 ,j (x)| ≤
2(n−s1 )α kFn−s1 ,j k`1 2nα 2nα
for |x − xs1 ,j | > CN + C2(n−s1 )α , where xs1 ,j denotes the center of Qs1 ,j . Moreover, for any x, |ρN ∗ Fn−s1 ,j (x)| ≤ N −α kFn−s1 ,j k`1 ≤ λN −α 2(n−s1 )α . Consequently, we have X X |ρN ∗ Fn−s1 ,j (x)| ≤ 2−nα kFn−s1 ,j k`1 j
{j:|x−xs1 ,j |≤CN +C2(n−s1 )α }
X
+ 2−s1 α
2−nα kFn−s1 ,j k`1
{j:|x−xs1 ,j |>CN +C2(n−s1 )α }
≤Cλ max{N 1−α , 2−s1 α } + Cλ2−s1 α ≤ Cλ2−s1 δ . Thus we estimate the second summand in (3.10) by
X
(N ) −s1 δ
kBn−s2 k`1 |ρN ∗ Fn−s1 ,j | kBn−s2 k`1 .
≤ Cλ2
j
∞ `
Finally, we get n
A ≤ Cλ2−s1 δ kBn−s2 k`1 + CλkBn−s2 k`1 2− 1000 ≤ CλkBn−s2 k`1 2−δs1 . (N )
The assumption supp Bn−s ⊂ [0, 4 · 2nα ] can be removed in a standard way. The estimates (3.7), (3.8) follow.
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R. URBAN AND J. ZIENKIEWICZ
Now we are ready to give Proof of Theorem 1.1. Using Lemma 3.6 we have, X (N ) X X (N ) λ2 |{x : max | Bn−s ∗ µN (x)| ≥ cλ}| ≤ max | Bn−s ∗ µN (x)|2 N
N
x
s≥0
s≥0
X X X (N ) ≤ | Bn−s ∗ µN (x)|2 x
(3.11)
≤
N
X
s≥0 (N )
≤
(N )
|hµN ∗ Bn−s1 , µN ∗ Bn−s2 i|
N,s1 >s2
N,s≥0
X
(N )
X
kµN ∗ Bn−s k2`2 + 2 (N ) CλkBn−s k`1 2−δs
(N )
+ 2−n kBn−s k2`2 + 2
X
(N )
CλkBn−s2 k`1 2−δs1
N,s1 >s2
N,s≥0
≤Cλkf k`1 , where the second summand in the last inequality is estimated by (3.5). By (3.1) we have, X {sup |µN ∗ f (x)| > 4Cλ} ⊂ {sup |µN ∗ (|g| + |gs(N ) |)(x)| > Cλ} N
N
s
X (N ) X (N ) |Bn+s |)(x)| > Cλ} ∪ {sup |µN ∗ ( |Bn−s |)(x)| > Cλ} ∪ {sup |µN ∗ ( N
N
s>0
∪ {sup N
X
µN ∗
|bs(N ) |
s>0
> Cλ} =: S1 ∪ S2 ∪ S3 ∪ S4 .
s
In the above sum, by (3.2), the first set is empty if the constant C is sufficiently (N ) ∗∗∗ large. Since supp µN ∗ Bs ⊂ ∪s,j Q∗∗∗ s,j for s ≥ n, the set S3 is a subset of ∪s,j Qs,j P C ∗∗∗ and consequently |S3 | ≤ | ∪s,j Qs,j | ≤ C s,j |Qs,j | ≤ λ kf k`1 . Moreover by (3.11) S (N ) we have |S2 | ≤ Cλ kf k`1 . The set S4 ⊂ N,s {µN ∗ |bs |(x) > 0}. Hence by (3.3) P (N ) |S4 | ≤ N,s |{µN ∗ |bs |(x) > 0}| ≤ Cλ kf k`1 . The theorem follows. Acknowledgements. The authors want to thank to J. Rosenblatt for communicating [4] and other related papers, and to M. Christ for a discussion on the subject of the paper. References [1] M. Boshernitzan, G. Kolesnik, A. Quas, and M. Wierdl, Ergodic averaging sequences, J. Anal. Math. 95 (2005), 63–103. [2] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), no. 1, 39–72. ´ [3] ———, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Etudes Sci. Publ. Math. (1989), no. 69, 5–45. [4] Z. Buczolich, Universally L1 good sequences with gaps tending to infinity, arXiv:math. [5] M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. (2) 128 (1988), no. 1, 19–42. [6] J.-M. Deshouillers, Probl` eme de Waring avec exposants non entiers, Bull. Soc. Math. France 101 (1973), 285–295. [7] ———, Un probl` eme binaire en th´ eorie additive, Acta Arith. 25 (1973–1974), 393–403.
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[8] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. [9] S. A. Gritsenko, Three additive problems, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 6, 1198–1216. Translation in Russian Acad. Sci. Izv. Math. 41 (1993), no. 3, 447–464. [10] A. A. Karatsuba, Basic analytic number theory, Springer-Verlag, Berlin (1993), ISBN 3-54053345-1. [11] A. Seeger, T. Tao, and J. Wright, Pointwise convergence of lacunary spherical means, in Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), Vol. 320 of Contemp. Math., 341–351, Amer. Math. Soc., Providence, RI (2003). [12] J. G. van der Corput, Neue zahlentheoretische Absch¨ atzungen, Math. Z. 29 (1929), no. 1, 397– 426. Institute of Mathematics, Wroclaw University, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland E-mail address:
[email protected] Same address in Wroclaw E-mail address:
[email protected] 
