The document summarizes quantitative estimates for the convergence of multiple ergodic averages of commuting transformations. Specifically, it presents a theorem that provides an explicit bound on the number of jumps in the Lp norm for double averages over commuting Aω actions on a probability space. The proof transfers the structure of the Cantor group AZ to R+ and establishes norm estimates for bilinear averages of functions on R2+. This allows bounding the variation of the double averages and proving the theorem.

1. Quantitative norm convergence
of some ergodic averages
Vjekoslav Kovaˇc (University of Zagreb)
Probabilistic Aspects of Harmonic Analysis
Bedlewo, April 30, 2014

2. 1. Motivation: Several commuting transformations
Multiple averages
Mn(f1, f2, . . . , fr ) :=
1
n
n−1
k=0
(f1 ◦ Tk
1 )(f2 ◦ Tk
2 ) · · · (fr ◦ Tk
r )

3. 1. Motivation: Several commuting transformations
Multiple averages
Mn(f1, f2, . . . , fr ) :=
1
n
n−1
k=0
(f1 ◦ Tk
1 )(f2 ◦ Tk
2 ) · · · (fr ◦ Tk
r )
(X, F, µ) a probability space
T1, T2, . . . , Tr : X → X commuting, measure preserving:
Ti Tj = Tj Ti , µ T−1
i (E) = µ(E) for E ∈ F
f1, f2, . . . , fr ∈ L∞

4. 1. Motivation: Several commuting transformations
Multiple averages
Mn(f1, f2, . . . , fr ) :=
1
n
n−1
k=0
(f1 ◦ Tk
1 )(f2 ◦ Tk
2 ) · · · (fr ◦ Tk
r )
(X, F, µ) a probability space
T1, T2, . . . , Tr : X → X commuting, measure preserving:
Ti Tj = Tj Ti , µ T−1
i (E) = µ(E) for E ∈ F
f1, f2, . . . , fr ∈ L∞
Motivation:
Furstenberg and Katznelson, 1978
(multidimensional Szemer´edi’s theorem)

5. 1. Motivation: Several commuting transformations
Multiple averages
Mn(f1, f2, . . . , fr ) :=
1
n
n−1
k=0
(f1 ◦ Tk
1 )(f2 ◦ Tk
2 ) · · · (fr ◦ Tk
r )
Convergence in L2
as n → ∞:
r = 1 von Neumann, 1930
r = 2 Conze and Lesigne, 1984
r ≥ 3 Tao, 2008
generalizations Austin, 2010; Walsh, 2012

6. 1. Motivation: Several commuting transformations
Multiple averages
Mn(f1, f2, . . . , fr ) :=
1
n
n−1
k=0
(f1 ◦ Tk
1 )(f2 ◦ Tk
2 ) · · · (fr ◦ Tk
r )
Convergence a.e. as n → ∞:
r = 1 Birkhoff, 1931
r ≥ 2 a long-standing open problem Calder´on?
r = 2 and T2 = Tm
1 , m ∈ Z Bourgain, 1990
many other partial results
We do not discuss a.e. convergence here

7. 1. Motivation: Several commuting transformations
Single averages
Mn(f ) :=
1
n
n−1
k=0
f ◦ Tk
Quantify L2
convergence of the sequence:
control the number of jumps in the norm
bound the norm-variation

8. 1. Motivation: Several commuting transformations
Single averages
Mn(f ) :=
1
n
n−1
k=0
f ◦ Tk
Theorem (Jones, Ostrovskii, and Rosenblatt, 1996, p = 2;
Jones, Kaufman, Rosenblatt, and Wierdl, 1998, p ≥ 2)
sup
n0<n1<···<nm
m
j=1
Mnj−1 (f ) − Mnj (f )
p
Lp ≤ Cp f p
Lp

9. 1. Motivation: Several commuting transformations
Single averages
Mn(f ) :=
1
n
n−1
k=0
f ◦ Tk
Theorem (Jones, Ostrovskii, and Rosenblatt, 1996, p = 2;
Jones, Kaufman, Rosenblatt, and Wierdl, 1998, p ≥ 2)
sup
n0<n1<···<nm
m
j=1
Mnj−1 (f ) − Mnj (f )
p
Lp ≤ Cp f p
Lp
Consequence:
Mn(f )
∞
n=0
has O ε−p f p
Lp jumps of size ≥ ε in the Lp
norm
Avigad and Rute, 2013 in more general Banach spaces

10. 1. Motivation: Several commuting transformations
Double averages
Mn(f , g) :=
1
n
n−1
k=0
(f ◦ Sk
)(g ◦ Tk
)
Avigad and Rute, 2012 asked for any (reasonable/explicit)
quantitative estimates of norm convergence of multiple averages

11. 1. Motivation: Several commuting transformations
Double averages
Mn(f , g) :=
1
n
n−1
k=0
(f ◦ Sk
)(g ◦ Tk
)
Avigad and Rute, 2012 asked for any (reasonable/explicit)
quantitative estimates of norm convergence of multiple averages
Partial results:
T = Sm, m ∈ Z Bourgain, 1990; Demeter, 2007
variational estimate for the dyadic model of BHT
Do, Oberlin, and Palsson, 2012

12. 2. The result: Commuting Aω
actions
”Cantor group” model of the integers
Aω
:=
∞
k=1
A = A ⊕ A ⊕ · · ·
for some finite abelian group A

13. 2. The result: Commuting Aω
actions
”Cantor group” model of the integers
Aω
:=
∞
k=1
A = A ⊕ A ⊕ · · ·
for some finite abelian group A
Typically: A = Z/dZ
Aω
= a = (ak)∞
k=1 : (∃k0)(∀k > k0)(ak = 0)

14. 2. The result: Commuting Aω
actions
”Cantor group” model of the integers
Aω
:=
∞
k=1
A = A ⊕ A ⊕ · · ·
for some finite abelian group A
Typically: A = Z/dZ
Aω
= a = (ak)∞
k=1 : (∃k0)(∀k > k0)(ak = 0)
Følner sequence (Fn)∞
n=1: limn→∞
|(a+Fn) Fn|
|Fn| = 0
The most natural choice:
Fn = a = (ak)∞
k=1 : (∀k > n)(ak = 0) ∼= An

15. 2. The result: Commuting Aω
actions
”Cantor group” model of the integers
Aω
:=
∞
k=1
A = A ⊕ A ⊕ · · ·
for some finite abelian group A
S = (Sa)a∈Aω and T = (Ta)a∈Aω commuting measure preserving
Aω-actions on a probability space (X, F, µ):
Sa, Ta : X → X are measurable
S0 = T0 = id, SaSb = Sa+b, TaTb = Ta+b, SaTb = TbSa
µ(SaE) = µ(E) = µ(TaE) for a ∈ Aω, E ∈ F

16. 2. The result: Commuting Aω
actions
Toy-model of double averages
Mn(f , g) :=
1
|Fn|
a∈Fn
(f ◦ Sa
)(g ◦ Ta
)

17. 2. The result: Commuting Aω
actions
Toy-model of double averages
Mn(f , g) :=
1
|Fn|
a∈Fn
(f ◦ Sa
)(g ◦ Ta
)
Such multiple averages have already appeared in the literature:
general countable amenable group Bergelson, McCutcheon,
and Zhang, 1997; Zorin-Kranich, 2011; Austin, 2013
“powers” of the same action of (Z/pZ)ω, p prime
Bergelson, Tao, and Ziegler, 2013
Note: All known convergence results are only qualitative or
extremely weakly quantitative in nature

18. 2. The result: Commuting Aω
actions
Toy-model of double averages
Mn(f , g) :=
1
|Fn|
a∈Fn
(f ◦ Sa
)(g ◦ Ta
)
Theorem (K., 2014) p ≥ 2
sup
n0<n1<···<nm
m
j=1
Mnj−1 (f , g) − Mnj (f , g)
p
Lp ≤ Cp f p
L2p g p
L2p

19. 2. The result: Commuting Aω
actions
Toy-model of double averages
Mn(f , g) :=
1
|Fn|
a∈Fn
(f ◦ Sa
)(g ◦ Ta
)
Theorem (K., 2014) p ≥ 2
sup
n0<n1<···<nm
m
j=1
Mnj−1 (f , g) − Mnj (f , g)
p
Lp ≤ Cp f p
L2p g p
L2p
Consequences:
f L2p = g L2p =1 ⇒ O ε−p ε-jumps in Lp
, p ≥ 2
f L∞ = g L∞ =1 ⇒ O(ε−max{p,2}) ε-jumps in Lp
, p ≥ 1

20. 3. The proof: Cantor group structure on R+
G := (ak)k∈Z ∈ AZ
: (∃k0)(∀k > k0)(ak = 0)
Ultrametric:
ρ (ak)k∈Z, (bk)k∈Z :=
dk if k is the largest s.t. ak = bk
0 if (ak)k∈Z = (bk)k∈Z
Haar measure: λG

21. 3. The proof: Cantor group structure on R+
G := (ak)k∈Z ∈ AZ
: (∃k0)(∀k > k0)(ak = 0)
Ultrametric:
ρ (ak)k∈Z, (bk)k∈Z :=
dk if k is the largest s.t. ak = bk
0 if (ak)k∈Z = (bk)k∈Z
Haar measure: λG
Transfer the structure to R+ = [0, ∞):
Φ: G → R+, Φ: (ak)k∈Z → k∈Z akdk,
Ψ: R+ → G, Ψ: t → d−kt mod d k∈Z
x ⊕ y := Φ Ψ(x) + Ψ(y)
A = Z/dZ ⇒ ⊕ is addition in base d without carrying over digits

22. 3. The proof: Averages for functions on R+
Bilinear averages d = |A|
Ak(F, G)(x, y) := dk
[0,d−k )
F(x ⊕ t, y)G(x, y ⊕ t) dt

23. 3. The proof: Averages for functions on R+
Bilinear averages d = |A|
Ak(F, G)(x, y) := dk
[0,d−k )
F(x ⊕ t, y)G(x, y ⊕ t) dt
Proposition p ≥ 2, k0 < k1 < . . . < km integers
m−1
j=0
Akj+1
(F, G) − Akj
(F, G)
p
Lp
(R2
+)
≤ Cp F p
L2p
(R2
+)
G p
L2p
(R2
+)

24. 3. The proof: Averages for functions on R+
Proposition p ≥ 2, k0 < k1 < . . . < km integers
m−1
j=0
Akj+1
(F, G) − Akj
(F, G)
p
Lp
(R2
+)
≤ Cp F p
L2p
(R2
+)
G p
L2p
(R2
+)
Reductions:
Fix k0 < k1 < . . . < km
Assume that p ≥ 2 is an integer
Otherwise use complex interpolation of
L2p
× L2p
→ p
(Lp
) = Lp
( p
)
Assume F, G ≥ 0 ⇒ Ak(F, G) ≥ 0
Otherwise split into positive/negative, real/complex parts

25. 3. The proof: Averages for functions on R+
Lemma
For p ≥ 2 and a, b ∈ R:
|a − b|p
p |a|p
− |b|p
− p(a − b)b|b|p−2

26. 3. The proof: Averages for functions on R+
Lemma
For p ≥ 2 and a, b ∈ R:
|a − b|p
p |a|p
− |b|p
− p(a − b)b|b|p−2
Proof
Assume a = b = 0 and p > 2; substitute t = a−b
b = 0
θ(t) :=
|1 + t|p − 1 − pt
|t|p
Bernoulli’s inequality: θ(t) > 0 for t = 0
L’Hˆopital’s rule: limt→0 θ(t) = +∞, limt→±∞ θ(t) = 1
⇒ θ(t) ≥ cp > 0 for t = 0

27. 3. The proof: Averages for functions on R+
Proposition p ≥ 2, k0 < k1 < . . . < km integers
m−1
j=0
Akj+1
(F, G) − Akj
(F, G)
p
Lp
(R2
+)
≤ Cp F p
L2p
(R2
+)
G p
L2p
(R2
+)
Apply the lemma with
a = Akj+1
(F, G)(x, y) ≥ 0, b = Akj
(F, G)(x, y) ≥ 0

28. 3. The proof: Averages for functions on R+
Proposition p ≥ 2, k0 < k1 < . . . < km integers
m−1
j=0
Akj+1
(F, G) − Akj
(F, G)
p
Lp
(R2
+)
≤ Cp F p
L2p
(R2
+)
G p
L2p
(R2
+)
Apply the lemma with
a = Akj+1
(F, G)(x, y) ≥ 0, b = Akj
(F, G)(x, y) ≥ 0
Sum over j = 0, 1, . . . , m − 1 and telescope:
m−1
j=0
Akj+1
(F, G)(x, y) − Akj
(F, G)(x, y)
p
p Akm (F, G)(x, y)p
− Ak0 (F, G)(x, y)p
− p
m−1
j=0
Akj+1
(F, G)(x, y) − Akj
(F, G)(x, y) Akj
(F, G)(x, y)p−1

29. 3. The proof: Averages for functions on R+
Proposition p ≥ 2, k0 < k1 < . . . < km integers
m−1
j=0
Akj+1
(F, G) − Akj
(F, G)
p
Lp
(R2
+)
≤ Cp F p
L2p
(R2
+)
G p
L2p
(R2
+)
Integrate in (x, y) over R2
+:
LHS p Akm (F, G) p
Lp
(R2
+)
+ |Λ(F, G)|
where
Λ(F, G) :=
m−1
j=0 R2
+
Akj+1
(F, G)(x, y) − Akj
(F, G)(x, y)
Akj
(F, G)(x, y)p−1
dxdy

30. 3. The proof: Averages for functions on R+
Proposition p ≥ 2, k0 < k1 < . . . < km integers
m−1
j=0
Akj+1
(F, G) − Akj
(F, G)
p
Lp
(R2
+)
≤ Cp F p
L2p
(R2
+)
G p
L2p
(R2
+)
Expand out, denoting ϕk := dk1[0,d−k ):
Λ(F, G) =
m−1
j=0 Rp+2
+
F(x⊕t1, y) · · · F(x⊕tp, y) G(x, y ⊕t1) · · · G(x, y ⊕tp)
ϕkj+1
(t1) − ϕkj
(t1) ϕkj
(t2) · · · ϕkj
(tp) dxdydt1 · · · dtp
It suffices to prove:
|Λ(F, G)| p F p
L2p(R2
+)
G p
L2p(R2
+)

31. 3. The proof: Averages for functions on R+
Claim
|Λ(F, G)| p F p
L2p
(R2
+)
G p
L2p
(R2
+)
Substitute:
zi = x ⊕ y ⊕ ti , F(z, y) := F(z y, y), G(z, x) := G(x, z x)
ti = zi x y, F(x ⊕ ti , y) := F(zi , y), G(x, y ⊕ ti ) := G(zi , x)
Write:
ϕkj+1
− ϕkj
=
kj+1−1
r=kj
d−1
s=1
dr
h
(s)
[0,d−r )
h
(s)
I , s = 0, 1, . . . , d −1 d-adic Haar wavelets (L∞
-normalized)

32. 3. The proof: Averages for functions on R+
Claim
|Λ(F, G)| p F p
L2p
(R2
+)
G p
L2p
(R2
+)
Λ(F, G) =
m−1
j=0
kj+1−1
r=kj
d−1
s=1 Rp+2
+
F(z1, y) · · · F(zp, y) G(z1, x) · · · G(zp, x)
dr+(p−1)kj
h
(s)
[0,d−r )(z1 x y)
1[0,d−kj )(z2 x y) · · · 1[0,d−kj )(zp x y)
dxdydz1 · · · dzp
Split the integrals in x and y using d-adic intervals I and J
Use the character property of 1I and h
(s)
I
separates zi , x, and y

33. 3. The proof: Averages for functions on R+
Claim
|Λ(F, G)| p F p
L2p
(R2
+)
G p
L2p
(R2
+)
Λ(F, G) =
m−1
j=0
kj+1−1
r=kj
d−1
s=1 I,J∈I, |I|=|J|=d−r
K:=I⊕J, L:=dr−kj K
dr+(p−1)kj
Rp+2
+
F(z1, y) · · · F(zp, y) G(z1, x) · · · G(zp, x)
h
(s)
I (x)h
(s)
J (y)h
(s)
K (z1)1L(z2) · · · 1L(zp) dxdydz1 · · · dzp
Λ is reminiscent of perfect dyadic entangled multilinear dyadic
Calder´on-Zygmund operators K. and Thiele, 2013

34. 3. The proof: Averages for functions on R+
Claim
|Λ(F, G)| p F p
L2p
(R2
+)
G p
L2p
(R2
+)
In short:
Cauchy-Schwarzing: |Λ(F, G)| ≤ Θ(F)1/2 Θ(G)1/2
Telescoping:
Θ(F) + Θ(F)
≥0
= Ξkm (F) − Ξk0 (F)
≥0
Bounding a single-scale average:
Ξkm (F) ≤ F 2p
L2p
(R2
+)
= F 2p
L2p
(R2
+)

35. 3. The proof: The transfer argument
Claim
X
m
j=1
Mnj−1 (f , g)−Mnj (f , g)
p
dµ p
X
|f |2p
+|g|2p
dµ
We use the Calder´on transference principle
≡ the reverse Furstenberg correspondence principle

36. 3. The proof: The transfer argument
Claim
X
m
j=1
Mnj−1 (f , g)−Mnj (f , g)
p
dµ p
X
|f |2p
+|g|2p
dµ
Use the ergodic decomposition of invariant measures for
general group actions Varadarajan, 1963
⇒ Enough to prove the claim when the action
(SaTb)(a,b)∈Aω×Aω of Aω ×Aω on (X, F, µ) is ergodic
Express the integrals as limits of discrete averages using the
pointwise ergodic theorem for Aω ×Aω Lindenstrauss, 2001
Define F(a, b) := f (SaTbx0), G(a, b) := g(SaTbx0),
x0 is a generic point; extend F and G to R+ as “step”
functions; apply the proposition

37. 4. Possible future work
Open problem: double averages, Z+-actions or Z-actions
Mn(f , g) :=
1
n
n−1
k=0
(f ◦ Sk
)(g ◦ Tk
)
sup
n0<n1<···<nm
m
j=1
Mnj−1 (f , g) − Mnj (f , g)
p
Lp ≤ Cp f p
L2p g p
L2p

38. 4. Possible future work
Open problem: double averages, Z+-actions or Z-actions
Mn(f , g) :=
1
n
n−1
k=0
(f ◦ Sk
)(g ◦ Tk
)
sup
n0<n1<···<nm
m
j=1
Mnj−1 (f , g) − Mnj (f , g)
p
Lp ≤ Cp f p
L2p g p
L2p
Open problem: triple averages, “powers” of a single Aω-action
Mn(f , g, h) :=
1
|Fn|
a∈Fn
(f ◦ Tc1a
)(g ◦ Tc2a
)(h ◦ Tc3a
), c1, c2, c3 ∈ Z
sup
n0<···<nm
m
j=1
Mnj−1
(f , g, h) − Mnj
(f , g, h)
p
Lp ≤ Cp f p
L3p g p
L3p h p
L3p

39. Thank you!
Thank you for your attention!