1) Averaging operators take the average of functions transformed by isometries. For discrete polynomial averages, the functions are transformed by integer shifts defined by a polynomial. 2) The theorem proved that for certain ranges of exponents p and q, discrete polynomial averages of order N map lp(Z) to lq(Z) with a decay rate of N−d(1/p - 1/q), where d is the degree of the polynomial. 3) The proof involves viewing the averages as projections of universal polynomial averages on Zd and applying Hausdorff-Young inequalities and estimates of exponential sums.

1. Improving estimates for discrete polynomial averaging operators
Vjekoslav Kovač (U. of Zagreb)
Joint work with
R. Han, M. T. Lacey, F. Yang (Georgia Tech), J. Madrid (UCLA)
Supported by HRZZ UIP-2017-05-4129 (MUNHANAP)
Zagreb Workshop on Operator Theory, Zagreb, June 29–30, 2020
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2. What are averaging operators?
Finite averages
ANf := 1
N
∑︀N
k=1
Tkf; N ∈ N
Assumptions: X normed space, Tk : X → X isometries
Obvious property: AN X→X ≤ 1
More general assumptions: X, Y normed spaces, Tk : X → Y, Tk X→Y = 1
Again: AN X→Y ≤ 1
Can the constant 1 be improved? Sometimes
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3. Norm-improving property
Finite averages
ANf := 1
N
∑︀N
k=1
Tkf; N ∈ N
Occasional additional property
AN X→Y ≤ c(N) < 1, c(N) → 0 as N → ∞
Appreciated in harmonic analysis, ergodic theory, etc.
We might want to find the optimal asymptotics of c(N)
All depends on the choice of Tk
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4. Example: Translations on Z
1 ≤ p ≤ q ≤ ∞, ak ∈ Z
Tk : ℓp(Z) → ℓq(Z), (Tkf)(m) := f(m + ak)
Finite averages on Z
(ANf)(m) = 1
N
∑︀N
k=1
f(m + ak); N ∈ N, m ∈ Z
Norm-improving property now depends on the choice of numbers ak
Several features comes into play, coming from the Fourier analysis, number theory, etc.
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5. Discrete polynomial averages
P: Z → Z polynomial with integer coefficients of degree d ≥ 2
Polynomial averages on Z
(ANf)(m) = 1
N
∑︀N
k=1
f(m + P(k)); N ∈ N, m ∈ Z
Theorem (R. Han, V. K., M. T. Lacey, J. Madrid, F. Yang (2019))
ANf ℓq(Z) ≤ C(P, p, q) N
−d( 1
p
− 1
q
)
f ℓp(Z)
in a certain range of exponents 1 ≤ p ≤ q ≤ ∞
Previously only the case P(x) = x2 was known — R. Han, M. T. Lacey, F. Yang (2019)
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6. Trying out easy examples
Theorem
(ANf)(m) = 1
N
∑︀N
k=1
f(m + P(k))
ANf ℓq(Z) ≤ C(P, p, q) N
−d( 1
p
− 1
q
)
f ℓp(Z)
Take f = 1{1,2,3,...,2P(N)} =⇒ optimality of the decay N
−d( 1
p
− 1
q
)
Take f = 1{P(1),P(2),...,P(N)} =⇒ necessary condition d
q
≥ d−1
p
Take f = 1{0} =⇒ necessary condition d−1
q
≥ d
p
− 1
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7. Range of exponents
Case d = 2
The theorem holds for
{︀
(p, q) : 1
q
≤ 1
p
,
2
q
>
1
p
,
1
q
>
2
p
− 1
}︀
When q = p the range specializes to 3
2
< p ≤ 2
The range in optimal modulo its boundary
Case d ≥ 3
The theorem holds for
{︀
(p, q) : 1
q
≤ 1
p
,
d2+d+1
q
>
d2+d−1
p
,
d2+d−1
q
>
d2+d+1
p
− 2
}︀
When q = p the range specializes to 2 − 2
d2+d+1
< p ≤ 2
No estimates outside the range
{︀
(p, q) : 1
q
≤ 1
p
,
d
q
≥ d−1
p
,
d−1
q
≥ d
p
− 1
}︀
When q = p this specializes to 2 − 1
d
≤ p ≤ 2
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8. Range of exponents
(ANf)(m) = 1
N
∑︀N
k=1
f(m + P(k))
ANf ℓq(Z) ≤ C(P, p, q) N
−d( 1
p
− 1
q
)
f ℓp(Z)
Best explained visually
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9. Sketch of the proof in the case d ≥ 3
Regard AN as “projections” of
Higher-dimensional “universal” polynomial averages
(̃︀ANf)(m1, m2, . . . , md) := 1
N
∑︀N
k=1
f(m1 + k, m2 + k2, . . . , md + kd);
N ∈ N, (m1, m2, . . . , md) ∈ Zd
We turn to proving
Theorem
̃︀ANf ℓq(Zd) d,p,q N
− d(d+1)
2
( 1
p
− 1
q
)
f ℓp(Zd)
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10. Sketch of the proof in the case d ≥ 3
Write (̃︀ANf)(m1, m2, . . . , md) as
1
N
N∑︁
k=1
∫︁
Td
̂︀f(t1, t2, . . . , td)e2πi((m1+k)t1+(m2+k2)t2+···+(md+kd)td)
dt1dt2 · · · dtd
=
∫︁
Td
̂︀f(t1, t2, . . . , td)SN(t1, t2, . . . , td)e2πi(m1t1+m2t2+···+mdtd)
dt1dt2 · · · dtd,
where
SN(t1, t2, . . . , td) := 1
N
∑︀N
k=1
e2πi(kt1+k2t2+···+kdtd); (t1, t2, . . . , td) ∈ Td = (R/Z)d are
the normalized exponential sums
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11. Sketch of the proof in the case d ≥ 3
Take q = p
Apply the Hausdorff–Young inequality twice to ̃︀ANf = −1
(︀
( f)SN
)︀
:
̃︀ANf ℓp (Zd) ≤ ̂︀fSN Lp(Td) ≤ ̂︀f Lp (Td) SN Ls(Td) ≤ f ℓp(Zd) SN Ls(Td),
where 1
s
= 1
p
− 1
p
= 2
p
− 1
Theorem (Vinogradov’s mean value conj. – J. Bourgain, C. Demeter, L. Guth (2016))
SN Ls(Td) ≤ C(d, s)N− d(d+1)
2s for d ≥ 3 and s > d(d + 1)
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12. Thank you for your attention!
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