ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

8. Analysis

1991 MS Classification: 42B 20, 42B 25, 11L 15 Wainger, Stephen, University of Wisconsin, Madison Discrete Analogues of Singular and Maximal Radon Transforms

For each x in R^{ℓ}, let γ(x, t) be a smooth k-dimensional surface through x in R^{k}. That is γ(x, t) is a smooth mapping from R^{ℓ} × R^{k} into R^{ℓ} with γ(x, 0) = x. Let f(x) be a function on R^{ℓ}. Then for suitably small η we define the singular and maximal Radon transforms respectively as

Sf(x) = ∫_{|t|≤η} f(γ(x, t))K(t)dt

and

Mf(x) = sup_{0<ϵ≤η} \frac{1}{|B(ϵ)|} ∫_{|t|≤ϵ} |f(γ(x, t))|dt

where K is a Calderón-Zymund kernel on R^{k}, and |B(ϵ)| denotes the measure of the ball of radius ϵ in R^{k}. The following is a rough statement of the result we have in mind for singular and maximal Radon transforms.

Theorem 1: (Christ, Nagel, Stein and Wainger) S and M are locally bounded in L^{p}, 1 < p < ∞, provided γ(x, t) satisfies an appropriate curvature hypothesis.

Corollary: If γ(x, t) is real analytic

lim_{ϵ→0} \frac{1}{|B(ϵ)|} ∫_{|t|<ϵ} f(γ(x, t))dt → f(x) a. e.

for f in L^{p}(R^{ℓ}). We are interested in discrete analogues of Theorem 1: We let P be a polynomial mapping from R^{ℓ} × R^{k} → R^{ℓ} with integer coefficients. Denote by Z^{ℓ} and Z^{k} the lattice points in R^{ℓ} and R^{k}, that is points with integer coordinates. For m ∈ Z^{ℓ}, set

Sf(m) = ∑_{n}′f(P (m, n))K(n)

where the prime denotes the term n = 0 is to be omitted, and

Mf(m) = sup_{N} \frac{1}{N^{k}} ∑_{|n|≤N} |f(P (m, n))|.

We are interested in estimates for S and M on ℓ^{p}(ℤ^{ℓ}). There are the following results:

Theorem 2: (Arkhipov and Oskolkov) (1987) If

P (m, n) = m + Q(n)
∥Sf∥_{ℓ^{2}} ≤ A∥f∥_{ℓ^{2}}.

Theorem 3: (Bourgain) (1988-1989) If

P (m, n) = m + Q(n)
∥Mf∥_{ℓ^{p}} ≤ A∥f∥_{ℓ^{p}}, 1 < p ≤ ∞.

Theorem 4: (Stein and Wainger) (1990) If

P (m, n) = m + Q(n)

∥Sf∥_{ℓ^{p}} ≤ A∥f∥_{ℓ^{p}}, \frac{3}{2} < p < 3.

Theorem 5: (Stein and Wainger) (1997) Let

m = (m′, m′′) with m′ ∈ z^{k}, m′′ ∈ z^{ℓ−k}.

If

P (m, n) = (m′ + n, m′′ + Q(m′, n))
∥Sf∥_{ℓ^{2}} ≤ A∥f∥_{ℓ^{2}}.

A related theorem is due to M. Wierdl.

Theorem 6: (1988) Let p_{n} denote the nth prime, and for m in Z set

Mf(m) = sup_{R} \frac{1}{R} ∑_{n=1}^{R} |f(m − p_{n})|.

Then

∥Mf∥_{ℓ^{p}} ≤ A∥f∥_{ℓ^{p}}, 1 < p ≤ ∞,

Theorems 3) and 6) have applications to ergodic theory. In our lecture we shall discuss the difficulties that arise in attacking the discrete problems and how methods of analytic number theory are used in the attack.