2024/12/20 by Daskalakis, Leonidas, Fragkos, Anastasios
#37A46 #42A45 #42B20 #42B25 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.15766
For c∈(1,2) we consider the following operators Ccf(x) = supλ∈ [-1/2,1/2)| ∑n ≠ 0f(x-n) \frace^2πiλ\lfloor |n|c \rfloorn|, Csgncf(x) = supλ∈ [-1/2,1/2)| ∑n ≠ 0f(x-n) \frace^2πiλsign(n) \lfloor |n|c \rfloorn| , and prove that both extend boundedly on ℓp(ℤ), p∈(1,∞). The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages ANf(x)=(1)/(N)∑n=1Nf(TnS\lfloor nc\rfloorx), where T,S\colon X→ X are commuting measure-preserving transformations on a σ-finite measure space (X,μ), and f∈ Lμp(X), p∈(1,∞). The point of departure for both proofs is the study of exponential sums with phases ξ2 \lfloor |nc|\rfloor+ ξ1n through the use of a simple variant of the circle method.