Oscillation Estimates for Truncated Singular Radon Operators

Abstract

In this paper we prove uniform oscillation estimates on $$L^p$$ L p , with $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , for truncated singular integrals of the Radon type associated with the Calderón–Zygmund kernel, both in continuous and discrete settings. In the discrete case we use the Ionescu–Wainger multiplier theorem and the Rademacher–Menshov inequality to handle the number-theoretic nature of the discrete singular integral. The result we obtained in the continuous setting can be seen as a generalisation of the results of Campbell, Jones, Reinhold and Wierdl for the continuous singular integrals of the Calderón–Zygmund type.

Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes

“…and, Słomian [37] later proved the counterpart of (1.24) in the case of the operators H P,k,0 t related to Calderón-Zygmund kernels satisfying (1.23). Lastly, in 2023, Słomian and the author [20] proved the oscillation and jump cases of Proposition 3.…”

“…In the proof of the above theorem, we use methods developed in [25,29,40] and very recently in [20,22,37]. We follow Bourgain's approach [5] to use the Calderón transference principle [7] which reduces the problem to the integer shift system (see Section 2.4) and then exploit the Hardy-Littlewood circle method to analyze the appropriate Fourier multipliers.…”

“…In 2003, Jones, Rosenblatt, and Wierdl [16] proved uniform oscillation inequalities on L p (X, µ) with p ∈ (1, 2] for the Birkhoff averages over cubes. However, the case of polynomial averages, even one-dimensional, was open until recent works [22,37], see [20] for the case of averages over primes.…”