In this work, we extend the Euclidean theory of oscillating singular integrals due to Fefferman and Stein in [Fef70; FS72] to arbitrary graded Lie groups. Our approach reveals the strong compatibility between the geometric measure theory of a graded Lie group and the Fourier analysis associated with Rockland operators. Our criteria are presented in terms of the oscillating Fefferman condition of the kernel of the operator and its group Fourier transform. One of the novelties of this work is that we use the infinitesimal representation of a Rockland operator to measure the decay of the Fourier transform of the kernel.
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Keywords: Calderón–Zygmund operator, Weak (1,1) inequality, Oscillating singular integrals
Cardona, Duván 1 ; Ruzhansky, Michael 1 , 2
Cardona, Duván; Ruzhansky, Michael. Oscillating singular integral operators on graded Lie groups revisited. Journal of Lie Theory, Volume 36 (2026) no. 2-3, pp. 417-437. doi: 10.5802/jolt.1433
@article{10_5802_jolt_1433,
author = {Cardona, Duv\'an and Ruzhansky, Michael},
title = {Oscillating singular integral operators on graded {Lie} groups revisited},
journal = {Journal of Lie Theory},
pages = {417--437},
year = {2026},
publisher = {XXXX},
volume = {36},
number = {2-3},
doi = {10.5802/jolt.1433},
language = {en},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1433/}
}
TY - JOUR AU - Cardona, Duván AU - Ruzhansky, Michael TI - Oscillating singular integral operators on graded Lie groups revisited JO - Journal of Lie Theory PY - 2026 SP - 417 EP - 437 VL - 36 IS - 2-3 PB - XXXX UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1433/ DO - 10.5802/jolt.1433 LA - en ID - 10_5802_jolt_1433 ER -
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