L^p-Estimates for maximal averages along mixed homogeneous hypersurfaces in ℝ³

@misc{buschenhenke2026lpestimates,
  title = {{$L^p$-Estimates for maximal averages along mixed homogeneous hypersurfaces in $\mathbbR^3$}},
  author = {Stefan Buschenhenke and Wenjuan Li and Detlef Müller and Huiju Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.22012},
  eprint = {2608.22012},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22012},
}

TY  - UNPB
TI  - L^p-Estimates for maximal averages along mixed homogeneous hypersurfaces in ℝ^3
AU  - Stefan Buschenhenke
AU  - Wenjuan Li
AU  - Detlef Müller
AU  - Huiju Wang
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.22012
UR  - https://arxiv.org/abs/2608.22012
AB  - In this paper, we study L^p-estimates for maximal averaging operators ℳ along hypersurfaces S in ℝ^3 which are the graph of a mixed homogeneous function Φ which is analytic away from the origin. The closure of such a surface will pass through the origin, so that the usual transversality condition that had been imposed in many previous works on maximal averages along hypersurfaces will not hold even when Φ is analytic at the origin. As our main result, under mild assumptions which are satisfied for instance for every mixed homogeneous polynomial Φ, we determine the critical Lebesgue exponent p_c for which ℳ is L^p-bounded for every p>p_c, but unbounded for p<p_c, in terms of multiplicities of the real roots of the Hessian determinant of Φ. It turns out that the study of the contributions by neighborhoods of a certain type of roots is closely related to recent work by Dendrinos, Ikromov and the first and third author on sharp estimates for a maximal averaging operator along a transversal hypersurface of an ``exceptional'' class, whose L^p-boundedness had been an open problem for a long time and which has recently been established by means of their new theory of FIO-cone multipliers.
ER  -