Asymmetric L^p-improving and maximal estimates for the triangle averaging operator

Abstract

Let d≥2 and t>0. We study the bilinear triangle averaging operator 𝒯_t(f,g)(x)=∫_ℐf(x−ty)g(x−tz) dσ(y,z), where σ is the normalized surface measure on the manifold ℐ={(y,z)∈ℝ^d×ℝ^d:|y|=|z|=|y−z|=1} of equilateral triangles of side length one with a vertex at the origin. We prove the strong-type asymmetric L^p-improving estimate 𝒯₁:L^{(d+1)/(d−1)}(ℝ^d)× L^{(d²−1)/(d²−3d+4)}(ℝ^d)⟶ L^(d+1)/2(ℝ^d) in every dimension d≥4. This enlarges the range of exponents (p,q,r) for which the triangle averaging operator is known to map L^p(ℝ^d)× L^q(ℝ^d) to L^r(ℝ^d), and symmetrization produces new symmetric estimates in every such dimension as well. The proof combines Sobolev smoothing estimates, obtained through a microlocal decomposition of the triangle incidence relation, with complex interpolation in an analytic family of averages over triangles of varying shape that contains the triangle averaging operator. We also establish maximal estimates for triangle averages. Let E⊂[1,2] be a nonempty set of upper Minkowski dimension β, and let 𝒯_E(f,g)=sup_{t∈ E}|𝒯_t(f,g)|. We prove that |𝒯_E(f,g)|₂≲|f|_H^−s₁|g|_H^−s₂ whenever s₁,s₂≥0, s₁+s₂<(d−3−β)/2 and d>3+β. This improves the previously known Sobolev smoothing estimate by half a derivative and lowers the dimension requirement. We also obtain asymmetric localized maximal estimates of the form 𝒯_E:L^p(ℝ^d)× L^q(ℝ^d)→ L^r(ℝ^d) in an enlarged range of exponents.

end point estimates for the bilinear circular averaging operator
Xuyi Wei
arXiv17 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Sharp weighted estimates for the multilinear maximal operator
Tomasz Gałązka, Yong Jiao, Adam Osękowski +1
arXiv7 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Sharp L^p estimates for the strong spherical maximal operator
Mingfeng Chen
arXiv5 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Sharp power-weight estimates for the fractional spherical maximal operator
Riju Basak, Surjeet Singh Choudhary, Chun-Yen Shen +1
arXiv9 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Sharp weighted norm estimates for the positive Bergman operator: the two-parameter case
Benoit F. Sehba
arXiv8 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{iosevich2026asymmetric,
  title = {{Asymmetric $L^p$-improving and maximal estimates for the triangle averaging operator}},
  author = {Alex Iosevich and Eyvindur Ari Palsson and Minh-Quy Pham},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.42061},
  eprint = {2610.42061},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.42061},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Asymmetric L^p-improving and maximal estimates for the triangle averaging operator
AU  - Alex Iosevich
AU  - Eyvindur Ari Palsson
AU  - Minh-Quy Pham
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.42061
UR  - https://arxiv.org/abs/2610.42061
AB  - Let d≥2 and t>0. We study the bilinear triangle averaging operator 𝒯_t(f,g)(x)=∫_ℐf(x−ty)g(x−tz) dσ(y,z), where σ is the normalized surface measure on the manifold ℐ={(y,z)∈ℝ^d×ℝ^d:|y|=|z|=|y−z|=1} of equilateral triangles of side length one with a vertex at the origin. We prove the strong-type asymmetric L^p-improving estimate 𝒯_1:L^(d+1)/(d−1)(ℝ^d)× L^(d^2−1)/(d^2−3d+4)(ℝ^d)⟶ L^(d+1)/2(ℝ^d) in every dimension d≥4. This enlarges the range of exponents (p,q,r) for which the triangle averaging operator is known to map L^p(ℝ^d)× L^q(ℝ^d) to L^r(ℝ^d), and symmetrization produces new symmetric estimates in every such dimension as well. The proof combines Sobolev smoothing estimates, obtained through a microlocal decomposition of the triangle incidence relation, with complex interpolation in an analytic family of averages over triangles of varying shape that contains the triangle averaging operator. We also establish maximal estimates for triangle averages. Let E⊂[1,2] be a nonempty set of upper Minkowski dimension β, and let 𝒯_E(f,g)=sup_t∈ E|𝒯_t(f,g)|. We prove that |𝒯_E(f,g)|_2≲|f|_H^−s_1|g|_H^−s_2 whenever s_1,s_2≥0, s_1+s_2<(d−3−β)/2 and d>3+β. This improves the previously known Sobolev smoothing estimate by half a derivative and lowers the dimension requirement. We also obtain asymmetric localized maximal estimates of the form 𝒯_E:L^p(ℝ^d)× L^q(ℝ^d)→ L^r(ℝ^d) in an enlarged range of exponents.
ER  -

Abstract

Related

Cite