Optimal dimension-dependent ℓ^p and ℓ^1,∞ estimates of the second-order discrete Riesz transforms

Abstract

In this paper we investigate the optimal dimension-dependent estimates of the second-order discrete Riesz transforms R_dis^(jk)f(n) = c_d∑_{m∈ℤ^d∖{0}} m_jm_k/(|m|^d+2)f(n−m), c_d=(Γ((d+2)/2))/π^d/2. For j≠ k and every fixed 1<p<∞, we prove that |R_dis^(jk)|_{ℓ^p→ℓ^p} = c_d[ 2/2^d/2 + (8/3+o(1))d/3^d/2 ] and 2c_d/2^d/2 ≤ |R_dis^(jk)|_{ℓ¹→ℓ^1,∞} ≤ c_d[ 2/2^d/2 + (8/3+o(1))d/3^d/2 ]. Since $ c_d\sim \sqrt{πd}\left(\frac{d}{2πe}\right)^{d/2} $ by Stirling's formula, thl ℓ^p estimates give a negative answer to the conjecture proposed by Bañuelos and Kim in \cite{BK2}. The diagonal case exhibits quite a different phenomenon: for every j and every 1<p<∞, R_dis^(jj) is neither bounded from ℓ^p(ℤ^d) to ℓ^p(ℤ^d) nor of weak type (1,1). Cancellation is restored for the operators R_dis^(jj−kk) =R_dis^(jj)−R_dis^(kk). For every fixed 1<p<∞, |R_dis^(jj−kk)|_{ℓ^p→ℓ^p} = 4c_d[ 1+(1+o(1))d/2^d/2 ], and 4c_d ≤ |R_dis^(jj−kk)|_{ℓ¹→ℓ^1,∞} ≤ 4c_d[ 1+(1+o(1))d/2^d/2 ].

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BibTeX RIS

BibTeX

@misc{tang2026optimal,
  title = {{Optimal dimension-dependent $\ell^p$ and $\ell^1,\infty$ estimates of the second-order discrete Riesz transforms}},
  author = {Hanli Tang and Zewei Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09770},
  eprint = {2609.09770},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09770},
}

RIS

TY  - UNPB
TI  - Optimal dimension-dependent ℓ^p and ℓ^1,∞ estimates of the second-order discrete Riesz transforms
AU  - Hanli Tang
AU  - Zewei Xu
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09770
UR  - https://arxiv.org/abs/2609.09770
AB  - In this paper we investigate the optimal dimension-dependent estimates of the second-order discrete Riesz transforms R_dis^(jk)f(n) = c_d∑_m∈ℤ^d∖{0} m_jm_k/(|m|^d+2)f(n−m), c_d=(Γ((d+2)/2))/π^d/2. For j≠ k and every fixed 1<p<∞, we prove that |R_dis^(jk)|_ℓ^p→ℓ^p = c_d[ 2/2^d/2 + (8/3+o(1))d/3^d/2 ] and 2c_d/2^d/2 ≤ |R_dis^(jk)|_ℓ^1→ℓ^1,∞ ≤ c_d[ 2/2^d/2 + (8/3+o(1))d/3^d/2 ]. Since $ c_d\sim \sqrt{πd}\left(\frac{d}{2πe}\right)^{d/2} $ by Stirling's formula, thl ℓ^p estimates give a negative answer to the conjecture proposed by Bañuelos and Kim in \cite{BK2}. The diagonal case exhibits quite a different phenomenon: for every j and every 1<p<∞, R_dis^(jj) is neither bounded from ℓ^p(ℤ^d) to ℓ^p(ℤ^d) nor of weak type (1,1). Cancellation is restored for the operators R_dis^(jj−kk) =R_dis^(jj)−R_dis^(kk). For every fixed 1<p<∞, |R_dis^(jj−kk)|_ℓ^p→ℓ^p = 4c_d[ 1+(1+o(1))d/2^d/2 ], and 4c_d ≤ |R_dis^(jj−kk)|_ℓ^1→ℓ^1,∞ ≤ 4c_d[ 1+(1+o(1))d/2^d/2 ].
ER  -

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