Weighted variational inequalities for the fractional Dunkl heat semigroup

Abstract

We investigate the convergence properties of the family of operators 𝒯_{bf N} f(x)=∑_j=N₁^N₂ v_j( e^{−a_j+1(−Δ_k)^s}f(x) − e^{−a_j(−Δ_k)^s}f(x) ), x∈ℝ^d, where {e^{−t(−Δ_k)^s}}_t>0 denotes the fractional heat semigroup generated by the Dunkl Laplacian Δ_k. Here bf N=(N₁,N₂)∈ℤ², N₁<N₂, the coefficients {v_j}_j∈ℤ form a bounded sequence of real numbers, and {a_j}_j∈ℤ is a monotone increasing sequence of nonnegative reals. The primary objective of this work is to establish boundedness results for these differential transform operators on weighted L^p(ℝ^d,dμ_k) spaces as well as on Dunkl BMO(ℝ^d) spaces. We also establish analogous boundedness properties for the associated maximal operator 𝒯^* f(x)=sup_{bf N} |𝒯_{bf N} f(x)| and study the pointwise convergence of the corresponding series. In addition, we prove that, for compactly supported functions, the maximal differential transform operator 𝒯^* exhibits local behaviour comparable to that of classical singular integral operators.

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BibTeX RIS

BibTeX

@misc{mukherjee2026weighted,
  title = {{Weighted variational inequalities for the fractional Dunkl heat semigroup}},
  author = {Suman Mukherjee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.03162},
  eprint = {2606.03162},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.03162},
}

RIS

TY  - UNPB
TI  - Weighted variational inequalities for the fractional Dunkl heat semigroup
AU  - Suman Mukherjee
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2606.03162
UR  - https://arxiv.org/abs/2606.03162
AB  - We investigate the convergence properties of the family of operators 𝒯_bf N f(x)=∑_j=N_1^N_2 v_j( e^−a_j+1(−Δ_k)^sf(x) − e^−a_j(−Δ_k)^sf(x) ), x∈ℝ^d, where {e^−t(−Δ_k)^s}_t>0 denotes the fractional heat semigroup generated by the Dunkl Laplacian Δ_k. Here bf N=(N_1,N_2)∈ℤ^2, N_1<N_2, the coefficients {v_j}_j∈ℤ form a bounded sequence of real numbers, and {a_j}_j∈ℤ is a monotone increasing sequence of nonnegative reals. The primary objective of this work is to establish boundedness results for these differential transform operators on weighted L^p(ℝ^d,dμ_k) spaces as well as on Dunkl BMO(ℝ^d) spaces. We also establish analogous boundedness properties for the associated maximal operator 𝒯^* f(x)=sup_bf N |𝒯_bf N f(x)| and study the pointwise convergence of the corresponding series. In addition, we prove that, for compactly supported functions, the maximal differential transform operator 𝒯^* exhibits local behaviour comparable to that of classical singular integral operators.
ER  -

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