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Riesz transform on manifolds with mixed ends for 1<q<2
Abstract
Let M1, ⋯, Mℓ be complete manifolds of the same dimension, where 2≤ ℓ∈ℕ. Suppose that each Mi satisfies two side Gaussian bounds. If some of these manifolds are parabolic and there exists a constant 1≤ ni≤ 2 such that for some xi∈ Mi with 1 ≤ r ≤ R < ∞, ci(R/r)ni≤ VolMi(B(xi,R))/VolMi(B(xi,r))≤ Ci(R/r)ni, by assuming that all the manifolds satisfy the relative connectedness of the annuli (RCA) condition introduced by Grigor'yan and Saloff-Coste, we show that the Riesz transform ∇ ℒ−1/2 is bounded on Lq(M) for each 1<q<2 on the gluing manifold M=M1 # M2 # ⋯ # Mℓ.
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BibTeX
@misc{li2026riesz,
title = {{Riesz transform on manifolds with mixed ends for $1<q<2$}},
author = {Bo Li and Tianjun Shen and Zihan Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23398},
eprint = {2609.23398},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23398},
}
RIS
TY - UNPB
TI - Riesz transform on manifolds with mixed ends for 1<q<2
AU - Bo Li
AU - Tianjun Shen
AU - Zihan Wang
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23398
UR - https://arxiv.org/abs/2609.23398
AB - Let M_1, ⋯, M_ℓ be complete manifolds of the same dimension, where 2≤ ℓ∈ℕ. Suppose that each M_i satisfies two side Gaussian bounds. If some of these manifolds are parabolic and there exists a constant 1≤ n_i≤ 2 such that for some x_i∈ M_i with 1 ≤ r ≤ R < ∞, c_i(R/r)^n_i≤ Vol_M_i(B(x_i,R))/Vol_M_i(B(x_i,r))≤ C_i(R/r)^n_i, by assuming that all the manifolds satisfy the relative connectedness of the annuli (RCA) condition introduced by Grigor'yan and Saloff-Coste, we show that the Riesz transform ∇ ℒ^−1/2 is bounded on L^q(M) for each 1<q<2 on the gluing manifold M=M_1 # M_2 # ⋯ # M_ℓ.
ER -