Local Tb theorems in non-homogeneous metric spaces, and applications to caloric measure
Abstract
We prove a local Tb theorem with testing conditions on metric balls, for possibly non-doubling measures of polynomial growth on geometrically doubling metric spaces. It resolves the maximal-truncation version of Hofmann's local Tb problem for all 1<p,q≤∞, without assuming antisymmetry of the Calderón-Zygmund kernel. Our theorem allows accretive systems of complex testing measures satisfying a quantitative small-set condition; these measures need not be absolutely continuous with respect to the underlying measure. Testing requires only a one-level L0 bound for maximal truncations, with an alternative for balls on which the underlying measure concentrates in a set of small relative diameter. We also remove antisymmetry from the known buffered and unbuffered fixed-truncation criteria. The proof combines a mixed corona decomposition adapted to the two testing systems with two-measure L2 estimates for suppressed operators. As an application in the parabolic space ℙn, local time-directed A∞-type conditions on caloric and adjoint caloric measure imply L2(σ)-boundedness of the spatial gradient of the heat single layer potential for boundary measures σ with upper parabolic (n+1)-growth. This gives a parabolic counterpart of the harmonic-measure theorem of Mourgoglou and Tolsa without Ahlfors regularity, corkscrew or Harnack chain assumptions.
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BibTeX
@misc{merlo2026local,
title = {{Local $Tb$ theorems in non-homogeneous metric spaces, and applications to caloric measure}},
author = {Andrea Merlo and Mihalis Mourgoglou and Carmelo Puliatti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40135},
eprint = {2610.40135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40135},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local Tb theorems in non-homogeneous metric spaces, and applications to caloric measure
AU - Andrea Merlo
AU - Mihalis Mourgoglou
AU - Carmelo Puliatti
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40135
UR - https://arxiv.org/abs/2610.40135
AB - We prove a local Tb theorem with testing conditions on metric balls, for possibly non-doubling measures of polynomial growth on geometrically doubling metric spaces. It resolves the maximal-truncation version of Hofmann's local Tb problem for all 1<p,q≤∞, without assuming antisymmetry of the Calderón-Zygmund kernel. Our theorem allows accretive systems of complex testing measures satisfying a quantitative small-set condition; these measures need not be absolutely continuous with respect to the underlying measure. Testing requires only a one-level L^0 bound for maximal truncations, with an alternative for balls on which the underlying measure concentrates in a set of small relative diameter. We also remove antisymmetry from the known buffered and unbuffered fixed-truncation criteria. The proof combines a mixed corona decomposition adapted to the two testing systems with two-measure L^2 estimates for suppressed operators. As an application in the parabolic space ℙ^n, local time-directed A_∞-type conditions on caloric and adjoint caloric measure imply L^2(σ)-boundedness of the spatial gradient of the heat single layer potential for boundary measures σ with upper parabolic (n+1)-growth. This gives a parabolic counterpart of the harmonic-measure theorem of Mourgoglou and Tolsa without Ahlfors regularity, corkscrew or Harnack chain assumptions.
ER -