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The semigroup generated by linear fractional divergence form operators in Lp(ℝN): the constant coefficient case
Abstract
Let 0<s<1 and A be a constant symmetric positive definite matrix. Given the quadratic form QA(u)=∫RN A∇s u(x)· ∇s u(x) dx, where ∇s denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that QA defines a Dirichlet form. In strike contrast with the local case s=1, positive definiteness of a constant matrix is not sufficient for the Markov property when 0<s<1. We also describe the associated symmetric Markov semigroup and its Lp realizations, prove the existence of a smooth nonnegative heat kernel and establish a bounded H∞ functional calculus.
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BibTeX
@misc{carbotti2026semigroup,
title = {{The semigroup generated by linear fractional divergence form operators in $L^p(\mathbbR^N)$: the constant coefficient case}},
author = {Alessandro Carbotti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15809},
eprint = {2609.15809},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15809},
}
RIS
TY - UNPB
TI - The semigroup generated by linear fractional divergence form operators in L^p(ℝ^N): the constant coefficient case
AU - Alessandro Carbotti
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15809
UR - https://arxiv.org/abs/2609.15809
AB - Let 0<s<1 and A be a constant symmetric positive definite matrix. Given the quadratic form Q_A(u)=∫_R^N A∇^s u(x)· ∇^s u(x) dx, where ∇^s denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that Q_A defines a Dirichlet form. In strike contrast with the local case s=1, positive definiteness of a constant matrix is not sufficient for the Markov property when 0<s<1. We also describe the associated symmetric Markov semigroup and its L^p realizations, prove the existence of a smooth nonnegative heat kernel and establish a bounded H^∞ functional calculus.
ER -