TY - JOUR
T1 - A maximal Function Approach to Two-Measure Poincaré Inequalities
AU - Kinnunen, Juha
AU - Korte, Riikka
AU - Lehrbäck, Juha
AU - Vähäkangas, Antti V.
PY - 2019/4
Y1 - 2019/4
N2 - This paper extends the self-improvement result of Keith and Zhong in Keith and Zhong (Ann. Math. 167(2):575–599, 2008) to the two-measure case. Our main result shows that a two-measure (p, p)-Poincaré inequality for 1 < p< ∞ improves to a (p, p- ε) -Poincaré inequality for some ε> 0 under a balance condition on the measures. The corresponding result for a maximal Poincaré inequality is also considered. In this case the left-hand side in the Poincaré inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincaré inequalities is used to characterize the self-improvement of two-measure Poincaré inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.
AB - This paper extends the self-improvement result of Keith and Zhong in Keith and Zhong (Ann. Math. 167(2):575–599, 2008) to the two-measure case. Our main result shows that a two-measure (p, p)-Poincaré inequality for 1 < p< ∞ improves to a (p, p- ε) -Poincaré inequality for some ε> 0 under a balance condition on the measures. The corresponding result for a maximal Poincaré inequality is also considered. In this case the left-hand side in the Poincaré inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincaré inequalities is used to characterize the self-improvement of two-measure Poincaré inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.
KW - Geodesic two-measure space
KW - Poincaré inequality
KW - Self-improvement
UR - http://www.scopus.com/inward/record.url?scp=85050191212&partnerID=8YFLogxK
UR - https://arxiv.org/pdf/1801.06978.pdf
U2 - 10.1007/s12220-018-0061-z
DO - 10.1007/s12220-018-0061-z
M3 - Article
AN - SCOPUS:85050191212
SN - 1050-6926
VL - 29
SP - 1763
EP - 1810
JO - Journal of Geometric Analysis
JF - Journal of Geometric Analysis
IS - 2
ER -