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L^p-Poincar\'e inequalities on nested fractals
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L^p-Poincar\'e inequalities on nested fractals
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We prove on some nested fractals scale invariant $L^p$-Poincar\'e inequalities on metric balls in the range $1 \le p \le 2$. Our proof is based on the development of the local $L^p$-theory of Korevaar-Schoen-Sobolev spaces on fractals using heat kernel methods. Applications to scale invariant Sobolev inequalities and to the study of maximal functions and Haj\l{}asz-Sobolev spaces on fractals are given. Results are illustrated and further developed in the case of the Vicsek set.
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