Abstract
We establish a covering lemma of Besicovitch type for metric balls in the setting of Hölder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions of power decay, critical density and double ball property and with the aid of the covering theorem we show how these notions are related. Next we present an axiomatic procedure to establish Harnack inequality that permits to handle both divergence and non divergence linear equations.
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Aimar, H., Forzani, L., Toledano, R.: Holder regularity of solutions of PDE’s: a geometrical view. Commun. Partial Differ. Equ. 26, 1145–1173 (2002)
Aimar, H.: Singular integral and approximate identities on spaces of homogeneous type. Trans. Am. Math. Soc. 292(1), 135–153 (1985)
Biroli, M., Mosco, U.: Sobolev inequalities on homogeneous spaces. Pot. Anal. 4, 311–324 (1995)
Buckley, S.M.: Inequalities of John–Nirenberg type in doubling spaces. J. Anal. Math. 79, 215–240 (1999)
Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear elliptic equations. Ann. Math. 130, 189–213 (1989)
Caffarelli, L.A., Cabrè, X.: Fully nonlinear elliptic equations, volume 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1995
Caffarelli, L.A., Gutièrrez, C.E.: Real analysis related to the Monge–Ampère equation. Trans. Am. Math. Soc. 348(3), 1075–1092 (1996)
Caffarelli, L.A., Gutièrrez, C.E.: Properties of the solutions of the linearized Monge–Ampère equation. Am. J. Math. 119(2), 423–465 (1997)
Coifman, R., Weiss, G.: Analyse Harmonique Non-commutative Sur Certains Espaces Homogenes, volume 242 of Lecture Notes in Mathematics. Springer, Berlin-New York, 1971
Danielli, D., Garofalo, N., M-Nhieu, D.: Trace inequality for Carnot-Carathèodory distance. Ann. Scuola Norm. Sup. Pisa 27, 195–152 (1998)
Franchi, B., Lanconelli, E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10(4), 523–541 (1983)
Gutièrrez, C.E., Lanconelli, E.: Maximum principle, Harnack inequality, and Liouville theorems for X-elliptic operators. Commun. Partial Differ. Equ. 28(11–12), 1883–1862 (2003)
Garofalo, N., M-Nhieu, D.: Isoperimetric and Sobolev inequalities for Carnot-Carathèodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)
Gutièrrez, C.E.: A covering lemma of Besicovitch type on the Heisenberg group H n. unpublished manuscript, 1998
Gutièrrez, C.E.: The Monge–Ampère equation. Birkhäuser, Boston (2001)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, Berlin-New York (2001)
Hajlasz, P., Koskela, P.: Sobolev met Poincarè. In Memoirs of the American Mathematical Society 145, AMS (2000)
Han, Q., Lin, F-H.: Elliptic partial differential equations. In: Courant Lecture Notes in Mathematics, volume 1. Courant Institute (1997)
Korànyi, A., Reimann, H.M.: Foundations for the theory of quasiconformal mappings on the Heisenberg group. Adv. Math. 111(1), 1–87 (1995)
Kinnunen, J., Shanmugalingam, N.: Regularity of quasi-minimizers on metric spaces. Manuscr. Math. 105, 401–423 (2001)
Mosco, U.: Energy functionals on certain fractal structures. J. Convex Anal. 9(2), 581–600 (2002)
Mosco, U.: Harnack inequalities on recurrent metric fractals. Proc. Steklov Inst. Math 326, 490–495 (2002)
Macias, R.A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)
Rigot, S.: Counter example to the Besicovitch covering property for some Carnot groups equipped with their Carnot Carath eodory metric. Math. Z. 248(4), 827–848 (2004)
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Di Fazio, G., Gutièrrez, C.E. & Lanconelli, E. Covering theorems, inequalities on metric spaces and applications to PDE’s. Math. Ann. 341, 255–291 (2008). https://doi.org/10.1007/s00208-007-0188-x
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DOI: https://doi.org/10.1007/s00208-007-0188-x