POINCARĂ INEQUALITIES BASED ON BANACH FUNCTION SPACES ON METRIC MEASURE SPACES

MARCELINA MOCANU âVasile Alecsandriâ University of BacÄu, Faculty of Engineering, Department of Power Engineering, Mechatronics and Computer Science, 157 Calea MÄrÄĆeĆti, BacÄu, 600115, Romania, e-mail:mmocanu@ub.ro

Issue:

SSRSMI, Number 1, Volume XXIII

Section:

Volume 23, Number 1

Abstract:

We introduce a new type of first order PoincarĂ© inequality for functions defined on a metric measure space, that is an useful tool in the study of Newtonian spaces based on Banach function spaces. This PoincarĂ© inequality extends the Orlicz-PoincarĂ© inequality introduced by AĂŻssaoui (2004) and the PoincarĂ© inequality based on Lorentz spaces, introduced by Costea and Miranda (2011), that in turn generalize the well-known weak (1,p)-PoincarĂ© inequality. Using very recent results of Durand-Cartagena, Jaramillo and Shanmugalingam (2012, 2013), it turns out that every complete metric space X, endowed with a doubling measure and supporting a weak PoincarĂ© inequality based on a Banach function space is (thick) quasiconvex. We prove that the validity of the PoincarĂ© inequality based on a Banach function space, on a doubling metric measure space, implies a pointwise estimate involving an appropriate maximal operator.

Keywords:

metric measure space , Banach function space , weak upper gradient , Newtonian space , PoincarĂ© inequality , maximal operator .

Code [ID]:

SSRSMI201323V23S01A0008 [0003825]

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