# How to get rid of one of the weights in a two-weight Poincaré inequality?

Annales Polonici Mathematici (2000)

- Volume: 74, Issue: 1, page 97-103
- ISSN: 0066-2216

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topFranchi, Bruno, and Hajłasz, Piotr. "How to get rid of one of the weights in a two-weight Poincaré inequality?." Annales Polonici Mathematici 74.1 (2000): 97-103. <http://eudml.org/doc/208379>.

@article{Franchi2000,

abstract = {We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.},

author = {Franchi, Bruno, Hajłasz, Piotr},

journal = {Annales Polonici Mathematici},

keywords = {weights; doubling measures; metric spaces; Poincaré inequality; weighted Poincaré inequality; doubling measure},

language = {eng},

number = {1},

pages = {97-103},

title = {How to get rid of one of the weights in a two-weight Poincaré inequality?},

url = {http://eudml.org/doc/208379},

volume = {74},

year = {2000},

}

TY - JOUR

AU - Franchi, Bruno

AU - Hajłasz, Piotr

TI - How to get rid of one of the weights in a two-weight Poincaré inequality?

JO - Annales Polonici Mathematici

PY - 2000

VL - 74

IS - 1

SP - 97

EP - 103

AB - We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.

LA - eng

KW - weights; doubling measures; metric spaces; Poincaré inequality; weighted Poincaré inequality; doubling measure

UR - http://eudml.org/doc/208379

ER -

## References

top- [1] D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), 1033-1074.
- [2] M. Biroli and U. Mosco, Sobolev inequalities on homogeneous spaces, Potential Anal. 4 (1995), 311-324. Zbl0833.46020
- [3] B. Bojarski, Remarks on Sobolev imbedding inequalities, in: Complex Analysis (Joensuu, 1987), Lecture Notes in Math. 1351, Springer 1988, 52-68.
- [4] J. Boman, ${L}^{p}$-estimates for very strongly elliptic systems, report no 29, Dept. of Math., Univ. of Stockholm, 1982.
- [5] S. M. Buckley, P. Koskela and G. Lu, Boman equals John, in: XVI Rolf Nevanlinna Colloquium (Joensuu,1995), de Gruyter, Berlin, 1996, 91-99.
- [6] L. Capogna, D. Danielli and N. Garofalo, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1997), 147-154. Zbl0893.35023
- [7] S. K. Chua, Weighted Sobolev inequalities on domains satisfying the chain condition, Proc. Amer. Math. Soc. 117 (1993), 449-457. Zbl0812.46020
- [8] B. Franchi, Weighted Sobolev-Poincaré inequalities and pointwise inequalities for a class of degenerate elliptic equations, Trans. Amer. Math. Soc. 327 (1991), 125-158. Zbl0751.46023
- [9] B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), 523-604. Zbl0822.46032
- [10] B. Franchi, P. Hajłasz and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 1903-1924. Zbl0938.46037
- [11] B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 10 (1983), 523-541. Zbl0552.35032
- [12] B. Franchi, G. Lu and R. L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), 577-604. Zbl0820.46026
- [13] B. Franchi, G. Lu and R. L. Wheeden, A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type, Internat. Mat. Res. Notices 1996, no. 1, 1-14. Zbl0856.43006
- [14] B. Franchi, C. Pérez and R. L. Wheeden, Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Funct. Anal. 153 (1998), 108-146. Zbl0892.43005
- [15] B. Franchi, C. Pérez and R. L. Wheeden, in preparation.
- [16] B. Franchi and R. L. Wheeden, Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type, J. Inequalities Appl. 3 (1999), 65-89 Zbl0934.46037
- [17] B. Franchi and R. L. Wheeden, Compensation couples and isoperimetric estimates for vector fields, Colloq. Math. 74 (1997), 1-27. Zbl0915.46028
- [18] J. García Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Math. Stud. 116, North-Holland, 1985.
- [19] N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081-1144. Zbl0880.35032
- [20] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403-415. Zbl0859.46022
- [21] P. Hajłasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris 320 (1995), 1211-1215. Zbl0837.46024
- [22] P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000). Zbl0954.46022
- [23] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523. Zbl0614.35066
- [24] R. V. Kohn, New integral estimates for the deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal. 78 (1982), 131-172. Zbl0491.73023
- [25] E. Lanconelli and D. Morbidelli, On the Poincaré inequality for vector fields, Ark. Mat. 38 (2000), 327-342. Zbl1131.46304
- [26] G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana 8 (1992), 367-439. Zbl0804.35015
- [27] G. Lu, The sharp Poincaré inequality for free vector fields: an endpoint result, ibid. 10 (1994), 453-466. Zbl0860.35006
- [28] P. Maheux et L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique, Math. Ann. 303 (1995), 713-740.
- [29] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Mat. Res. Notices 1992, no. 2, 27-38. Zbl0769.58054

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