## Revista Matemática Iberoamericana

### Strong $A_\infty$-weights are $A_\infty$-weights on metric spaces

#### Abstract

We prove that every strong $A_\infty$-weight is a Muckenhoupt weight in Ahlfors-regular metric measure spaces that support a Poincaré inequality. We also explore the relations between various definitions for $A_\infty$-weights in this setting, since some of these characterizations are needed in the proof of the main result.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 27, Number 1 (2011), 335-354.

Dates
First available in Project Euclid: 4 February 2011

https://projecteuclid.org/euclid.rmi/1296828837

Mathematical Reviews number (MathSciNet)
MR2815740

Zentralblatt MATH identifier
1223.42016

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis

#### Citation

Korte, Riikka; Kansanen, Outi Elina. Strong $A_\infty$-weights are $A_\infty$-weights on metric spaces. Rev. Mat. Iberoamericana 27 (2011), no. 1, 335--354. https://projecteuclid.org/euclid.rmi/1296828837

#### References

• Baldi, A. and Franchi, B.: Mumford-Shah-type functionals associated with doubling metric measures. Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 1, 1-23.
• Björn, A. and Björn, J.: Nonlinear potential theory in metric spaces. To appear in EMS Tracts in Mathematics. European Mathematical Society, Zurich.
• Björn, J.: Boundary continuity for quasiminimizers on metric spaces. Illinois J. Math. 46 (2002), no. 2, 383-403.
• Bonk, M., Heinonen, J. and Saksman, E.: The quasiconformal Jacobian problem. In In the tradition of Ahlfors and Bers, III, 77-96. Contemp. Math. 355. Amer. Math. Soc., Providence, RI, 2004.
• Bonk, M., Heinonen, J. and Saksman, E.: Logarithmic potentials, quasiconformal flows, and $Q$-curvature. Duke Math. J. 142 (2008), no. 2, 197-239.
• Costea, Ş.: Strong $A_\infty$-weights and scaling invariant Besov capacities. Rev. Mat. Iberoam. 23 (2007), no. 3, 1067-1114.
• Costea, Ş.: Strong $A_\infty$-weights and Sobolev capacities in metric measure spaces. Houston J. Math. 35 (2009), no. 4, 1233-1249.
• David, G. and Semmes, S.: Strong $A_\infty$ weights, Sobolev inequalities and quasiconformal mappings. In Analysis and partial differential equations, 101-111. Lecture Notes in Pure and Appl. Math. 122. Dekker, New York, 1990.
• Di Fazio, G. and Zamboni, P.: Regularity for quasilinear degenerate elliptic equations. Math. Z. 253 (2006), no. 4, 787-803.
• Di Fazio, G. and Zamboni, P.: Strong $A_\infty$ weights and quasilinear elliptic equations. Matematiche (Catania) 60 (2005), no. 2, 513-518.
• Franchi, B., Gutiérrez, C.E. and Wheeden, R.L.: Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5 (1994), no. 2, 167-175.
• García-Cuerva, J. and Rubio de Francia, J.L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studies 116. North-Holland, Amsterdam, 1985.
• Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer-Verlag, New York, 2001.
• Heinonen, J. and Koskela, P.: Weighted Sobolev and Poincaré inequalities and quasiregular mappings of polynomial type. Math. Scand. 77 (1995), no. 2, 251-271.
• Heinonen, J. and Semmes, S.: Thirty-three yes or no questions about mappings, measures, and metrics. Conform. Geom. Dyn. 1 (1997), 1-12 (electronic).
• Laakso, T.J.: Plane with $A_\infty$-weighted metric not bi-Lipschitz embeddable to $\mathbbR^N$. Bull. London Math. Soc. 34 (2002), no. 6, 667-676.
• Maasalo, O.E.: The Gehring lemma in metric spaces. Preprint available at arXiv:0704.3916v3 [math.CA], 2008.
• Semmes, S.: Bi-Lipschitz mappings and strong $A_\infty$ weights. Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 211-248.
• Semmes, S.: On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_\infty$-weights. Rev. Mat. Iberoamericana 12 (1996), no. 2, 337-410.
• Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16 (2000), no. 2, 243-279.
• Strömberg, J.-O. and Torchinsky, A.: Weighted Hardy spaces. Lecture Notes in Mathematics 1381. Springer-Verlag, Berlin, 1989.
• Zatorska-Goldstein, A.: Very weak solutions of nonlinear subelliptic equations. Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 2, 407-436.