## Revista Matemática Iberoamericana

### Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Șerban Costea

#### Abstract

This article studies strong $A_{\infty}$-weights and Besov capacities as well as their relationship to Hausdorff measures. It is shown that in the Euclidean space ${\mathbb{R}}^n$ with $n\ge 2$, whenever $n-1 < s \le n$, a function $u$ yields a strong $A_\infty$-weight of the form $w=e^{nu}$ if the distributional gradient $\nabla u$ has sufficiently small $||\cdot||_{{\mathcal L}^{s,n-s}}({\mathbb{R}}^n; {\mathbb{R}}^n)$-norm. Similarly, it is proved that if $2\le n < p < \infty$, then $w=e^{nu}$ is a strong $A_\infty$-weight whenever the Besov $B_p$-seminorm $[u]_{B_p({\mathbb{R}}^n)}$ of $u$ is sufficiently small. Lower estimates of the Besov $B_p$-capacities are obtained in terms of the Hausdorff content associated with gauge functions $h$ satisfying the condition $\int_0^1 h(t)^{p'-1} \frac{dt}{t} < \infty$.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 23, Number 3 (2007), 1067-1114.

Dates
First available in Project Euclid: 27 February 2008

https://projecteuclid.org/euclid.rmi/1204128311

Mathematical Reviews number (MathSciNet)
MR2414503

Zentralblatt MATH identifier
1149.46028

#### Citation

Costea, Șerban. Strong $A_{\infty}$-weights and scaling invariant Besov capacities. Rev. Mat. Iberoamericana 23 (2007), no. 3, 1067--1114. https://projecteuclid.org/euclid.rmi/1204128311

#### References

• Adams, D. R. and Hedberg, L. I.: Function spaces and potential theory. Fundamental Principles of Mathematical Sciences 314. Springer-Verlag, Berlin, 1996.
• Adams, D. R. and Hurri-Syrjänen, R.: Besov functions and vanishing exponential integrability. Illinois J. Math. 47 (2003), no. 4, 1137-1150.
• Björn, J.: Poincaré inequalities for powers and products of admissible weights. Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 1, 175-188.
• 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. and Lang, U.: Bi-Lipschitz parameterization of surfaces. Math. Ann. 327 (2003), 135-169.
• Bourdon, M.: Une caractérisation algébrique des homéomorphismes quasi-Möbius. Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 1, 235-250.
• Bourdon, M. and Pajot, H.: Cohomologie $l_p$ et espaces de Besov. J. Reine Angew. Math. 558 (2003), 85-108.
• Carleson, L.: Selected problems on exceptional sets. Van Nostrand Mathematical Studies 13D. Van Nostrand, Princeton, NJ-Toronto, Ont.-London, 1967.
• 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.
• David, G. and Semmes, S.: Fractured fractals and broken dreams. Self-similar geometry through metric and measure. Oxford Lecture Series in Mathematics and its Applications 7. The Clarendon Press, Oxford University Press, New York, 1997.
• Dolzmann, G., Hungerbühler, N. and Müller, S.: Uniqueness and maximal regularity for nonlinear elliptic systems of $n$-Laplace type with measure valued right hand side. J. Reine Angew. Math. 520 (2000), 1-35.
• Doob, J. L.: Classical potential theory and its probabilistic counterpart. Fundamental Principles of Mathematical Sciences 262. Springer-Verlag, New York, 1984.
• Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften 153. Springer-Verlag, New York, 1969.
• Folland, G.: Real analysis. Modern techniques and their applications. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, New York, 1984.
• García-Cuerva, J. and Rubio de Francia, J. L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studies 116. North-Holland Publishing Co., Amsterdam, 1985.
• Gehring, F. W.: The $L^p$-integrability of the partial derivatives of quasiconformal mappings. Acta Math. 130 (1973), 265-277.
• Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies 105. Princeton University Press, Princeton, NJ, 1983.
• Greco, L., Iwaniec, T. and Sbordone, C.: Inverting the $p$-harmonic operator. Manuscripta Math. 92 (1997), 249-258.
• Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.
• Heinonen, J., Kilpeläinen, T. and Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1993.
• Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 261-262.
• Kinnunen, J. and Martio, O.: The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 367-382.
• Kinnunen, J. and Martio, O.: Choquet property for the Sobolev capacity in metric spaces. In Proceedings on Analysis and Geometry held in Novosibirsk, 285-290. Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000.
• Martio, O.: Capacity and measure densities. Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 1, 109-118.
• Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics 44. Cambridge University Press, Cambridge, 1995.
• Netrusov, Yu.: Metric estimates of the capacities of sets in Besov spaces. In Proc. Steklov Inst. Math. 190, 167-192. American Mathematical Society, Providence, RI, 1992.
• Netrusov, Yu.: Estimates of capacities associated with Besov spaces. J. Math. Sci. 78 (1996), 199-217.
• Peetre, J.: New thoughts on Besov spaces. Duke University Mathematics Series 1. Mathematics Department, Duke University, Durham, NC, 1976.
• Reshetnyak, Yu.: On the conformal representation of Alexandrov surfaces. In Papers on analysis, 287-304. Rep. Univ. Jyväskylä Dep. Math. Stat. 83. Univ. Jyväskylä, Jyväskylä, 2001.
• 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.: Some novel types of fractal geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford Univ. Press, New York, 2001.
• Yosida, K.: Functional Analysis. Sixth edition. Fundamental Principles of Mathematical Sciences 123. Springer-Verlag, Berlin-New York, 1980.