×

zbMATH — the first resource for mathematics

Strong \(A_{\infty}\)-weights and scaling invariant Besov capacities. (English) Zbl 1149.46028
This article studies strong \(A_\infty\)-weights and Besov capacities as well as their relationship to Hausdorff measures. A weight \(\omega\) is said to be an \(A_\infty\)-weight if there exist constants \(C\geq1\) and \(q>1\) such that \(\left(\frac{1}{| B| }\int_B \omega(x)^q\,dx\right)^{1/q}\leq\frac{1}{| B| }\int_B \omega(x)\,dx\) for all balls \(B\subset{\mathbb R}^n\). An \(A_\infty\)-weight \(\omega\) is called a strong \(A_\infty\)-weight if there exists a distance function \(\delta^1_\mu\) on \({\mathbb R}^n\) and a positive constant \(C\) such that \(C^{-1}\delta_\mu(x,y)\leq\delta_\mu^1(x,y)\leq C\delta_\mu(x,y)\), where \(\mu\) is the measure on \({\mathbb R}^n\) with density \(\omega\) and \(\delta_\mu(x,y)=\mu(B_{x,y})^{1/n}\), where \(B_{x,y}\) denotes the smallest closed ball which contains the points \(x\) and \(y\).
The author proves that in the Euclidean space \({\mathbb R}^n\) with \(n\geq2\), whenever \(n-1<s\leq n\), a function \(u\) yields a strong \(A_\infty\)-weight of the form \(\omega=e^{nu}\) if the distributional gradient \(\nabla u\) has sufficiently small \(\| .\| _{\mathcal{L}^{n,n-s}({\mathbb R}^n;\, {\mathbb R}^n)}\)-norm, where \(\mathcal{L}^{n,n-s}({\mathbb R}^n)\) denotes the Morrey space of vector-valued measurable functions and
\[ \| \nabla u\| _{\mathcal{L}^{n,n-s}({\mathbb R}^n;\,{\mathbb R}^n)}=\sup_{x\in{\mathbb R}^n}\sup_{r>0}\left(r^{-(n-s)}\int_{B(x,r)}| \nabla u(y)| ^n\, dy\right)^{1/n}. \] As a corollary of this result, the author obtains strong \(A_\infty\)-weights of the form \(\omega=e^{nu}\), where \(u\) is a distributional solution of \(-\text{div}(| \nabla u| ^{n-2}\nabla u)=\mu\) whenever \(\mu\) is a signed Radon measure with small total variation. Similarly, the author also proves that if \(2\leq n<p<\infty\), then \(\omega=e^{nu}\) is a strong \(A_\infty\)-weight whenever the Besov \(B_p\)-seminorm \([u]_{B_p({\mathbb R}^n)}\) of \(u\) is sufficiently small, where \([u]_{B_p({\mathbb R}^n)}=(\int_{{\mathbb R}^n}\int_{{\mathbb R}^n}\frac{| u(x)-u(y)| ^p}{| x-y| ^{2n}}\,dx\,dy)^{1/p}\).
The author also develops a theory of the Besov \(B_p\)-capacity on \({\mathbb R}^n\) and proves that this capacity is a Choquet set function. Moreover, the author obtains lower estimates of the Besov \(B_p\)-capacities in terms of the Hausdorff content associated with gauge functions \(h\) satisfying the condition \(\int^1_0h(t)^{p'-1}\frac{dt}{t}<\infty\), where \(1/p+1/p'=1\).

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
31C99 Generalizations of potential theory
30C99 Geometric function theory
PDF BibTeX XML Cite
Full Text: DOI Euclid EuDML
References:
[1] Adams, D. R. and Hedberg, L. I.: Function spaces and potential theory . Fundamental Principles of Mathematical Sciences 314 . Springer-Verlag, Berlin, 1996. · Zbl 0834.46021
[2] Adams, D. R. and Hurri-Syrjänen, R.: Besov functions and vanishing exponential integrability. Illinois J. Math. 47 (2003), no. 4, 1137-1150. · Zbl 1044.46026 · www.math.uiuc.edu
[3] Björn, J.: Poincaré inequalities for powers and products of admissible weights. Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 1, 175-188. · Zbl 1002.46023 · emis:journals/AASF/Vol26/j-bjorn.html · eudml:122274
[4] 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. · Zbl 1069.30036
[5] Bonk, M. and Lang, U.: Bi-Lipschitz parameterization of surfaces. Math. Ann. 327 (2003), 135-169. · Zbl 1042.53044 · doi:10.1007/s00208-003-0443-8
[6] Bourdon, M.: Une caractérisation algébrique des homéomorphismes quasi-Möbius. Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 1, 235-250. · Zbl 1125.46042 · eudml:128132
[7] Bourdon, M. and Pajot, H.: Cohomologie \(l_p\) et espaces de Besov. J. Reine Angew. Math. 558 (2003), 85-108. · Zbl 1044.20026 · doi:10.1515/crll.2003.043
[8] Carleson, L.: Selected problems on exceptional sets . Van Nostrand Mathematical Studies 13D . Van Nostrand, Princeton, NJ-Toronto, Ont.-London, 1967. · Zbl 0189.10903
[9] 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. · Zbl 0752.46014
[10] 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. · Zbl 0887.54001
[11] 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. · Zbl 0937.35065 · doi:10.1515/crll.2000.022
[12] Doob, J. L.: Classical potential theory and its probabilistic counterpart . Fundamental Principles of Mathematical Sciences 262 . Springer-Verlag, New York, 1984. · Zbl 0549.31001
[13] Federer, H.: Geometric measure theory . Die Grundlehren der mathematischen Wissenschaften 153 . Springer-Verlag, New York, 1969. · Zbl 0176.00801
[14] 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. · Zbl 0549.28001
[15] 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. · Zbl 0578.46046
[16] Gehring, F. W.: The \(L^p\)-integrability of the partial derivatives of quasiconformal mappings. Acta Math. 130 (1973), 265-277. · Zbl 0258.30021 · doi:10.1007/BF02392268
[17] Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems . Annals of Mathematics Studies 105 . Princeton University Press, Princeton, NJ, 1983. · Zbl 0516.49003
[18] Greco, L., Iwaniec, T. and Sbordone, C.: Inverting the \(p\)-harmonic operator. Manuscripta Math. 92 (1997), 249-258. · Zbl 0869.35037 · doi:10.1007/BF02678192 · eudml:156263
[19] Heinonen, J.: Lectures on analysis on metric spaces . Universitext. Springer-Verlag, New York, 2001. · Zbl 0985.46008
[20] 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. · Zbl 0780.31001
[21] Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 261-262. · Zbl 0919.31006 · emis:journals/AASF/Vol23/vol23.html · eudml:230050
[22] Kinnunen, J. and Martio, O.: The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 367-382. · Zbl 0859.46023 · emis:journals/AASF/Vol21/vol21.html · eudml:227633
[23] 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. · Zbl 0992.46023
[24] Martio, O.: Capacity and measure densities. Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 1, 109-118. · Zbl 0408.31008
[25] Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability . Cambridge Studies in Advanced Mathematics 44 . Cambridge University Press, Cambridge, 1995. · Zbl 0819.28004
[26] 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. · Zbl 0789.46025
[27] Netrusov, Yu.: Estimates of capacities associated with Besov spaces. J. Math. Sci. 78 (1996), 199-217.
[28] Peetre, J.: New thoughts on Besov spaces . Duke University Mathematics Series 1 . Mathematics Department, Duke University, Durham, NC, 1976. · Zbl 0356.46038
[29] 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. · Zbl 1021.53041
[30] Semmes, S.: Bi-Lipschitz mappings and strong \(A_\infty\)-weights. Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 211-248. · Zbl 0742.46010 · emis:journals/AASF/Vol18/semmes.html · eudml:233152
[31] Semmes, S.: Some novel types of fractal geometry . Oxford Mathematical Monographs. The Clarendon Press, Oxford Univ. Press, New York, 2001. · Zbl 0970.28001
[32] Yosida, K.: Functional Analysis . Sixth edition. Fundamental Principles of Mathematical Sciences 123 . Springer-Verlag, Berlin-New York, 1980. · Zbl 0435.46002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.