Weighted variable Sobolev spaces and capacity. (English) Zbl 1241.46019
Summary: We define weighted variable Sobolev capacity and discuss properties of capacity in the space \(W^{1,p(\cdot)} (\mathbb R^n, w)\). We investigate the role of capacity in the pointwise definition of functions in this space if the Hardy-Littlewood maximal operator is bounded on the space \(W^{1,p(\cdot)} (\mathbb R^n, w)\). Also the relation between the Sobolev capacity and Bessel capacity is shown.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
References:
| [1] | O. Ková\vcik and J. Rákosník, “On spaces Lp(x) and Wk,p(x),” Czechoslovak Mathematical Journal, vol. 41, no. 116, pp. 592-618, 1991. · Zbl 0784.46029 |
| [2] | L. Diening, “Maximal function on generalized Lebesgue spaces Lp(.),” Mathematical Inequalities and Applications, vol. 7, no. 2, pp. 245-253, 2004. · Zbl 1071.42014 |
| [3] | X. Fan and D. Zhao, “On the spaces Lp(x)(\Omega ) and Wm,p(x)(\Omega ),” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424-446, 2001. · Zbl 1028.46041 · doi:10.1006/jmaa.2000.7617 |
| [4] | L. Diening, P. Hästö, and A. Nekvinda, “Open problems in variable exponent Lebesgue and Sobolev spaces,” in Proceedings of the Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA ’04), pp. 38-58, Milovy, Czech, 2004. |
| [5] | S. Samko, “On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators,” Integral Transforms and Special Functions, vol. 16, no. 5-6, pp. 461-482, 2005. · Zbl 1069.47056 · doi:10.1080/10652460412331320322 |
| [6] | V. G. Maz’ja, Sobolev Spaces, Springer, Berlin, Germany, 1985. |
| [7] | L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1992. · Zbl 0804.28001 |
| [8] | J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford University Press, 1993. · Zbl 0780.31001 |
| [9] | T. Kilpeläinen, “Weighted Sobolev spaces and capacity,” Annales Academiæ Scientiarium Fennicæ. Mathematica, vol. 19, no. 1, pp. 95-113, 1994. · Zbl 0801.46037 |
| [10] | P. Harjulehto, P. Hästö, M. Koskenoja, and S. Varonen, “Sobolev capacity on the space W1,p(.)(\Bbb Rn),” Journal of Function Spaces and Applications, vol. 1, no. 1, pp. 17-33, 2003. · Zbl 1078.46021 · doi:10.1155/2003/895261 |
| [11] | P. Harjulehto and P. Hästö, “Lebesgue points in variable exponent spaces,” Annales Academiæ Scientiarium Fennicæ. Mathematica, vol. 29, no. 2, pp. 295-306, 2004. · Zbl 1199.46079 |
| [12] | P. Gurka, P. Harjulehto, and A. Nekvinda, “Bessel potential spaces with variable exponent,” Mathematical Inequalities and Applications, vol. 10, no. 3, pp. 661-676, 2007. · Zbl 1129.46025 |
| [13] | P. Hästö and L. Diening, “Muckenhoupt weights in variable exponent spaces,” preprint, http://www.helsinki.fi/ pharjule/varsob/publications.shtml. |
| [14] | P. A. Hästö, “Local-to-global results in variable exponent spaces,” Mathematical Research Letters, vol. 16, no. 2, pp. 263-278, 2009. · Zbl 1184.46033 · doi:10.4310/MRL.2009.v16.n2.a5 |
| [15] | S. G. Samko, “Convolution type operators in Lp(x),” Integral Transforms and Special Functions, vol. 7, no. 1-2, pp. 123-144, 1998. · Zbl 1023.31009 · doi:10.1080/10652469808819204 |
| [16] | B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Transactions of the American Mathematical Society, vol. 165, pp. 207-226, 1972. · Zbl 0236.26016 · doi:10.2307/1995882 |
| [17] | N. Miller, “Weighted Sobolev spaces and pseudodifferential operators with smooth symbols,” Transactions of the American Mathematical Society, vol. 269, no. 1, pp. 91-109, 1982. · Zbl 0482.35082 · doi:10.2307/1998595 |
| [18] | J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2000. · Zbl 0969.42001 |
| [19] | V. Kokilashvili and S. Samko, “Singular integrals in weighted Lebesgue spaces with variable exponent,” Georgian Mathematical Journal, vol. 10, no. 1, pp. 145-156, 2003. · Zbl 1046.42006 |
| [20] | E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, USA, 1970. · Zbl 0207.13501 |
| [21] | A. Almeida and S. Samko, “Characterization of Riesz and Bessel potentials on variable Lebesgue spaces,” Journal of Function Spaces and Applications, vol. 4, no. 2, pp. 113-144, 2006. · Zbl 1129.46022 · doi:10.1155/2006/610535 |
| [22] | R. Hunt, B. Muckenhoupt, and R. Wheeden, “Weighted norm inequalities for the conjugate function and Hilbert transform,” Transactions of the American Mathematical Society, vol. 176, pp. 227-251, 1973. · Zbl 0262.44004 · doi:10.2307/1996205 |
| [23] | G. Choquet, “Theory of capacities,” Annales de l’Institut Fourier, vol. 5, pp. 131-295, 1953-1954. · Zbl 0064.35101 · doi:10.5802/aif.53 |
| [24] | P. Hajłasz and J. Onninen, “On boundedness of maximal functions in Sobolev spaces,” Annales Academiæ Scientiarium Fennicæ. Mathematica, vol. 29, no. 1, pp. 167-176, 2004. · Zbl 1059.46020 |
| [25] | T. Kilpeläinen, “A remark on the uniqueness of quasi continuous functions,” Annales Academiæ Scientiarium Fennicæ. Mathematica, vol. 23, no. 1, pp. 261-262, 1998. · Zbl 0919.31006 |
| [26] | D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, vol. 314, Springer, New York, NY, USA, 1996. · Zbl 1039.76518 · doi:10.1063/1.868916 |
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