Abstract
In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev–Poincaré inequality. We show that this method gives a Sobolev–Poincaré inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev–Poincaré inequality depends on the domain. The Sobolev–Poincaré inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy–Littlewood maximal operator.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. In: Pure and Applied Mathematics Series, 2nd edition vol 140. Elsevier/Academic Press, Amsterdam (2003)
Cianchi, A.: A sharp embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45, 39–65 (1996)
Cianchi, A.: Strong and weak type inequalities for some classical operators in Orlicz spaces. J. Lond. Math. Soc. 2(60), 187–202 (1999)
Cianchi, A., Stroffolini, B.: An extension of Hedberg’s convolution inequality and applications. J. Math. Anal. Appl. 227, 166–186 (1998)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin (1998)
Garling, D.J.H.: Inequalities. A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)
Hajłasz, P.: Sobolev inequalities, truncation method, and John domains, Papers on Analysis: A volume dedicated to Olli Martio on the occasion of his 60th birthday. Edited by J. Heinonen, T. Kilpeläinen, and P. Koskela, Report Univ. Jyväskylä, 83, University of Jyväskylä, Jyväskylä, pp. 109–126 (2001)
Hajłasz, P., Koskela, P.: Isoperimetric inequalities and imbedding theorems in irregular domains. J. Lond. Math. Soc. 58(2), 425–450 (1998)
Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)
Harjulehto, P., Hurri-Syrjänen, R.: An embedding into an Orlicz space for \(L^1_1\)-functions from irregular domains, proceedings of Complex Analysis and Dynamical Systems VI. Part 1, 177–189, Contemp. Math. 653, Amer. Math. Soc., Providence, RI (2015)
Harjulehto, P., Hurri-Syrjänen, R., Kapulainen, J.: An embedding into an Orlicz space for irregular John domains. Comput. Methods Funct. Theory 14, 257–277 (2014)
Harjulehto, P., Hurri-Syrjänen, R., Vähäkangas, A.V.: On the \((1, p)\)-Poincaré inequality. Ill. J. Math. 56, 905–930 (2012)
Hedberg, L.I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36, 505–510 (1972)
Kilpeläinen, T., Maly, J.: Sobolev inequalities on sets with irregular boundaries. Z. Anal. Angew. 19, 369–380 (2000)
Kokilashvili, V., Krbec, M.: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, Singapore (1991)
Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Boundedness of maximal operators and Sobolev’s inequality on Musielak–Orlicz–Morrey spaces. Bull. Sci. Math. 137(1), 76–96 (2013)
Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd revised and augmented Edition, A Series of Comprehensive Studies in Mathematics, 342. Springer, Heidelberg Dordrecht London New York (2011)
Maz’ya, V., Poborchi, S.: Differentiable Functions on Bad Domains. World Scientific, Singapore (1997)
Moser, J.: A sharp form of an inequality of N. Trudinger’s inequality. Indiana Univ. Math. J. 11, 1077–1092 (1971)
Ohno, T., Shimomura, T.: Trudinger’s inequality for Riesz potentials of functions in Musielak–Orlicz spaces. Bull. Sci. Math. 138, 225–235 (2014)
O’Neil, R.: Fractional integration in Orlicz spaces. Trans. Am. Math. Soc. 115, 300–328 (1965)
Peetre, J.: Espaces d’interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16, 279–317 (1966)
Pohozhaev, S.I.: On the imbedding Sobolev theorem for pl=n. In: Doklady Conference, Section Math. Moscow Power Inst.(1965), pp. 158–170 (Russian)
Reshetnyak, Yu.G.: Integral representations of differentiable functions in domains with nonsmooth boundary (Russian), Sibirsk. Mat. Zh. 21(1980), 108–116; translation. Sib. Math. J. 21, 833–839 (1981)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Strichartz, R.S.: A note on Trudinger’s extension of Sobolev’s inequality. Indiana Univ. Math. J. 21, 841–842 (1972)
Torchinsky, A.: Interpolation of operators and Orlicz classes. Studia Math. 59, 177–207 (1976)
Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)
Yudovich, V.I.: On some estimates connected with integral operators and with solutions of elliptic equations (Russian), Dokl. Akad. Nauk SSSR 138 (1961), 805–808; translation in Sov. Math. Dokl. 2(1961), 746–749
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Harjulehto, P., Hurri-Syrjänen, R. Pointwise estimates to the modified Riesz potential. manuscripta math. 156, 521–543 (2018). https://doi.org/10.1007/s00229-017-0983-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-017-0983-y