Illinois Journal of Mathematics

Newtonian Lorentz metric spaces

Şerban Costea and Michele Miranda Jr.

Full-text: Open access


This paper studies Newtonian Sobolev–Lorentz spaces. We prove that these spaces are Banach. We also study the global $p,q$-capacity and the $p,q$-modulus of families of rectifiable curves. Under some additional assumptions (that is, $X$ carries a doubling measure and a weak Poincaré inequality), we show that when $1\le q<p$ the Lipschitz functions are dense in those spaces; moreover, in the same setting we show that the $p,q$-capacity is Choquet provided that $q>1$. We also provide a counterexample to the density result in the Euclidean setting when $1<p\le n$ and $q=\infty$.

Article information

Illinois J. Math., Volume 56, Number 2 (2012), 579-616.

First available in Project Euclid: 22 November 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31C15: Potentials and capacities 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems


Costea, Şerban; Miranda Jr., Michele. Newtonian Lorentz metric spaces. Illinois J. Math. 56 (2012), no. 2, 579--616. doi:10.1215/ijm/1385129966.

Export citation


  • C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Boston, MA, 1988.
  • J. Björn, Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math. 46 (2002), 383–403.
  • A. Björn, J. Björn and N. Shanmugalingam, Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions in metric measure spaces, Houston J. Math. 34 (2008), 1197–1211.
  • J. Cheeger, Differentiability of Lipschitz functions on metric spaces, Geom. Funct. Anal. 9 (1999), 428–517.
  • H. M. Chung, R. A. Hunt and D. S. Kurtz, The Hardy–Littlewood maximal function on ${L}(p,q)$ spaces with weights, Indiana Univ. Math. J. 31 (1982), 109–120.
  • \c S. Costea, Scaling invariant Sobolev–Lorentz capacity on ${\mathbb{R}}^n$, Indiana Univ. Math. J. 56 (2007), 2641–2669.
  • \c S. Costea, Sobolev capacity and Hausdorff measures in metric measure spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), 179–194.
  • \c S. Costea and V. Maz'ya, Conductor inequalities and criteria for Sobolev–Lorentz two-weight inequalities, Sobolev Spaces in Mathematics II. Applications in Analysis and Partial Differential Equations, Springer, New York, 2009, pp. 103–121.
  • J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer, New York, 1984.
  • G. B. Folland, Real analysis, 2nd ed. Wiley, New York, 1999.
  • B. Franchi, P. Hajłasz and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 1903–1924.
  • P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403–415.
  • P. Hajłasz, Sobolev spaces on metric-measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 173–218.
  • P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc., vol. 145, Amer. Math. Soc., Providence, RI, 2000.
  • J. Heinonen, Lectures on analysis on metric spaces, Springer, New York, 2001.
  • J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Univ. Press, New York, 1993.
  • J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61.
  • R. Hunt, On $L(p,q)$ measure spaces, Enseignement Math. 12 (1966), 249–276
  • S. Kallunki and N. Shanmugalingam, Modulus and continuous capacity, Ann. Acad. Sci. Fenn. Math. 26 (2001), 455–464.
  • J. Kauhanen, P. Koskela and J. Malý, On functions with derivatives in a Lorentz space, Manuscripta Math. 100 (1999), 87–101.
  • J. Kinnunen and O. Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn. Math. 21 (1996), 367–382.
  • J. Kinnunen and O. Martio, Choquet property for the Sobolev capacity in metric spaces, Proceedings of the conference on Analysis and Geometry held in Novosibirsk, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000, pp. 285–290.
  • P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1–17.
  • V. Maz'ya, Sobolev spaces, Springer, Berlin, 1985.
  • E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837–842.
  • P. Podbrdsky, Fine properties of Sobolev functions (in Czech), preprint, 2004.
  • N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), 243–279.
  • N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), 1021–1050.
  • E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1975.
  • K. Yosida, Functional Analysis, Springer, Berlin, 1980.