Arkiv för Matematik

  • Ark. Mat.
  • Volume 50, Number 2 (2012), 201-230.

Morrey spaces in harmonic analysis

David R. Adams and Jie Xiao

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Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.


Jie Xiao was in part supported by NSERC of Canada.

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Ark. Mat., Volume 50, Number 2 (2012), 201-230.

Received: 6 July 2010
First available in Project Euclid: 31 January 2017

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2011 © Institut Mittag-Leffler


Adams, David R.; Xiao, Jie. Morrey spaces in harmonic analysis. Ark. Mat. 50 (2012), no. 2, 201--230. doi:10.1007/s11512-010-0134-0.

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  • Adams, D. R., A note on Riesz potentials, Duke Math. J. 42 (1975), 765–778.
  • Adams, D. R., Lecture Notes onLp-Potential Theory, Dept. of Math., University of Umeå, Umeå, 1981.
  • Adams, D. R., A note on Choquet integrals with respect to Hausdorff capacity, in Function Spaces and Applications (Lund, 1986 ), Lecture Notes in Math. 1302, pp. 115–124, Springer, Berlin–Heidelberg, 1988.
  • Adams, D. R., Choquet integrals in potential theory, Publ. Mat. 42 (1998), 3–66.
  • Adams, D. R. and Hedberg, L. I., Function Spaces and Potential Theory, Springer, Berlin, 1996.
  • Adams, D. R. and Xiao, J., Nonlinear analysis on Morrey spaces and their capacities, Indiana Univ. Math. J. 53 (2004), 1629–1663.
  • Alvarez, J., Continuity of Calderón–Zygmund type operators on the predual of a Morrey space, in Clifford Algebras in Analysis and Related Topics (Fayetteville, AR, 1993 ), Stud. Adv. Math. 5, pp. 309–319, CRC, Boca Raton, FL, 1996.
  • Anger, B., Representation of capacities, Math. Ann. 229 (1977), 245–258.
  • Bennett, C. and Sharpley, R., Interpolation of Operators, Pure and Applied Math., 129, Academic Press, New York, 1988.
  • Bensoussan, A. and Frehse, J., Regularity Results for Nonlinear Elliptic Systems and Applications, Springer, Berlin, 2002.
  • Blasco, O., Ruiz, A. and Vega, L., Non-interpolation in Morrey–Campanato and block spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 28 (1999), 31–40.
  • Caffarelli, L. A., Salsa, S. and Silvestre, L., Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), 425–461.
  • Campanato, S., Proprietá di inclusione per spazi di Morrey, Ricerche Mat. 12 (1963), 67–86.
  • Carleson, L., Selected Problems on Exceptional Sets, Van Nostrand, Princeton, NJ, 1967.
  • Chiarenza, F. and Frasca, M., Morrey spaces and Hardy–Littlewood maximal function, Rend. Mat. Appl. 7 (1988), 273–279.
  • Choquet, G., Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953–54), 131–295.
  • Duong, X., Xiao, J. and Yan, L. X., Old and new Morrey spaces with heat kernel bounds, J. Fourier Anal. Appl. 13 (2007), 87–111.
  • Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006.
  • John, F. and Nirenberg, L., On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.
  • Fefferman, C. and Stein, E. M., Hp spaces of several variables, Acta Math. 129 (1972), 137–193.
  • Fefferman, R., A theory of entropy in Fourier analysis, Adv. Math. 30 (1978), 171–201.
  • Harrell II, E. M. and Yolcu, S. Y., Eigenvalue inequalities for Klein–Gordon operators, J. Funct. Anal. 256 (2009), 3977–3995.
  • Kalita, E. A., Dual Morrey spaces, Dokl. Akad. Nauk 361 (1998), 447–449 (Russian). English transl.: Dokl. Math. 58 (1998), 85–87.
  • Malý, J. and Ziemer, W. P., Fine Regularity of Solutions of Elliptic Partial Differential Equations, Math. Surveys and Monographs 51, Amer. Math. Soc., Providence, RI, 1997.
  • Maz′ya, V. G. and Verbitsky, I. E., Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator, Invent. Math. 162 (2005), 81–136.
  • Morrey, C. B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166.
  • Orobitg, J. and Verdera, J., Choquet integrals, Hausdorff content and the Hardy–Littlewood maximal operator, Bull. Lond. Math. Soc. 30 (1998), 145–150.
  • Peetre, J., On the theory of ${\mathcal{L}}_{p,\lambda}$ spaces, J. Funct. Anal. 4 (1969), 71–87.
  • Ruiz, A. and Vega, L., Corrigenda to unique …, and a remark on interpolation on Morrey spaces, Publ. Mat. 39 (1995), 405–411.
  • Sadosky, C., Interpolation of Operators and Singular Integrals, Pure and Appl. Math., Marcel Dekker, New York–Basel, 1979.
  • Sarason, D., Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405.
  • Stampacchia, G., The spaces ${\mathcal{L}}^{(p,\lambda)}$, N(p, λ) and interpolation, Ann. Sc. Norm. Super. Pisa 19 (1965), 443–462.
  • Stein, E. M., Singular Integrals and Differentiability of Functions, Princeton Univ. Press, Princeton, NJ, 1970.
  • Stein, E. M., Harmonic Analysis : Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.
  • Stein, E. M. and Zygmund, A., Boundedness of translation invariant operators on Hölder spaces and Lp-spaces, Ann. of Math. 85 (1967), 337–349.
  • Taylor, M., Microlocal analysis on Morrey spaces, in Singularities and Oscillations (Minneapolis, MN, 1994/1995 ), IMA Vol. Math. Appl. 91, pp. 97–135, Springer, New York, 1997.
  • Torchinsky, A., Real-variable Methods in Harmonic Analysis, Dover, New York, 2004.
  • Xiao, J., Homothetic variant of fractional Sobolev spaces with application to Navier–Stokes system, Dyn. Partial Differ. Equ. 4 (2007), 227–245.
  • Yang, D. and Yuan, W., A note on dyadic Hausdorff capacities, Bull. Sci. Math. 132 (2008), 500–509.
  • Zorko, C. T., Morrey spaces, Proc. Amer. Math. Soc. 98 (1986), 586–592.