## Arkiv för Matematik

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

### Morrey spaces in harmonic analysis

#### Abstract

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.

#### Note

Jie Xiao was in part supported by NSERC of Canada.

#### Article information

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

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907172

Digital Object Identifier
doi:10.1007/s11512-010-0134-0

Mathematical Reviews number (MathSciNet)
MR2961318

Zentralblatt MATH identifier
1254.31009

Rights

#### Citation

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. https://projecteuclid.org/euclid.afm/1485907172

#### References

• 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.