Microlocal Analysis on Morrey Spaces

  • Michael E. Taylor
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 91)


The spaces now called Morrey spaces were introduced by C.B. Morrey to study regularity properties of solutions to quasi-linear elliptic PDE, but since then they have been useful in other areas of PDE. Before saying more on this, let us first define the Morrey spaces M q p (ℝ n


Pseudodifferential Operator Ricci Tensor Morrey Space Nonlinear Elliptic System MICROLOCAL Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ad]
    D. Adams, A note on Riesz potentials, Duke Math. J. 42(1975),765–778.MathSciNetMATHCrossRefGoogle Scholar
  2. [An]
    M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102(1990), 429–445.MathSciNetMATHCrossRefGoogle Scholar
  3. [AC]
    M. Anderson and J. Cheeger, Ca-compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Diff. Geom. 35(1992),265–281.MathSciNetMATHGoogle Scholar
  4. [AT]
    P. Auscher and M. Taylor, Paradifferential operators and commutator estimates, Preprint, 1994.Google Scholar
  5. [Be]
    M. Beals, Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems, Birkhauser, Boston, 1989.MATHCrossRefGoogle Scholar
  6. [BL]
    J. Bergh and J. Lofström, Interpolation Spaces, Springer-Verlag, New York, 1976.MATHCrossRefGoogle Scholar
  7. [Bo]
    J. Bony, Calcul symbolique et propagation des singularities pour les equations aux derivees nonlineaires, Ann. Sci. Ecole Norm. Sup. 14(1981),209–246.MathSciNetMATHGoogle Scholar
  8. [Caf]
    L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 58(1988),253–284.MathSciNetMATHCrossRefGoogle Scholar
  9. [C]
    S.S. Chern (ed.), Seminar on Nonlinear Partial Differential Equations, MSRI Publ. #2, Springer-Verlag, New York, 1984.CrossRefGoogle Scholar
  10. [CF]
    F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. S. 7(1987), 273–279.Google Scholar
  11. [CFL1]
    F. Chiarenza, M. Frasca, and P. Longo,InteriorW2.Pestimates for non divergence elliptic equations with discontinuous coefficients,Ricerche Mat. 40(1991), 149–168.Google Scholar
  12. [CFL2]
    F.Chiarenza, M.Frasca, and P.Longo,W2’P-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,Trans.AMS 336(1993), 841–853.MathSciNetMATHGoogle Scholar
  13. [Chr]
    M. Christ, Lectures on Singular Integral Operators, CBMS Reg. Conf. Ser. in Math. #77, AMS, Providence, RI, 1990.Google Scholar
  14. [CLMS]
    R. Coifman, P. Lions, Y. Meyer, and S. Semmes,Compensated compactness and Hardy spaces, J. Math. Pure Appl.72(1993),247–286.MathSciNetMATHGoogle Scholar
  15. [CM]
    R. Coffman and Y. Meyer, Au-dela des Operateurs Pseudo-differentiels. Asterisque, #57, Soc. Math. de France, 1978.Google Scholar
  16. [CRW]
    R. Coifman, R. Rochberg, and G. Weiss,Factorization theorems for Hardy spaces in several variables, Ann. of Math.103(1976),611–635.MathSciNetMATHCrossRefGoogle Scholar
  17. [CH]
    H.O. Cordes and E. Herman,Gelfand theory of pseudo-differential operators, Amer. J. Math.90(1968),681–717.MathSciNetMATHCrossRefGoogle Scholar
  18. [DK]
    D. DeTurck and J. Kazdan,Some regularity theorems in Riemannian geometry,Ann. Sci. Ecole Norm. Sup.14(1980),249–260.Google Scholar
  19. [DR1]
    G. DiFazio and M. Ragusa, Commutators and Morrey spaces, Boll. Un. Mat. Ital. 5-A(1991), 323–332.Google Scholar
  20. [DR2]
    G. DiFazio and M. Ragusa,Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, Jour. Funct. Anal.112(1993),241–256.Google Scholar
  21. [Dou]
    R. Douglas, On the spectrum of Toeplitz and Wiener-Hopf operators, Abstract Spaces and Approximation (Proc. Conf. Oberwolfach 1968), 53–66, Birkhauser, Basel, 1969.Google Scholar
  22. [Fed]
    P. Federbush,Navier and Stokes meet the wavelet, Commun.Math. Phys. 155(1993), 219–248.Google Scholar
  23. [FS]
    C. Fefferman and E. SteinHPspaces of several variables,Acta Math. 129(1972),137–193.MathSciNetMATHCrossRefGoogle Scholar
  24. [Fre]
    J. Frehse,A discontinuous solution to a mildly nonlinear elliptic system,Math. Zeit. 134(1973),229–230.MathSciNetMATHCrossRefGoogle Scholar
  25. [Gia]
    M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, 1983.Google Scholar
  26. [Gia2]
    M. Giaquinta, Nonlinear elliptic systems with quadratic growth, Manuscripts Math. 24(1978), 323–349.MathSciNetMATHCrossRefGoogle Scholar
  27. [GH]
    M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 336(1982), 124–164.MathSciNetGoogle Scholar
  28. [GM]
    Y. Giga and T. Miyakawa, Navier-Stokes flows in il83with measures as initial vorticity and Morrey spaces, Comm. PDE 14(1989),577–618.MathSciNetMATHCrossRefGoogle Scholar
  29. [JR]
    J. Joly and J. Rauch, Justification of multidimensional single phase semilinear geometrical optics, Trans. AMS 330(1992),599–623.MathSciNetMATHCrossRefGoogle Scholar
  30. [K]
    T. Kato, Strong solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Brasil. Math. 22(1992), 27–155.Google Scholar
  31. [KP]
    T. Kato and G. Ponce, Commutator estimates and the Euler and NavierStokes equations, CPAM 41(1988), 891–907.MathSciNetMATHGoogle Scholar
  32. [KY]
    H. Kozono and M. Yamazaki, Semilinear heat equations and the NavierStokes equation with distributions in new function spaces as initial data, Preprint, 1993.Google Scholar
  33. [Mey]
    Y. Meyer, Regularite des solutions des equations aux derivees partielles nonlineaires, Springer LNM #842(1980), 293–302.Google Scholar
  34. [P]
    J. Peetre, On the theory of Lpa spaces, J. Funct. Anal. 4(1969), 71–87.MathSciNetMATHCrossRefGoogle Scholar
  35. [RR]
    J. Rauch and M. Reed, Bounded, stratified, and striated solutions of hyperbolic systems, Nonlinear Partial Differential Equations and their Applications, Vol.9 (H. Brezis and J. Lions, eds.), Research Notes in Math. #181, Pitman, New York, 1989.Google Scholar
  36. [Sar]
    D. Sarason, Functions of vanishing mean oscillation, Trans. AMS 207(1975), 391–405.MathSciNetMATHCrossRefGoogle Scholar
  37. [Sch]
    R. Schoen, Analytic aspects of the harmonic map problem, 321–358 in [C].Google Scholar
  38. [St]
    E. Stein, Singular Integrals and Pseudo-Differential Operators, Graduate Lecture Notes, Princeton Univ.,1972.Google Scholar
  39. [TI]
    M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhauser, Boston, 1991.MATHCrossRefGoogle Scholar
  40. [T2]
    M. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. PDE 17(1992),1407–1456.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Michael E. Taylor
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaUSA

Personalised recommendations