×

Non interpolation in Morrey-Campanato and block spaces. (English) Zbl 0955.46013

Morrey-Campanato spaces are generalizations of Hölder spaces and BMO. The authors show that the preduals of these spaces can represented by block spaces in almost all cases. In the next section they prove that if the underlaying space is the real line then there are linear operators between interpolation couples of Morrey and \(L^p\) spaces as well as \(L^p\) spaces and block spaces with the property that these operators do not map between the interpolation spaces.

MSC:

46B70 Interpolation between normed linear spaces
46A20 Duality theory for topological vector spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] J. Alvarez , Continuity of Calderon-Zygmund type operators on the predual of a Morrey space , To appear in the Proc. of ”Clifford Algebras in Analysis”, Studies in Advanced Math. , CRC Press . MR 1383112 | Zbl 0842.42008 · Zbl 0842.42008
[2] J. Bergh - J. Lofstrom , ” Interpolation Spaces ”, Springer-Verlag New York , 1976 . MR 482275 | Zbl 0344.46071 · Zbl 0344.46071
[3] S. Campanato , Proprietà di hölderianità di alcune classi difunzioni , Ann. Scuola Norm. Sup. Cl. Sci. 17 ( 1963 ), 175 - 188 . Numdam | MR 156188 | Zbl 0121.29201 · Zbl 0121.29201
[4] A. Campanato - M.K.V. Murthy , Una generalizzazione del teorema di Riezs-Thorin , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19 ( 1965 ), 87 - 100 . Numdam | MR 180860 | Zbl 0145.16301 · Zbl 0145.16301
[5] S. Chanillo - E. Sawyer , Unique continuation for \Delta + V and the C. Fefferman-Phong class , Trans. Amer. Math. Soc. 318 ( 1990 ), 275 - 300 . Zbl 0702.35034 · Zbl 0702.35034 · doi:10.2307/2001239
[6] F. Chiarenza - A. Ruiz , Uniform l2-weighted Sobolev inequalities , Proc. Amer. Math. Soc. 112 ( 1991 ), 53 - 64 . MR 1055768 | Zbl 0745.35007 · Zbl 0745.35007 · doi:10.2307/2048479
[7] C. Fefferman - D.H. Phong , Lower bounds for Schrödinger equations, J. Eq. aux Derivees Partielles . Saint Jean de Monts. Soc. Mat. de France , 1982 . Numdam | MR 672274 | Zbl 0492.35057 · Zbl 0492.35057
[8] C. Kenig , Restriction theorems, Carleman Estimates, Uniform Sobolev inequalities and unique continuation , Proceedings of Harmonic Analysis and PDEs, 69 - 91 . E1 Escorial, 1987 . Lectures Notes in Math. 1384 . MR 1013816 | Zbl 0685.35003 · Zbl 0685.35003
[9] A. Kufner - O. John - S. Fucik , ” Function Spaces ”. Noordhoff International Publishing , Leyden , 1977 . MR 482102 | Zbl 0364.46022 · Zbl 0364.46022
[10] G.N. Meyers , Mean oscillation over cubes and Hölder continuity , Proc. Amer. Math. Soc. 15 ( 1964 ), 717 - 721 . MR 168712 | Zbl 0129.04002 · Zbl 0129.04002 · doi:10.2307/2034586
[11] J. Peetre , On the theory of ,Lp,\lambda Spaces , J. Funct. Anal. 4 ( 1969 ), 71 - 87 . Zbl 0175.42602 · Zbl 0175.42602 · doi:10.1016/0022-1236(69)90022-6
[12] A. Ruiz - L. Vega , Unique continuation for the solutions of the Laplacian plus a drift , Ann. Inst. Fourier ( Grenoble ) 41 , 3 ( 1991 ), 651 - 663 . Numdam | MR 1136598 | Zbl 0772.35008 · Zbl 0772.35008 · doi:10.5802/aif.1268
[13] A. Ruiz - L. Vega , Local regularity of solutions to wave equations with time-dependent potentials , Duke Math. J. 76 , 3 ( 1994 ), 913 - 940 . Article | MR 1309336 | Zbl 0826.35014 · Zbl 0826.35014 · doi:10.1215/S0012-7094-94-07636-9
[14] A. Ruiz - L. Vega , Corrigenda to unique ... , and a remark on interpolation on Morrey spaces , Publicacions Matematiques 39 ( 1995 ), 405 - 411 . MR 1370896 | Zbl 0849.47022 · Zbl 0849.47022 · doi:10.5565/PUBLMAT_39295_15
[15] M. Schechter , ” Spectra of Partial Differential Operators ”, second edition, North Holland , 1986 . MR 869254 | Zbl 0607.35005 · Zbl 0607.35005
[16] G. Soares De Souza - R. O’Neil - G. Sampson , Several characterization for the special atom spaces with applications , Revista Mat. Iberoamericana 2 ( 1986 ), 333 - 355 . MR 908057 | Zbl 0642.46031 · Zbl 0642.46031 · doi:10.4171/RMI/37
[17] F. Soria , Characterizations of classes of functions generated by blocks and associated Hardy Spaces , Indiana Univ. Math. J. 34 , 3 ( 1985 ), 463 - 491 . MR 794573 | Zbl 0573.42015 · Zbl 0573.42015 · doi:10.1512/iumj.1985.34.34027
[18] G. Stampacchia , L(p,\lambda ) -Spaces and interpolation , Comm. Pure Appl. Math. 17 ( 1964 ), 293 - 306 . Zbl 0149.09201 · Zbl 0149.09201 · doi:10.1002/cpa.3160170303
[19] E.M. Stein - A. Zygmund , Boundedness of translation invariant operators on Hölder spaces and Lp -spaces , Ann. Math. 85 ( 1967 ), 337 - 349 . MR 215121 | Zbl 0172.40102 · Zbl 0172.40102 · doi:10.2307/1970445
[20] M. Taylor , Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations , Comm. Partial Differential Equations 17 ( 1992 ), 1407 - 1456 . MR 1187618 | Zbl 0771.35047 · Zbl 0771.35047 · doi:10.1080/03605309208820892
[21] T. Wolff , Unique continuation for |\Delta u| \leq V |\nabla u| and related problems , Revista Mat. Iberoamericana 6 , 3 ( 1990 ), 155 - 200 . Zbl 0735.35024 · Zbl 0735.35024 · doi:10.4171/RMI/101
[22] C. Zorko , The Morrey space , Proc. Amer. Math. Soc. 98 ( 1986 ), 586 - 592 . MR 861756 | Zbl 0612.43003 · Zbl 0612.43003 · doi:10.2307/2045731
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.