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Tytuł książki

On Lorentz-Zygmund spaces

Seria

Rozprawy Matematyczne tom/nr w serii: 175 wydano: 1980

Zawartość

Warianty tytułu

Abstrakty

EN

CONTENTS

I. Introduction

 1. The Marcinkiewicz interpolation theorem..................................... 5
 2. The classical results: Theorem B.......................................................... 11
 3. The classical results: Theorem C......................................................... 13
 4. Remarks..................................................................................................... 21

II. Preliminaries

 5. Lorentz spaces; operators of strong and weak types (p, g)...... 22
 6. Variations on Hardy's inequalities......................................................... 24
 7. Averaging operators $A_p$, $B_p$, $G_p$ and $D_p$................... 28

III. The Lorentz-Zygmund spaces

 8. Lorentz-Zygmund spaces Lila{logL)a.......................................... 29
 9. Inclusion relations.................................................................................... 30
 10. The classical function spaces............................................................. 34
 11. The auxiliary spaces $ℒ^{pa}(logℒ)^a$ and $M^{pa}(logM)^a$... 38
 12. The embedding theorem....................................................................... 42

IV. Operators of weak type (p, q; r, s)

 13. Definition and elementary properties.......................................... 43
 14. The Fourier transform............................................................................. 49
 15. The Hardy-Littlewood maximal operator............................................. 49
 16. The (maximal) Hilbert transform........................................................... 50
 17. The fractional integrals........................................................................... 58

V. Proof of the main results

 18. Proof of Theorems B and C........................................................... 59
 19. Connections with interpolation space theory..................................... 62

References........................................................................................................... 65

Słowa kluczowe

Tematy

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 175

Liczba stron

67

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CLXXV

Daty

wydano
1980

Twórcy

  • Department of Mathematics University of South Carolina Columbia, South Carolina 29 208
autor
  • Department of Mathematics Texas A & M University College Station, Texas 77843

Bibliografia

  • [1] C. Bennett, Intermediate spaces and the class Llog+ L, Ark, Mat. 11 (1973) pp. 215-228.
  • [2] C. Bennett, Estimates for weak-type operators, Bull, Amer. Math. Soc. 79 (1973), pp, 933-935.
  • [3] C. Bennett, Banach function spaces and interpolation methods. I. The abstract theory, J. Functional Anal. 17 (1974), pp. 409-440.
  • [4] C. Bennett, Banach function spaces and interpolation methods. II. Interpolation of weak-type operators. "Linear Operators and Approximation. II", Proc. Confer. Oberwolfach, P. L. Bufczer & B. Sz. Nagy, Eds., ISNM 25, Birkhauser Verlag, Basel-Stuttgart, 1974, pp. 129-139.
  • [5] C. Bennett, Banach function spaces and interpolation methods. III. Hausdorff-Young estimates, J. Approx. Theory 13 (1975), pp. 267-275.
  • [6] C. Bennett, A best constant for Zygmund's conjugate-function inequality, to appear in Proc. Amer. Math. Soc.
  • [7] D. W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. Math. J. 21 (1969), pp. 1245-1254.
  • [8] P. L. Butzor and H. Borons, Semigroups of operators and approximation, Springer Verlag, New York 1967.
  • [9] A. P. Caldorón, Spaces between $L^1$ and and the theorem of Marcinkiewicz, Studia Math. 26 (1966), pp. 273-299.
  • [10] C. Fefferman and E. M, Stein, $H^p$ spaces of several variables, Acta Math. 19 (1972), pp. 137-193.
  • [11] A. Garsia, Topics in almost everywhere convergence, Markham, Chicago 1970.
  • [12] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, preprint.
  • [13] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, Math. Ann. 97 (1926), pp. 159-209.
  • [14] G. H. Hardy and J. E. Littlewood, Some new properties of fractional integrals (I), Math. Z. 28 (1928), pp. 565-606; (II), ibid,, 34 (1931-32), pp. 403-439.
  • [15] G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), pp. 81-116.
  • [16] G. H. Hardy and J. E. Littlewood, Notes on the theory of series XIII: Some new properties of Fourier constants, J. London Math. Soc. 6 (1931), pp. 3-9.
  • [17] G. H. Hardy and J. E. Littlewood, Notes on the theory of series XVIII: On the convergence of Fourier series, Proc. Camb. Phil. Soc. 31 (1935), pp. 317-323.
  • [18] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge 1967.
  • [19] F. Hausdorff, Eine Ausdehnung des Parsevalschen Satzes über Fourier reihen, Math. Z. 16 (1923), pp. 163-169.
  • [20] C. Herz, The Hardy-Littlewood maximal theorem, Symposium on Harmonic Analysis, Univ. of Warwick, 1968.
  • [21] C. Herz, A best possible Marcinkiewicz interpolation theorem with applications to martingale inequalities, preprint.
  • [22] T. Holmstedt, Interpolation of quasinormed spaces, Math. Scand. 26 (1970), pp. 177-199.
  • [23] R. A. Hunt, On L(p, q) spaces, L'Ens. Math. 12 (1966), pp. 249-275.
  • [24] Y. Katznelson, An introduction to harmonic analysis, Wiley, Now York 1968.
  • [25] S. Koizumi, Contributions to the theory of interpolation of operations, Osaka Math. J. 8 (1971), pp. 135-149.
  • [26] A. N. Kolmogorov, Sur les fonctions harmoniques conjuguées el les séries de Fourier, Fund. Math. 7 (1925), pp. 23-28.
  • [27] M. A. Kraanoselskiǐ and Ja. B. Rutickiǐ, Convex functions and Orlicz spaces, Noordhoff, Groningen 1961.
  • [28] G. G. Lorentz, Some new functional spaces, Ann. Math. 51 (1950), pp. 37-55.
  • [29] G. G. Lorentz, On the theory of spaces A, Pacific J, Math. 1 (1951), pp. 411-429.
  • [30] G. G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961), pp. 127-132.
  • [31] W. A. J. Luxemburg, Rearrangement-invariant Banach function spaces, Queen's Papers in Pure and Applied Math., Queen's Univ., 10 (1967), pp. 83-144.
  • [32] J. Marcinkiewicz, Sur Interpolation d'opérations, C. R. Acad. Sc. Paris 208 (1939), pp. 139-140.
  • [33] E. T. Oklander, Interpolacion, espacios de Lorents y teorema de Marcinkiewicz, Cursos y sem, de mat., fasc. 20, Buenos Aires 1965.
  • [34] R. O'Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), pp. 129-142.
  • [35] R. O'Neil, Fractional integration in Orlicz spaces, I, Trans. Amer. Math. Soc. 115 (1965), pp. 300-328.
  • [36] R. O'Neil, Les fonctions conjuguées et les intégrales fractionnaires de la classe $L(log^+L)^s$, C. R. Acad. Sc. Paris 263 (1966), pp. 463-466.
  • [37] R. O'Neil and G. Weiss, The Hilbert transform and rearrangement of functions, Studia Math. 23 (1963), pp. 189-198.
  • [38] R. E. A. C. Paley, Some theorems on orthogonal functions, ibid. 3 (1931), pp. 226-245.
  • [39] S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, ibid. 44 (1972), pp. 165-179.
  • [40] M. Riesz, Sur les fonctions conjuguées, Math. Z. 27 (1927), pp. 218-244.
  • [41] N. M. Rivière and Y. Sagher, Interpolation between $L^∞$ and $H^1$, the real method, J. Functional Anal. 14 (1973), pp. 401-409.
  • [42] K. Rudnick, Lorentz-Zygmund spaces and interpolation of weak-type operators, thesis, California Institute of Technology, 1976.
  • [43] R. C. Sharpley, Spaces $A_a(X)$ and interpolation, J. Functional Anal. 11 (1972), pp. 479-513.
  • [44] E. M. Stein, Note on the class LlogL, Studia Math. 32(1969), pp. 305-310
  • [45] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton 1970.
  • [46] E. M. Stein and G. Weiss, An extension of a theorem, of Marcinkiewicz and some, of its applications, J. Math. Much. 8 (1959), pp. 263-284.
  • [47] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton 1971.
  • [48] E. C. Titchmarsh, On conjugate functions, Proc. London Math. Soc,. 29 (1928), pp. 49-80.
  • [49] E. C. Titchmarsh, Additional note on conjugate functions, J. London Math. Soc. 4 (1929), pp. 204-206.
  • [50] E. C. Titchmarsh, Theory of Fourier integrals, Oxford 1937.
  • [51] A. Torchinsky, Interpolation of operators and Orlicz classes, preprint.
  • [52] N. Wiener, The ergodic theorem, Duke Math. J. 5 (1939), pp. 1-18.
  • [53] A. Zygmund, Sur les fonctions conjuguées, C. R. Acad. Sc. Paris 187 (1928), pp. 1025-1026; Fund. Math. 13 (1929), pp. 284-303.
  • [54] A. Zygmund, Some points in the theory of trigonometric and power series, Trans. Amer. Math. Soc. 36 (1934), pp. 586-617.
  • [55] A. Zygmund, On a theorem of Marcinkiewicz concerning interpolation of operations, J. Math. Pures Appl. 35 (1956), pp. 223-248.
  • [56] A. Zygmund, Trigonometric series, Vol. I, Cambridge 1959.
  • [57] A. Zygmund, Trigonometric series, Vol, II, Cambridge 1969.

Języki publikacji

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Identyfikator YADDA

bwmeta1.element.zamlynska-401a9cd7-4787-4b21-8aa9-89494eb535c7

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ISBN
83-01-01111-4
ISSN
0012-3862

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DML-PL
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