On Lorentz-Zygmund spaces

Bennett, Colin; Rudnick, Karl

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Tytuł książki

On Lorentz-Zygmund spaces

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Colin Bennett Karl Rudnick

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

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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

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 175

Liczba stron

67

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Opis fizyczny

Dissertationes Mathematicae, Tom CLXXV

Daty

wydano

1980

Twórcy

autor

Colin Bennett

Department of Mathematics University of South Carolina Columbia, South Carolina 29 208

autor

Karl Rudnick

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.

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Książka - szczegóły

On Lorentz-Zygmund spaces

Autorzy

Seria

Zawartość

Warianty tytułu

Abstrakty

Słowa kluczowe

Tematy

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Copyright

Seria

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Bibliografia

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