# $BM{O}_{\psi}$-spaces and applications to extrapolation theory

Studia Mathematica (1997)

- Volume: 122, Issue: 3, page 235-274
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topGeiss, Stefan. "$BMO_ψ$-spaces and applications to extrapolation theory." Studia Mathematica 122.3 (1997): 235-274. <http://eudml.org/doc/216374>.

@article{Geiss1997,

abstract = {We investigate a scale of $BMO_ψ$-spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $BMO_ψ$-$L_∞$-estimates, and arrives at $L_p$-$L_p$-estimates, or more generally, at estimates between K-functionals from interpolation theory.},

author = {Geiss, Stefan},

journal = {Studia Mathematica},

keywords = {scale of -spaces; Lorentz norms; extrapolation techniques; operators defined on adapted sequences; --estimates; K-functionals; interpolation theory},

language = {eng},

number = {3},

pages = {235-274},

title = {$BMO_ψ$-spaces and applications to extrapolation theory},

url = {http://eudml.org/doc/216374},

volume = {122},

year = {1997},

}

TY - JOUR

AU - Geiss, Stefan

TI - $BMO_ψ$-spaces and applications to extrapolation theory

JO - Studia Mathematica

PY - 1997

VL - 122

IS - 3

SP - 235

EP - 274

AB - We investigate a scale of $BMO_ψ$-spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $BMO_ψ$-$L_∞$-estimates, and arrives at $L_p$-$L_p$-estimates, or more generally, at estimates between K-functionals from interpolation theory.

LA - eng

KW - scale of -spaces; Lorentz norms; extrapolation techniques; operators defined on adapted sequences; --estimates; K-functionals; interpolation theory

UR - http://eudml.org/doc/216374

ER -

## References

top- [1] N. H. Asmar and S. J. Montgomery-Smith, On the distribution of Sidon series, Ark. Mat. 31 (1993), 13-26. Zbl0836.43011
- [2] N. L. Bassily and J. Mogyoródi, On the $BM{O}_{\Phi}$-spaces with general Young function, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 27 (1984), 215-227. Zbl0581.60036
- [3] J. Bastero and F. J. Ruiz, Interpolation of operators when the extreme spaces are ${L}^{\infty}$, Studia Math. 104 (1993), 133-150. Zbl0814.46063
- [4] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988. Zbl0647.46057
- [5] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, 1976.
- [6] J. Bourgain, On the behaviour of the constant in the Littlewood-Paley inequality, in: J. Lindenstrauss and V. D. Milman (eds.), Israel Seminar, GAFA 1987/88, Lecture Notes in Math. 1376, Springer, 1989, 202-208.
- [7] J. Bourgain and W. J. Davis, Martingale transforms and complex uniform convexity, Trans. Amer. Math. Soc. 294 (1986), 501-515. Zbl0638.46011
- [8] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42. Zbl0301.60035
- [9] D. L. Burkholder, Martingale transforms and the geometry of Banach spaces, in: Probability in Banach spaces III, 1980, Lecture Notes in Math. 860, Springer, 1981, 35-50.
- [10] D. L. Burkholder, B. J. Davis, and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proc. 6th Berkeley Sympos. Math. Statist. Probab., Vol. 2, Univ. of California Press, 1972, 223-240. Zbl0253.60056
- [11] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasilinear operators on martingales, Acta Math. 124 (1970), 249-304. Zbl0223.60021
- [12] S. Chang, M. Wilson and J. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60 (1985), 217-246. Zbl0575.42025
- [13] V. H. de la Peña, S. J. Montgomery-Smith, and J. Szulga, Contraction and decoupling inequalities for multilinear forms and U-statistics, Ann. Probab. 22 (1994), 1745-1765. Zbl0861.60008
- [14] A. M. Garsia, Martingale Inequalities, Seminar Notes on Recent Progress, Benjamin, Reading, 1973.
- [15] E. Giné and J. Zinn, Central limit theorems and weak laws of large numbers in certain Banach spaces, Z. Wahrsch. Verw. Gebiete 62 (1983), 323-354. Zbl0488.60009
- [16] P. Hitczenko, Upper bounds for the ${L}_{p}$-norms of martingales, Probab. Theory Related Fields 86 (1990), 225-238. Zbl0677.60017
- [17] F. John, Quasi-isometric mappings, in: Seminari 1962-1963 di Analisi, Algebra, Geometria e Topologia, Vol. II, Ed. Cremonese, Rome, 1965, 462-473.
- [18] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. Zbl0102.04302
- [19] W. B. Johnson and G. Schechtman, Sums of independent random variables in rearrangement invariant function spaces, Ann. Probab. 17 (1989), 789-808. Zbl0674.60051
- [20] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, 1991.
- [21] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980.
- [22] G. Pisier, Martingales with values in uniformly convex spaces, Isreal J. Math. 20 (1975), 326-350. Zbl0344.46030
- [23] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993. Zbl0821.42001
- [24] J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511-544. Zbl0429.46016
- [25] G. Wang, Some sharp inequalities for conditionally symmetric martingales, PhD thesis, Univ. of Illinois, 1989.
- [26] F. Weisz, Martingale operators and Hardy spaces generated by them, Studia Math. 114 (1995), 39-70. Zbl0822.60043

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.