Geodesic
Extrapolation of operators acting into quasi-Banach spaces
Sbornik. Mathematics, Tome 207 (2016) no. 1, pp. 85-112 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Linear and sublinear operators acting from the scale of $L_p$ spaces to a certain fixed quasinormed space are considered. It is shown how the extrapolation construction proposed by Jawerth and Milman at the end of 1980s can be used to extend a bounded action of an operator from the $L_p$ scale to wider spaces. Theorems are proved which generalize Yano's extrapolation theorem to the case of a quasinormed target space. More precise results are obtained under additional conditions on the quasinorm. Bibliography: 35 titles.
Keywords: extrapolation of operators, Yano's theorem, symmetric space, Lorentz space
Mots-clés : quasi-Banach space. 
@article{SM_2016_207_1_a3,
     author = {K. V. Lykov},
     title = {Extrapolation of operators acting into {quasi-Banach} spaces},
     journal = {Sbornik. Mathematics},
     pages = {85--112},
     year = {2016},
     volume = {207},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2016_207_1_a3/}
}
TY  - JOUR
AU  - K. V. Lykov
TI  - Extrapolation of operators acting into quasi-Banach spaces
JO  - Sbornik. Mathematics
PY  - 2016
SP  - 85
EP  - 112
VL  - 207
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_2016_207_1_a3/
LA  - en
ID  - SM_2016_207_1_a3
ER  - 
%0 Journal Article
%A K. V. Lykov
%T Extrapolation of operators acting into quasi-Banach spaces
%J Sbornik. Mathematics
%D 2016
%P 85-112
%V 207
%N 1
%U http://geodesic.mathdoc.fr/item/SM_2016_207_1_a3/
%G en
%F SM_2016_207_1_a3
K. V. Lykov. Extrapolation of operators acting into quasi-Banach spaces. Sbornik. Mathematics, Tome 207 (2016) no. 1, pp. 85-112. http://geodesic.mathdoc.fr/item/SM_2016_207_1_a3/

[1] Sh. Yano, “Notes on Fourier analysis (XXIX): An extrapolation theorem”, J. Math. Soc. Japan, 3:2 (1951), 296–305 | DOI | MR | Zbl

[2] B. Jawerth, M. Milman, Extrapolation theory with applications, Mem. Amer. Math. Soc., 89, no. 440, Amer. Math. Soc., Providence, RI, 1991, iv+82 pp. | DOI | MR | Zbl

[3] B. Jawerth, M. Milman, “New results and applications of extrapolation theory”, Interpolation spaces and related topics (Haifa, 1990), Israel Math. Conf. Proc., 5, Bar-Ilan Univ., Ramat Gan, 1992, 81–105 | MR | Zbl

[4] M. Milman, Extrapolation and optimal decompositions with applications to analysis, Lecture Notes in Math., 1580, Springer-Verlag, Berlin, 1994, xii+162 pp. | MR | Zbl

[5] G. E. Karadzhov, M. Milman, “Extrapolation theory: new results and applications”, J. Approx. Theory, 133:1 (2005), 38–99 | DOI | MR | Zbl

[6] E. I. Berezhnoĭ, “A sharp extrapolation theorem for Lorentz spaces”, Siberian Math. J., 54:3 (2013), 406–418 | DOI | MR | Zbl

[7] E. I. Berezhnoi, “Can Yano's extrapolation theorem be strengthened?”, Funct. Anal. Appl., 49:2 (2015), 145–147 | DOI | DOI | MR | Zbl

[8] E. I. Berezhnoi, A. A. Perfil'ev, “A sharp extrapolation theorem for operators”, Funct. Anal. Appl., 34:3 (2000), 211–213 | DOI | DOI | MR | Zbl

[9] S. V. Astashkin, “Some new extrapolation estimates for the scale of $L_p$-spaces”, Funct. Anal. Appl., 37:3 (2003), 221–224 | DOI | DOI | MR | Zbl

[10] S. V. Astashkin, “Extrapolation properties of the scale of $L_p$-spaces”, Sb. Math., 194:6 (2003), 813–832 | DOI | DOI | MR | Zbl

[11] S. V. Astashkin, K. V. Lykov, “Extrapolatory description for the Lorentz and Marcinkiewicz spaces “close” to $L_\infty$”, Siberian Math. J., 47:5 (2006), 797–812 | DOI | MR | Zbl

[12] S. V. Astashkin, K. V. Lykov, “Strong extrapolation spaces and interpolation”, Siberian Math. J., 50:2 (2009), 199–213 | DOI | MR | Zbl

[13] S. V. Astashkin, K. V. Lykov, M. Mastyło, “On extrapolation of rearrangement invariant spaces”, Nonlinear Anal.: Theory, Methods and Applications, 75:5 (2012), 2735–2749 | DOI | MR | Zbl

[14] E. I. Berezhnoi, “A simple proof of an extrapolation theorem for Marcinkiewicz spaces”, Math. Notes, 93:6 (2013), 923–927 | DOI | DOI | MR | Zbl

[15] S. Astashkin, K. Lykov, “Extrapolation description of rearrangement invariant spaces and related problems”, Banach and function spaces III (ISBFS 2009) (Kitakyushu, Japan, 2009), Yokohama Publisher, Yokohama, 2011, 1–52 | MR | Zbl

[16] S. V. Astashkin, “Extrapolation functors on a family of scales generated by the real interpolation method”, Siberian Math. J., 46:2 (2005), 205–225 | DOI | MR | Zbl

[17] M. J. Carro, J. Martín, “Extrapolation theory for the real interpolation method”, Collect. Math., 53:2 (2002), 165–186 | MR | Zbl

[18] S. G. Kreĭn, Yu. I. Petunin, E. M. Semenov, Interpolation of linear operators, Transl. Math. Monogr., 54, Amer. Math. Soc., Providence, R.I., 1982, xii+375 pp. | MR | MR | Zbl | Zbl

[19] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, v. II, Ergeb. Math. Grenzgeb., 97, Function spaces, Springer-Verlag, Berlin–New York, 1979, x+243 pp. | MR | Zbl

[20] C. Bennett, R. Sharpley, Interpolation of operators, Pure Appl. Math., 129, Academic Press, Inc., Boston, MA, 1988, xiv+469 pp. | MR | Zbl

[21] N. J. Kalton, “Convexity, type and the three space problem”, Studia Math., 69:3 (1980/81), 247–287 | MR | Zbl

[22] E. M. Stein, N. J. Weiss, “On the convergence of Poisson integrals”, Trans. Amer. Math. Soc., 140 (1969), 35–54 | DOI | MR | Zbl

[23] G. G. Lorentz, “Relations between function spaces”, Proc. Amer. Math. Soc., 12:1 (1961), 127–132 | DOI | MR | Zbl

[24] N. Yu. Antonov, “Convergence of Fourier series”, East J. Approx., 2:2 (1996), 187–196 | MR | Zbl

[25] J. Arias-De-Reyna, “Pointwise convergence of Fourier series”, J. London Math. Soc. (2), 65:1 (2002), 139–153 | DOI | MR | Zbl

[26] J. Arias de Reyna, Pointwise convergence of Fourier series, Lecture Notes in Math., 1785, Springer-Verlag, Berlin, 2002, xviii+175 pp. | DOI | MR | Zbl

[27] M. J. Carro, M. Mastyło, L. Rodríguez-Piazza, “Almost everywhere convergent Fourier series”, J. Fourier Anal. Appl., 18:2 (2012), 266–286 | DOI | MR | Zbl

[28] P. Sjölin, F. Soria, “Remarks on a theorem by N. Yu. Antonov”, Studia Math., 158:1 (2003), 79–97 | DOI | MR | Zbl

[29] M. J. Carro, “From restricted weak type to strong type estimates”, J. London Math. Soc. (2), 70:3 (2004), 750–762 | DOI | MR | Zbl

[30] M. J. Carro, J. Martín, “Endpoint estimates from restricted rearrangement inequalities”, Rev. Mat. Iberoamericana, 20:1 (2004), 131–150 | DOI | MR | Zbl

[31] M. J. Carro, P. Tradacete, “Extrapolation on $L^{p,\infty}(\mu)$”, J. Funct. Anal., 265:9 (2013), 1840–1869 | DOI | MR | Zbl

[32] K. H. Moon, “On restricted weak type $(1,1)$”, Proc. Amer. Math. Soc., 42:1 (1974), 148–152 | DOI | MR | Zbl

[33] P. A. Hagelstein, R. L. Jones, “On restricted weak type $(1,1)$: the continuous case”, Proc. Amer. Math. Soc., 133:1 (2005), 185–190 | DOI | MR | Zbl

[34] P. A. Hagelstein, “Problems in interpolation theory related to the almost everywhere convergence of Fourier series”, Topics in harmonic analysis and ergodic theory, Contemp. Math., 444, Amer. Math. Soc., Providence, RI, 2007, 175–183 | DOI | MR | Zbl

[35] M. Carro, L. Colzani, G. Sinnamon, “From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces”, Studia Math., 182:1 (2007), 1–27 | DOI | MR | Zbl