Dichotomies for Lorentz spaces

• Autorzy: Szymon Głąb , Filip Strobin , Chan Yang
• Języki publikacji: EN

Abstrakty

EN

Assume that L p,q, $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \{ (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \} $. We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L p spaces.

Słowa kluczowe

EN

Lorentz spaces; Integration; Baire category; Porosity

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46G99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 7
• Strony: 1228-1242

Daty

wydano

2013-07-01

online

2013-04-26

Twórcy

autor

Szymon Głąb

• Institute of Mathematics, Łódź University of Technology, Faculty of Technical Physics, Information Technology and Applied Mathematics, Wólczanska 215, 93-005, Łódz, Poland

autor

Filip Strobin

autor

Chan Yang

• Department of Mathematics, Korea University, Seoul, 136-701, Republic of Korea

Bibliografia

• [1] Balcerzak M., Wachowicz A., Some examples of meager sets in Banach spaces, Real Anal. Exchange, 2000/01, 26(2), 877–884
• [2] Grafakos L., Classical Fourier Analysis, 2nd ed., Grad. Texts in Math., 249, Springer, New York, 2008
• [3] GŁab S., Strobin F., Dichotomies for L p spaces, J. Math. Anal. Appl., 2010, 368(1), 382–390 http://dx.doi.org/10.1016/j.jmaa.2010.02.011[Crossref]
• [4] Jachymski J., A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces, Studia Math., 2005, 170(3), 303–320 http://dx.doi.org/10.4064/sm170-3-7[Crossref]
• [5] Zajíček L., Porosity and σ-porosity, Real Anal. Exchange, 1987/1988, 13(2), 314–350
• [6] Zajíček L., On σ-porous sets in abstract spaces, Abstr. Appl. Anal., 2005, 5, 509–534 http://dx.doi.org/10.1155/AAA.2005.509[Crossref]

Identyfikatory

DOI

10.2478/s11533-013-0241-9