TY - JOUR

T1 - Some inequalities in classical spaces with mixed norms

AU - Boccuto, Antonio

AU - Bukhvalov, Alexander V.

AU - Sambucini, Anna Rita

N1 - Boccuto, A. Some inequalities in classical spaces with mixed norms / A. Boccuto, A. V. Bukhvalov, A. R. Sambucini // Positivity. - 2002. - Volume 6, Issue 4. - P. 393-411.

PY - 2002/12/1

Y1 - 2002/12/1

N2 - We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from Lp. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov-Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that Lp is the only space where it is possible to change this order.

AB - We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from Lp. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov-Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that Lp is the only space where it is possible to change this order.

KW - Banach Function Spaces

KW - Lorentz spaces

KW - Mixed norm inequalities for functions of many variables

KW - Spaces with mixed norms

KW - SCOPUS

KW - WOS

KW - SCOPUS

KW - WOS

UR - http://www.scopus.com/inward/record.url?scp=0141566563&partnerID=8YFLogxK

U2 - 10.1023/A:1021353215312

DO - 10.1023/A:1021353215312

M3 - Article

AN - SCOPUS:0141566563

VL - 6

SP - 393

EP - 411

JO - Positivity

JF - Positivity

SN - 1385-1292

IS - 4

ER -