On the structure of variable exponent spaces
Julio Flores, Francisco L. Hernández, César Ruiz, Mauro Sanchiz
TL;DR
The paper investigates the geometry of variable exponent spaces $L^{p( olinebreak[0]( olinebreak[0]( olinebreak[0]( olinebreak[0]( Omega)}$ (Nakano spaces) by studying lattice structure, notably the existence of lattice-isomorphic and complemented copies of $\ell_q$ for $q$ in the essential range $R_{p( olinebreak[0]( olinebreak[0]( olinebreak[0]( olinebreak[0]( ) )]+ and subprojectivity; these results hold without assuming regularity on $p( olinebreak[0]( olinebreak[0]( olinebreak[0]( olinebreak[0]( )$. In addition, it analyzes strictly singular and disjointly strictly singular operators between such spaces, establishing criteria for disjoint strict singularity of inclusions $L^{p( olinebreak[0]( olinebreak[0]( olinebreak[0]( olinebreak[0]( Omega)}$ into $L^{q( olinebreak[0]( olinebreak[0]( olinebreak[0]( olinebreak[0]( Omega)}$, in terms of the essential range $R_{p( olinebreak[0]( linebreak[0]( ) and $p( olinebreak[0]( linebreak[0]( )$. The results include lattice-complemented copies of $\ell_q$, subprojectivity and superprojectivity characterizations, and sufficient/necessary conditions for DSS of embeddings, accompanied by explicit examples and open problems. Together, they advance the understanding of the geometry and operator theory of Nakano spaces and highlight how the lack of rearrangement invariance complicates projection techniques.
Abstract
The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) $\lpv$. In the second part strictly singular and disjointly strictly singular operators between spaces $\lpv$ are studied. New results on the disjoint strict singularity of the inclusions $ L^{p(\cdot)}(Ω) \hookrightarrow L^{q(\cdot)}(Ω)$ are given.