Browsing by Author "Micherda, Bartosz"

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Access status: Open Access ,
A characterization of convex φ-functions
(2012) Micherda, Bartosz
The properties of four elements $(LPFE)$ and $(UPFE)$, introduced by Isac and Persson, have been recently examined in Hilbert spaces, $L^p$-spaces and modular spaces. In this paper we prove a new theorem showing that a modular of form $\rho_{\Phi}(f)=\int_{\Omega}\Phi(t,|f(t)|)d\mu(t)$ satisfies both $(LPFE)$ and $(UPFE)$ if and only if $\Phi$ is convex with respect to its second variable. A connection of this result with the study of projections and antiprojections onto latticially closed subsets of the modular space $L^{\Phi}$ is also discussed.
Access status: Open Access ,
A new characterization of convex φ-functions with a parameter
(2015) Micherda, Bartosz
We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by $\Phi$ are isotonic if and only if $\Phi$ is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012.