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Singular integral operators on Nakano spaces with weights having finite sets of discontinuities

Alexei Yu. Karlovich (2011)

Banach Center Publications

In 1968, Gohberg and Krupnik found a Fredholm criterion for singular integral operators of the form aP + bQ, where a,b are piecewise continuous functions and P,Q are complementary projections associated to the Cauchy singular integral operator, acting on Lebesgue spaces over Lyapunov curves. We extend this result to the case of Nakano spaces (also known as variable Lebesgue spaces) with certain weights having finite sets of discontinuities on arbitrary Carleson curves.

Smoothness in Musielak-Orlicz spaces equipped with the Orlicz norm.

Henryk Hudzik, Zenon Zbaszyniak (1997)

Collectanea Mathematica

A formula for the distance of an arbitrary element x in Musielak-Orlicz space L^Phi from the subspace E^Phi of order continuous elements is given for both (the Luxemburg and the Orlicz) norms. A formula for the norm in the dual space of L^Phi is given for any of these two norms. Criteria for smooth points and smoothness in L^Phi and E^Phi equipped with the Orlicz norm are presented.

Some approximation results in Musielak-Orlicz spaces

Ahmed Youssfi, Youssef Ahmida (2020)

Czechoslovak Mathematical Journal

We prove the continuity in norm of the translation operator in the Musielak-Orlicz L M spaces. An application to the convergence in norm of approximate identities is given, whereby we prove density results of the smooth functions in L M , in both the modular and norm topologies. These density results are then applied to obtain basic topological properties.

Some classical function systems in separable Orlicz spaces

C. Finet, G. Tkebuchava (1996)

Studia Mathematica

The boundedness of (sub)sequences of partial Fourier and Fourier-Walsh sums in subspaces of separable Orlicz spaces is studied. The boundedness of the shift operator and Paley function with respect to the Haar system is also investigated. These results are applied to get the analogues of the classical theorems on basicness of the trigonometric and Walsh systems in nonreflexive separable Orlicz spaces.

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