A new concept of smoothness in Orlicz spaces

D. E. FERREYRA; F. E. LEVIS; M.V. ROLDÁN

COLLECTANEA MATHEMATICA

UNIV BARCELONA

Año: 2022 vol. 73 p. 505 - 520

0010-0757

In a 2015 article Cuenya and Ferreyra defined a class of functions in Lp-spaces, denoted by c_n^p(x). The class c_n^p(x) contains the class of Lp-differentiability functions, denoted by t_n^p(x), introduced in a 1961 article by Calderón-Zygmund. A more recent paper by Acinas, Favier and Zó introduced a new class of functions in Orlicz spaces LPhi, called LPhi-differentiable functions in the present article. The class of LPhi-differentiable functions is closely related to the class t_n^p(x). In this work, we define a class of functions in LPhi, denoted by c_n^Phi(x). The class c_n^Phi(x) is more general than the class of LPhi-differentiable functions. We prove the existence of the best local Phi-approximation for functions in c_n^Phi(x) and study the convexity of the set of cluster points of the set of best Phi-approximations to a function on an interval when their measures tend to zero.