A new concept of smoothness in Orlicz spaces

Abstract

In a 2015 article Cuenya and Ferreyra defined a class of functions in $L^p$-spaces, denoted by $c_n^p(x)$. The class $c_n^p(x)$ contains the class of $L^p$-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 $L^\Phi$, called $L^\Phi$-differentiable functions in the present article. The class of $L^\Phi$-differentiable functions is closely related to the class $t_n^p(x)$. In this work, we define a class of functions in $L^\Phi$, denoted by $c_n^{\Phi }(x)$. The class $c_n^{\Phi }(x)$ is more general than the class of $L^{\varPhi}$-differentiable functions. We prove the existence of the best local $\Phi$-approximation for functions in $c_n^{\varPhi }(x)$ 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.