Resumen de A new concept of smoothness in Orlicz spaces

David E. Ferreyra , F. E. Levis, Marina Vanesa Roldán

• 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.