Extensions in spaces with variable exponents — the half space

Abstract. In this article we study the Hardy–Littlewood max-imal operator in variable exponent spaces. For a given variable exponent p on a domain Ω for which the maximal operator is con-tinuous on Lp(·)(Ω) we construct an extension of p to Rn such that the maximal operator is continuous on Rn. The variable exponent will be constructed by means of a Whitney type extension. In this paper we restrict ourselves to the case where Ω is the half space. But the technique applies to less regular domains. The application to ε-δ-domains (Jones domains) is the subject of a forthcoming article. 1