Type and Cotype of Musielak–Orlicz Sequence Spaces

Abstract

Let φ=(φn) be a Musielak–Orlicz function andlφthe Musielak–Orlicz sequence space generated by φ. It is shown that φ satisfies condition δqiff φ is equivalent to aq-concave function ψ=(ψn). It is also proved thatlφis of cotypeq(typep) if and only if φ satisfies condition δq(δ2and δ*pconditions, respectively). As applications we obtain characterization of the type and cotype of various sequence spaces: Nakano, weighted Orlicz, and Orlicz sequence spaces. Furthermore, a relation betweenq-concavity (p-convexity), lowerq-estimate (upperp-estimate), and cotype (type) oflφis given.