Convexity and concavity of Banach ideal spaces and imbedding theorems (Q807935)

A Banach functional space A is \((\theta,\tau)\)-convex \((\theta,\tau \in [1,\infty])\) if \[ \| (\sum^{i}_{k=1}| f_ k|^{\tau})^{1/\tau}\|_ A\leq c(\sum^{i}_{k=1}\| f_ k\|_ A^{\theta})^{1/\theta}\text{ for any }\{f_ k\}^ i_{k=1}\subset A. \] The author proves the imbedding theorem for certain functional spaces that are analogons of the Sobolev spaces. The imbedding of this spaces is equivalent to certain convexity.