Embeddings of the classes \(H^\omega\)
The author proves two theorems giving necessary and sufficient conditions for embedding relations. One of them states the following attractive result: For the embedding \(H^\omega\subset\Lambda BV\), it is necessary and sufficient that for any sequence \(\{t_k\}\) such that \(t_k\geq 0\) and \(\sum^\infty_{k=1} t_k\leq 1\) the series \(\sum^\infty_{k=1} \omega(t_k) \lambda^{-1}_k\) be convergent, where \(\lambda_k\nearrow \infty\) and \(\sum \lambda^{-1}_k= \infty\), moreover \(\Lambda BV\) denotes the functions satisfying the condition \[ \sup_G \sum^N_{k=1}|f(b_k)- f(a_k)|\lambda^{-1}_k< \infty, \] the supremum is taken over all systems \(G\) of nonintersecting intervals \((a_k,b_k)\) belonging to the closed interval \([a,b]\) where \(f\) is given.
scientific article; zbMATH DE number 1407580