Imbedding theorems for generalized Hölder classes of one variable (Q1066051)

The author continues his interesting investigations pertaining to imbedding theory. In the present paper he proves nine new theorems. Here we recall only two of them as sample theorems. For notions and notations we refer to the paper. Theorem 4. Let \(\omega\) (\(\delta)\) be an arbitrary modulus of continuity and \(\phi\) (u) a \(\phi\)-function satisfying \(\Delta^ 2\)-condition. Then the imbedding \(H^{\omega}_{{\mathcal M}_ p}\subset \phi (L)\) holds if and only if \[ \sum^{\infty}_{n=1}n^{-2}\phi (n^{1/p}\omega (\frac{1}{n}))<\infty. \] Theorem 6. If \(1\leq p<\infty\), and the modulus of continuity \(\omega\) (\(\delta)\) satisfies conditions (L) and \((L_ 1)\), then the class \(H_ p^{\omega}\) can be imbedded into the symmetric class X of measurable functions if and only if \(\int^{1}_{x}t^{-1-(1/p)}\omega (t)dt\in X.\) The author considers the following symmetric spaces X: Lebesgue spaces \(L_ 1\) (1\(\leq p\leq \infty)\), Orlicz spaces \(L^*_{\phi}\), Lorentz spaces \(\Lambda\) (\(\phi)\) and \(L_{p,q}\), and Marcinkiewicz spaces \({\mathcal M}(\phi)\).