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Weighted theorems on fractional integrals in the generalized Hölder spaces via indices ${m}_{\omega }$ and ${M}_{\omega }$. (English) Zbl 1114.26005

The numbers ${m}_{\omega }$ and ${M}_{\omega }$ of a function $\omega \left(x\right)\in W$ are defined by

${m}_{\omega }=\underset{x>1}{sup}\frac{ln\left[\underline{{lim}_{h\to 0}}\frac{\omega \left(xh\right)}{\omega \left(h\right)}\right]}{lnx},\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{M}_{\omega }=\underset{x>1}{inf}\frac{ln\left[\overline{{lim}_{h\to 0}}\frac{\omega \left(xh\right)}{\omega \left(h\right)}\right]}{lnx}·$

In order to answer the question whether it is possible to derive results on mapping properties of operators of fractional integration in generalized Hölder spaces in terms of the direct numerical interval for the exponents of the weight, with boundaries depending on the indices ${m}_{\omega }$ and ${M}_{\omega }$, the authors establish the following main theorem, which provides the equivalence of the integral Zygmund conditions to certain direct numerical inequalities for the aforesaid indices.

Theorem. A function $\omega \in W$ belongs to ${Z}^{\beta }$, $\beta \ge 0$, iff ${m}_{\omega }>\beta$ and it belongs to ${Z}_{\gamma }$, $\gamma >0$, iff ${M}_{\omega }<\gamma$, so that in the case $0\le \beta <\gamma$, there holds the formula $\omega \in {{\Phi }}_{\gamma }^{\beta }⇔\beta <{m}_{\omega }\le {M}_{\omega }<\gamma$ and for $\omega \in {{\Phi }}_{\gamma }^{\beta }$ and for any $\epsilon >0$ there exist constants ${c}_{1}={c}_{1}\left(\epsilon \right)>0$ and ${c}_{2}={c}_{2}\left(\epsilon \right)>0$ such that ${c}_{1}{x}^{{M}_{\omega }+\epsilon }\le \omega \left(x\right)\le {c}_{2}{x}^{{m}_{\omega }-\epsilon }$, $0\le x\le \ell ·$

Based on this theorem, the authors further establish a series of new theorems on the action of fractional integrals in the generalized Hölder spaces for various types of weights including one- dimensional and multi-dimensional cases.

##### MSC:
 26A33 Fractional derivatives and integrals (real functions) 26A16 Lipschitz (Hölder) classes, etc. (one real variable)