Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces

Abstract

We consider non-standard H\"older spaces $H^{\lb(\cdot)}(X)$ of functions $f$ on a metric measure space $(X,d,\mu)$, whose H\"older exponent $\lb(x)$ is variable, depending on $x\in X$. We establish theorems on mapping properties of potential operators of variable order $\al(x)$, from such a variable exponent H\"older space with the exponent $\lb(x)$ to another one with a ``better'' exponent $\lb(x)+\al(x)$, and similar mapping properties of hypersingular integrals of variable order $\al(x)$ from such a space into the space with the ``worse'' exponent $\lb(x)-\al(x)$ in the case $\al(x)<\lb(x)$. These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their ensities. These estimates allow us to treat not only the case of the spaces $H^{\lb(\cdot)}(X)$, but also the generalized H\"older spaces $H^{w(\cdot,\cdot)}(X)$ of functions whose continuity modulus is dominated by a given function $w(x,h), x\in X, h>0$. We admit variable complex valued orders $\al(x)$, where $\Re\al(x)$ may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized H\"older spaces with the weight $\al(x)$.