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

We consider non-standard Holder spaces H(lambda(.))(X) of functions f on a metric measure space (X, d, mu), whose Holder exponent lambda(x) is variable, depending on x is an element of X. We establish theorems on mapping properties of potential operators of variable order alpha(x), from such a variable exponent Holder space with the exponent lambda(x) to another one with a "better" exponent lambda(x) + alpha(x), and similar mapping properties of hypersingular integrals of variable order alpha(x) from such a space into the space with the "worse" exponent lambda(x) - alpha(x) in the case alpha(x) 0. We admit variable complex valued orders alpha(x), where R alpha(x) may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized Holder spaces with the weight alpha(x).FCT, Portugal [SFRH/BPD/34258/2006