Fractional integrals of imaginary order in the space of Hölder functions with a power weight on an interval. (Q1889504)

Recently, the authors have studied an analog of fractional integrals of purely imaginary order in the Marchaud form [cf.~Dokl. Math. 59, No. 1, 110--112 (1999); transl.~from Dokl. Akad. Nauk, Ross. Akad. Nauk 364, No. 6, 738--740 (1999; Zbl 0969.26006)]. The use of the Marchaud form is convenient to establish the semigroup property of operators of imaginary order. In the present paper, the authors study fractional integrals of imaginary order in the space of Hölder functions with polynomial weight. Fractional integrals of purely imaginary order behave like singular integrals and hence the earlier procedure cannot be applied here. The authors obtain conditions on the orders of the weight for which this operator is invertible in the above mentioned space.