Weighted generalized Hölder spaces as willposednedd classes for Sonine integral Equations

The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions

“…are discussed. However, we stress that the results proved in the present paper are more general since they take into account also kernels that are more singular in a neighborhood of the origin, such as, indeed, the Volterra function I, whose asymptotyc expansion near 0 is given by 1 x log 2 (1/x) (see (8)).…”

“…Remark 3.5. It is interesting to note that in the paper [33] (see also [8]) the authors prove a result of the same type as Theorem 3.1, where kernels more general than powers are considered. However, in [33] the assumption to belong to a certain class V λ , 0 < λ < 1, implies that the kernel enjoys a higher integrability property (in fact, if k ∈ V λ , it is k(x) ≤ c x −λ around zero), hence kernels like our model example I, discussed in Section 2, cannot be considered.…”

“…Proof. The second part is immediate since I is continuous, positive and coercive by (8). On the other hand, in order to prove the second part, it is sufficient to show d 2 dx 2 I(x) ≥ 0; namely, by (9), that d 3 dx 3 µ(x, 0, 0) ≥ 0.…”

“…It is also worth mentioning that a first discussion on the operator I is present in [32], whereas similar integral operators, but with more regular kernels, have been investigated more recently by [8,31]. More in detail, in [8] it is analyzed the case of a certain class of almost decreasing Sonine kernels in terms of weighted generalized Hölder spaces, while in [31] an "inverse" operator is discussed within the framework of L p spaces. We also recall that in [18] some relevant features of Volterra functions are pointed out, such as asymptotic expansions and some striking relations with the Ramanujan integrals.…”