Some equivalent criteria for the boundedness of Hardy-type operators on the cone of quasimonotone functions

Let beta and gamma be non-negative Borel measures on [0, infty), p,q>0, let Omega be a certain cone of non-negative Borel measurable functions on [0, infty), and let A be a positive operator. par Introduce H_Omega (A)= sup_{fin Omega}left[left(int_0^infty (Af)^q dgammaright)^{1/q} left(int_0^infty f^p dbeta right)^{-1/p} right]. par Let us consider the cones of functions that are monotone with respect to the prescribed positive continuous functions k and m: Omega_k = {f geq 0 : f(tau)/k(tau)downarrow};quad Omega^m = {f geq 0 : f(tau)/m(tau)uparrow}. Let us consider the generalized Hardy operators A_mu and B_mu, where mu is a non-negative Borel measure on [0, infty) defined as (A_mu f)(t) = int_{(0,t]} f dmu, quad (B_mu f)(t) = int_{[t, infty)} f dmu. par In this paper, two-sided estimates are established for H_{Omega^m} (A_mu) and H_{Omega_k} (B_mu). Moreover, the author presents some other equivalent descriptions, considers some particular cases and establishes results in the case of a degenerate measure.