Hausdorff operators on holomorphic Hardy spaces and applications

Abstract

The aim of this paper is to characterize the nonnegative functions ϕ defined on (0, ∞) for which the Hausdorff operatoris bounded on the Hardy spaces of the upper half-plane H p a (C + ), p ∈ [1, ∞]. The corresponding operator norms and their applications are also given.2010 Mathematics Subject Classification. 47B38 (42B30, 46E15).

Hausdorff Operators on Some Spaces of Holomorphic Functions on the Unit Disc

“…Then formula (10) is valid. In view of (9) this implies (8). In turn, (8) implies (7) and then for every algebraic polynomial q n we get…”

“…It should be noted that different operators of Hausdorff type on spaces of holomorphic functions in the disc or the half-plane were considered in [10], [2], [6], [7], [8], [9], [5] but all of them are not special cases of the definition 1.…”

On Hardy kernels as reproducing kernels

“…The boundedness of H + on L p (R + ) for 1 < p < ∞ immediately follows from the boundedness of the Hilbert transform on L p (R) (see, for example, [5]). The following theorem has been inspired by [11,14]. Proof Let f ∈ L p (R + ).…”

“…where F θ (r) ∶= F(re iθ ), for r > 0, θ ∈ (−π/2, π/2). Indeed, the last term in (1.2) shows that D H F is holomorphic (see, for example, [11]), and the boundedness of D H F follows by an application of Hardy's inequality together with the realization of the norm of H p (C + ) given in [19] by…”

“…Hardy’s inequality also allows us to define a bounded operator on the Hardy spaces on the half plane , where , by where , for . Indeed, the last term in (1.2) shows that is holomorphic (see, for example, [11]), and the boundedness of follows by an application of Hardy’s inequality together with the realization of the norm of given in [19] by We will refer to these families of bounded operators on and as Hardy operators. These families have been actively studied, and are often labeled as Hausdorff operators due to its relation to the Hausdorff summability method through the function for (see the survey articles [4, 15] for more details).…”