A Derivative-Hilbert operator acting on Hardy spaces

Abstract

Let µ be a positive Borel measure on the interval [0, 1). The Hankel matrix H µ = (µ n,k ) n,k≥0 with entries µ n,k = µ n+k , where µ n = [0,1) t n dµ(t), induces formally the operator(1−tz) 2 dµ(t) for all in Hardy spaces H p (0 < p < ∞), and among them we describe those for which DH µ is a bounded(resp.,compact) operator from H p (0 < p < ∞) into H q (q > p and q ≥ 1). We also study the analogous problem in Hardy spaces H p (1 ≤ p ≤ 2).

Application on a Factor Derived from Hilbert and Carleson Measure on Hardy Spaces

“…And since is 1-Carleson measure by Theorem 3 in [7], we know which implies that is a well defined for all and (see [9]).…”

“…We obtain the sufficient condition such this are well-defined in and obtain that for all with the certain condition (see [9]).…”

“…We characterize the positive Borel measures for which the operator which and is well defined in the Hardy spaces Then we give the necessary and sufficient conditions such that operator is bounded from the Hardy space into the space or from into (see [9]).…”

“…The space of all Bloch functions is denoted by The classic reference for Bloch's functions theory is [3,14]. The relationship between these spaces which we gave above is well known, As usual, during this paper, refers to a positive constant that depends only on the displayed parameters but is not necessarily the same from case to case, for any given will denote the conjugate index of that is, (see [9]).…”

“…The Hankel matrix with entries where for analytic functions the generalized Hilbert operator define as When the right side is logical and defines an analytical function in In recent decades, the generalized Hilbert operator which is induced by the Hankel matrix has been studied extensively [1][2][3][4][5]. Galanopoulos and Pel ez [6], characterized the Borel measure for which the Hankel operator is a bounded operator on Then Chatzifountas, Girela, and Peláez [7] extended this with Hardy spaces with In [8], Girela and Merchán studied factorization that operates on certain fixed areas of analytic functions on disk, we follow S. Ye and G. Feng [9].…”