On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque L p,α spaces

Abstract

Abstract In this paper, we consider the square function ( S f ) ( x ) = ( ∫ 0 ∞ | ( f ⊗ Φ t ) ( x ) | 2 d t t ) 1 / 2 $$\begin{array}{} \displaystyle (\mathcal{S}f)(x)=\left( \int\limits_{0}^{\infty }|(f\otimes {\it\Phi}_{t})\left( x\right) |^{2}\frac{dt}{t}\right) ^{1/2} \end{array} $$ associated with the Bessel differential operator B t = d 2 d t 2 + ( 2 α + 1 ) t d d t , $\begin{array}{} B_{t}=\frac{d^{2}}{dt^{2}}+\frac{(2\alpha+1)}{t}\frac{d}{dt}, \end{array} $ α > −1/2, t > 0 on the half-line ℝ+ = [0, ∞). The aim of this paper is to obtain the boundedness of this function in L p,α , p > 1. Firstly, we proved L 2,α -boundedness by means of the Bessel-Plancherel theorem. Then, its weak-type (1, 1) and L p,α , p > 1 boundedness are proved by taking into account vector-valued functions.