New Estimates for the Higher Order Riesz Bessel Transform

Abstract

In this paper, we study the B-Riesz transformation $R^{(k)}_{\gamma}$ generated
by a generalized translation operator $\mathbf{T}^{y}$ ($y\in\mathbb{R}^{n}_{+}$).
We prove that the Cotlar-type inequality holds for these operators. In particular,
we prove results related to the Cotlar inequality for the even and odd cases of
the kernel of the B-Riesz transform. Thus, new estimates for the
B-Riesz transform in weighted Lebesgue spaces $L_{p,\gamma,w}$ are obtained.