Two-weighted inequality for $p$-admissible $B_{k,n}$–singular operators in weighted Lebesgue spaces

In this paper, we study the boundedness of $p$-admissible singular operators, associated with the Laplace-Bessel differential operator $B_{k,n}=\sum\limits_{i=1}^{n} \frac{\partial ^2}{\partial x_i^2} + \sum\limits_{j=1}^k \frac{\gamma_j}{x_j}\frac \partial {\partial x_j}$ ($p$-admissible $B_{k,n}$–singular operators) on a weighted Lebesgue spaces $L_{p,\omega,\gamma}(\Rnk)$ including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the $B_{k,n}$–maximal operator, $B_{k,n}$–singular integral operators and so on. Sufficient conditions on weighted functions $\omega$ and $\omega_1$ are given so that $p$-admissible $B_{k,n}$–singular operators are bounded from $L_{p,\omega,\gamma}(\Rnk)$ to $L_{p,\omega_1,\gamma}(\Rnk)$ for $1<p<\infty$ and weak $p$-admissible $B_{k,n}$–singular operators are bounded from $L_{p,\omega,\gamma}(\Rnk)$ to $L_{p,\omega_1,\gamma}(\Rnk)$ for $1\le p<\infty$.