Abstract
In this article, we consider the Laplace-Bessel differential operator
{\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0.
Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator
{M}_{b,\gamma }
and the commutator
{[}b,{A}_{\gamma }]
of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator
{[}b,{I}_{\alpha ,\gamma }]
of the B-Riesz potential on B-Morrey spaces
{L}_{p,\lambda ,\gamma }
, when
b\in {\text{BMO}}_{\gamma }
.
Sobolev-type inequalities and eigenvalue growth on graphs with finite measure
