In this treatise, the boundedness of the multilinear fractional Hardy operators is scrutinized within the context of variable exponent Morrey-Herz spaces, denoted as $M{\dot{K}}_{q,p(\cdot )}^{\alpha (\cdot ),\lambda }({\mathbb{R}}^n)$. Analogous estimations are derived for their commutators, contingent upon the symbol functions residing in the space of bounded mean oscillation (BMO) with variable exponents.

This paper demonstrates the boundedness of the fractional bilinear Hardy operator and its adjoint on weighted $-\lambda$-central Morrey spaces with a variable exponent. Analogous outcomes for their commutators were derived when the symbol functions are elements of the weighted $-\lambda$-central bounded mean oscillation ( $-\lambda$-central BMO) spaces.

This paper intends to demonstrate the boundedness of the fractional bilinear Hardy operator and its adjoint on the $\lambda$-central Morrey space with variable exponents. Analogous outcomes for their commutators are derived when the symbol functions are elements of the $\lambda$-central bounded mean oscillation ( $\lambda$-central BMO) space.