The boundedness of commutators of Calderón–Zygmund operators in grand variable exponent Morrey spaces is established. The operators and spaces are defined on quasi-metric measure spaces with doubling measure. The obtained results are applied to study regularity properties of solutions of the second-order partial differential equations with discontinuous coefficients in the frame of grand variable exponent Morrey spaces.

In this study, the concept of $(m,n)-$polynomial $(p_1,p_2)$- convex functions on the co-ordinates has been established with some basic properties. Dependent on this new concept, a new Hermite-Hadamard type inequality has been proved, then some new integral inequalities have been obtained for partial differentiable $(m,n)-$polynomial $(p_1,p_2)$- convex functions on the co-ordinates. Several special cases that some of them proved in earlier works have been considered.