LAUSR

This paper aims to further explore the existing theory of weighted fractional operators with respect to functions. This theory extends some fundamental results of classical Riemann–Liouville and Caputo fractional derivatives to their weighted counterparts involving fractional differentiation and integration with respect to functions. By investigating the fundamental principles of these operators, we establish mean value theorems, Taylor's theorems, and integration by parts formulae. The Leibniz rule is extended for weighted Riemann–Liouville derivatives with respect to functions. Also, we present necessary conditions for the existence and uniqueness of solutions for a class of initial value problems, involving weighted Caputo fractional derivatives with respect to functions, in a Sobolev space.