  The Sonine kernel described by the classical Sonine condition of convolution
form is an important class of kernels used in integral equations and nonlocal
differential equations. This work extends this idea to introduce weighted
Sonine conditions where the non-convolutional weight functions accommodate the
inhomogeneity in practical applications. We characterize tight relations
between classical Sonine condition and its weighted versions, which indicates
that the non-degenerate weight functions may not introduce significant changes
on the set of Sonine kernels. To demonstrate the application of weighted Sonine
conditions, we employ them to derive equivalent but more feasible formulations
of weighted integral equations and nonlocal differential equations to prove
their well-posedness, and discuss possible application to corresponding partial
differential equation models.
