  In this paper we introduce Besov-Morrey spaces with all indices variable and
study some fundamental properties. This includes a description in terms of
Peetre maximal functions and atomic and molecular decompositions. This new
scale of non-standard function spaces requires the introduction of variable
exponent mixed Morrey-sequence spaces, which in turn are defined within the
framework of semimodular spaces. In particular, we obtain a convolution
inequality involving special radial kernels, which proves to be a key tool in
this work.
