Categories of results in variable Lebesgue spaces theory

Variable (exponent) Lebesgue spaces represent a relevant research area within the theory of Banach function spaces. Much attention is devoted to look for sufficient conditions on the variable exponent to establish the assertions within the the- ory. In this Note we try to show the beauty of the research in this field, mainly quoting some known results organized into "categories", each of them characterized by a com- mon typology of conditions on the variable exponent. New results involve the failure of rearrangement-invariant property, the rearrangement of the exponent, and a generaliza- tion of a formula known for constant exponents.