In specific acid catalysis, protonated solvent is the catalyst.  The reaction rate is proportional to the concentration of the protonated solvent molecules SH. The acid catalyst itself (AH) only contributes to the rate acceleration by shifting the chemical equilibrium between solvent S and AH in favor of the SH species. This kind of catalysis is common for strong acids in polar solvents, such as water.
For example, in an aqueous buffer solution the reaction rate for reactants R depends on the pH of the system but not on the concentrations of different acids.
This type of chemical kinetics is observed when reactant R is in a fast equilibrium with its conjugate acid RH which proceeds to react slowly with R to the reaction product; for example, in the acid catalysed aldol reaction.