Uncertainty principles and Calderón's formulas for the deformed Hankel $L^2_\alpha$-multiplier operators

The main purpose of this paper is to introduce the deformed Hankel $L^2_\alpha$-multiplier operators and to give some new results related to these operators as Plancherel’s, Calderón's reproducing formulas and Heisenberg's, Donoho-Stark's uncertainty principles. Next, using the theory of reproducing kernels, we give best estimates and an integral representation of the extremal functions related to these operators on weighted Sobolev spaces.