Heisenberg uncertainty principle and Hausdorff–Young theorem associated to refined (a,b)-Fourier transform

Source abstract

In this paper, we consider a class of refined [Formula: see text]-Fourier integral transforms defined as follows: [Formula: see text] where [Formula: see text] are nonzero arbitrary complex coefficients such that [Formula: see text]. This transform was introduced in a recent work [Asian-Eur. J. Math. 15(8) (2022) 2250151]. First, we establish an analog version of Heisenberg’s classical uncertainty principle associated to [Formula: see text] transform on the real line and derive conditions on the function that involves the equal sign in the uncertainty principle. Then, we formulate Hausdorff–Young inequalities adapted to this family of transforms and their corresponding reverse transforms. These inequalities are employed to demonstrate the boundedness of a Hermite-weighted convolution operator defined via the [Formula: see text] transform. Finally, we apply these inequalities to analyze the solvability of a particular class of Fredholm integral equations and heat source problems. Two explicit examples are provided to illustrate and validate the effectiveness of the obtained results.