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

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In this paper, we consider a class of refined (a,b)-Fourier integral transforms defined as [Formula: see text] where a, b are nonzero arbitrary complex coefficients such that [Formula: see text]. This transform was introduced in recent work [Asian-Euro. J. Math. 15 (2022), no. 08, 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 integral equations and heat source problems. An explicit example is provided to illustrate and validate the effectiveness of the obtained results