q–DEFORMED HILBERT TRANSFORM AND ITS RELATED PROPERTIES AND INEQUALITIES
Shaimardan S. Tokmagambetov N.S.
July 2025Element D.O.O.
Mathematical Inequalities and Applications
2025#28Issue 3493 - 518 pp.
We present a new formulation of the Hilbert transform constructed via the q-deformation of convolution, which is called the q-deformed Hilbert transform. We also find the q-deformed Hilbert transform of some basic functions and examine its connection with the qFourier transform. In particular, a number of new related inequalities and embeddings are proved such as a q-analuge of the Chebyshev inequality and a Hardy-type inequality. In additionally, we present a direct application of Hardy-type inequality to study some inequalities for the qdeformed Hilbert transform on Lp(Rq) and Lp,r(Rq). Finally, we prove a weak (1.1) inequality for the q-deformed Hilbert transform.
boundedness , Hardy inequality , Hilbert transform , inequality , q-calculus , q-derivative
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Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Institute of Applied Mathematics, Karaganda Buketov University, Karaganda, Kazakhstan
Institute of Mathematics and Mathematical Modeling
Institute of Applied Mathematics
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