On the Boundedness of q-Hausdorff Operators on q-Hardy Spaces
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Authors: O. TYR
DOI: 10.46793/KgJMat2608.1261T
Abstract:
E. Liflyand and F. Móricz proved that the Hausdorff operator generated by a function φ ∈ L1(ℝ) is a linear operator bounded on the real Hardy space H1(ℝ) by using the classical Fourier transform and the Hilbert transform, they also proved that this operator commutes with Hilbert transform. In this work, we extend these results to the context of q-harmonic analysis associated with the q-Rubin’s operator, we introduce the q-Hilbert transform on the real line, we study some of its main properties. Next, we define the q-Hardy spaces ℍq1(ℝq) by means of the q-Hilbert transforms, we finally study the q-Hausdorff operator and we prove the boundedness property and the commuting relation of this operator and q-Hilbert transform in q-Hardy spaces.
Keywords:
q-Fourier transform, q-translation operator, q-Hilbert transform, q-Hardy spaces, q-Hausdorff operator.
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