Abstract
Abstract We generalize the Zygmund inequality for the conjugate function to the Morrey type spaces defined on the unit circle T. We obtain this extended Zygmund inequality by introducing the Morrey-Zygmund space on T.
Highlights
This paper aims to extend the celebrated Zygmund inequality to the Morrey-Zygmund space on the unit circle T = {eiθ : −π < θ π}
Inspired by the recent developments of the studies of Morrey spaces, we investigate the extension of the Zygmund inequality on Morrey spaces
Since we study the conjugate function operator, we consider the Morrey type spaces defined on the unit circle [11]
Summary
This paper aims to extend the celebrated Zygmund inequality to the Morrey-Zygmund space on the unit circle T = {eiθ : −π < θ π}. The classical Zygmund inequality gives the borderline behavior of the conjugate function operator on L1. The importance of the conjugate function operator stems from its role in the study of Fourier series. Since the introduction of the classical Morrey spaces on Rn in [3], several important results in Lebesgue spaces have been extended to Morrey spaces. Since we study the conjugate function operator, we consider the Morrey type spaces defined on the unit circle [11]. The main result of this paper establishes the boundedness of the conjugate function operator as a mapping from the Morrey spaces built on Zygmund space to Morrey spaces.
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