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Paper IPM / M / 17679  


Abstract:  
This contribution is based on joint work with Hormozi et al. (J. Fourier Anal. Appl. 29:49, 2023). We simplify and shorten the proofs of the main results of Shi et al. (J. Math. Pures Appl. 101:394Ã?ÃÂ¢??413, 2014) significantly. In particular, the new proof for Shi et al. (J. Math. Pures Appl. 101:394Ã?ÃÂ¢??413, 2014; Theorem 1.1) is quite short and, unlike the original proof, does not rely on the properties of the Ã?ÃÂ¢??Marcinkiewicz functionÃ?ÃÂ¢?ÃÂ¯ÃÂ¿ÃÂ½. This allows us to get a precise linear dependence on Dini constants with a subsequent application to LittlewoodÃ?ÃÂ¢??Paley operators by the wellknown techniques.
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