Two weight L^p-inequalities for dyadic shifts and the dyadic square\n function

2015/04/22 by Emil Vuorinen, Vuorinen, Emil
Mathematics · #42B20 #42B25 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1504.05759

openalex publication_date 2015/04/22 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider two weight L^p\→ L^q-inequalities for dyadic shifts and\nthe dyadic square function with general exponents 1<p,q<\∞. It is shown\nthat if a so-called quadratic mathscrA_p,q-condition related to the\nmeasures holds, then a family of dyadic shifts satisfies the two weight\nestimate in an \R-bounded sense if and only if it satisfies the\ndirect- and the dual quadratic testing condition. In the case p=q=2 this\nreduces to the result by T. Hyt "onen, C. P 'erez, S. Treil and A. Volberg.\n The dyadic square function satisfies the two weight estimate if and only if\nit satisfies the quadratic testing condition and the quadratic\n mathscrA_p,q-condition holds. Again in the case p=q=2 we recover the\nresult by F. Nazarov, S. Treil and A. Volberg.\n An example shows that in general the quadratic mathscrA_p,q-condition\nis stronger than the Muckenhoupt type A_p,q-condition.\n

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