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A TWO-WEIGHT INEQUALITY BETWEEN $L^{p}(\ell ^{2})$ AND $L^{p}$

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  14 February 2018

Tuomas Hytönen  and
Emil Vuorinen
Tuomas Hytönen
Affiliation:
Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland email [email protected]
Emil Vuorinen
Affiliation:
Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland email [email protected]
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Abstract

We consider boundedness of a certain positive dyadic operator

$$\begin{eqnarray}T^{\unicode[STIX]{x1D70E}}:L^{p}(\unicode[STIX]{x1D70E};\ell ^{2})\rightarrow L^{p}(\unicode[STIX]{x1D714}),\end{eqnarray}$$
that arose during our attempts to develop a two-weight theory for the Hilbert transform in $L^{p}$. Boundedness of $T^{\unicode[STIX]{x1D70E}}$ is characterized when $p\in [2,\infty )$ in terms of certain testing conditions. This requires a new Carleson-type embedding theorem that is also proved.

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 284 - 302
Copyright
Copyright © University College London 2018 

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