Collectanea Mathematica

, Volume 69, Issue 2, pp 297–314 | Cite as

On generalized Littlewood–Paley functions

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Abstract

We study the \(L^{p}\) boundedness of certain classes of generalized Littlewood–Paley functions \(\mathcal{S}_{\Phi }^{(\lambda )}(f)\). We obtain \(L^{p}\) estimates of \(\mathcal{S}_{\Phi }^{(\lambda )}(f)\) with sharp range of p and under optimal conditions on \(\Phi \). By using these estimates along with an extrapolation argument we obtain some new and improved results on generalized Littlewood–Paley functions. The approach in proving our results is mainly based on proving vector-valued inequalities and in turn the proof of our results (in the case \(\lambda =2)\) provides us with alternative proofs of the results obtained by Duoandikoetxea as his approach is based on proving certain weighted norm inequalities.

Keywords

Littlewood–Paley functions Triebel–Lizorkin spaces Orlicz spaces Block spaces Extrapolation \(L^{p}\)boundedness 

Mathematics Subject Classification

Primary 42B20 Secondary 42B25 42B35 42B99 

Notes

Acknowledgements

The authors would like to express their gratitude to the referee for his/her very careful reading and for many important valuable comments.

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Copyright information

© Universitat de Barcelona 2017

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsQatar UniversityDohaQatar
  2. 2.Department of MathematicsBryn Mawr CollegeBryn MawrUSA
  3. 3.Department of MathematicsUniversity of PittsburghPittsburghUSA

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