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Boundedness of some bilinear operators on variable Herz-type Hardy spaces

  • Douadi Drihem
  • Rabah Heraiz
Article
  • 8 Downloads

Abstract

This paper is concerned with proving some estimate on variable Herz-type Hardy spaces of bilinear operators
$$\begin{aligned} B(f,g)(x)=\overset{N}{\underset{\gamma =1}{\sum }}\left( T_{\gamma }^{1}f\right) (x)\left( T_{\gamma }^{2}g\right) (x),\quad x\in {\mathbb {R}}^{n}, \end{aligned}$$
where \(N\in {\mathbb {N}}\), \(T_{\gamma }^{1}\) and \(T_{\gamma }^{2}\) are operators satisfying certain conditions. More precisely we prove the boundedness of B from \(H\dot{K}_{p_{1}(\cdot ) }^{\alpha _{1}(\cdot ) ,q_{1}(\cdot ) }\left( {\mathbb {R}} ^{n}\right) \times \dot{K}_{p_{2}(\cdot ) }^{\alpha _{2}(\cdot ) ,q_{2}(\cdot ) }\left( {\mathbb {R}} ^{n}\right) \) into \(H\dot{K}_{p(\cdot ) }^{\alpha (\cdot ) ,q(\cdot ) }\left( {\mathbb {R}} ^{n}\right) \) and from \(H\dot{K}_{p_{1}(\cdot ) }^{\alpha _{1}(\cdot ) ,q_{1}(\cdot ) }\left( {\mathbb {R}} ^{n}\right) \times H\dot{K}_{p_{2}(\cdot ) }^{\alpha _{2}(\cdot ) ,q_{2}(\cdot ) }\left( {\mathbb {R}} ^{n}\right) \) into \(H\dot{K}_{p(\cdot ) }^{\alpha (\cdot ) ,q(\cdot ) }\left( {\mathbb {R}} ^{n}\right) \), with some appropriate assumptions on the parameters \(\alpha (\cdot ) \), \(\alpha _{i}(\cdot ) \), \(p(\cdot ) \), \(p_{i}(\cdot ) \), \(q(\cdot ) \) and \( q_{i}(\cdot ) \), \(i=1,2\). Our results cover the results on Herz-type Hardy spaces with fixed exponents.

Keywords

Herz-type Hardy space Atom Variable exponent Sublinear operator 

Mathematics Subject Classification

Primary 42B20 Secondary 42B25 42B30 

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Laboratory of Functional Analysis and Geometry of Spaces, Department of MathematicsM’sila UniversityM’silaAlgeria

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