On Discrete Weighted Lorentz Spaces and Equivalent Relations between Discrete <i>ℓ_p</i>-Classes

Abstract

In this paper, we study some relations between different weights in the classes Bp,Bp*,Mp and Mp* that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space Λp. These results will be proven by using the properties of classes Bp,Bp*,Mp and Mp*, including the self-improving properties and also the properties of the generalized Hardy operator Hp, the adjoint operator Sq and some fundamental relations between them connecting their composition to their sum.

Keywords

• discrete operators
• weighted norm inequalities
• self-improving properties
• A p –weights%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm470"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi mathvariant="script">A</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>–weights
• B p –weights%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm471"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="script">B</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>–weights
• B p * –weights%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm472"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="script">B</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mo>*</mml:mo> </mml:msubsup> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>–weights