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A weighted inequality for the Hardy operator involving suprema

Pavla Hofmanová (2016)

Commentationes Mathematicae Universitatis Carolinae

Let u be a weight on ( 0 , ) . Assume that u is continuous on ( 0 , ) . Let the operator S u be given at measurable non-negative function ϕ on ( 0 , ) by S u ϕ ( t ) = sup 0 < τ t u ( τ ) ϕ ( τ ) . We characterize weights v , w on ( 0 , ) for which there exists a positive constant C such that the inequality 0 [ S u ϕ ( t ) ] q w ( t ) d t 1 q 0 [ ϕ ( t ) ] p v ( t ) d t 1 p holds for every 0 < p , q < . Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

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