The Weighted Hardy Inequality: New Proofs and the Case  p  = 1 | Zendy

Sinnamon Gord | Zendy; Stepanov Vladimir D. | Zendy

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The Weighted Hardy Inequality: New Proofs and the Case p = 1

Author(s) -

Sinnamon Gord,

Stepanov Vladimir D.

Publication year - 1996

Publication title -

journal of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.441

H-Index - 62

eISSN - 1469-7750

pISSN - 0024-6107

DOI - 10.1112/jlms/54.1.89

Subject(s) - mathematics , mathematical proof , monotone polygon , inequality , monotonic function , combinatorics , pure mathematics , discrete mathematics , mathematical analysis , geometry

An elementary proof is given of the weight characterisation for the Hardy inequality 1.1( ∫ 0 ∞( ∫ 0 ∞ f ) q υ ( x ) d x ) 1 / q ⩽ C ( ∫ 0 ∞ f p u ) 1 / p for f ⩾ 0 , in the case 0 < q < p , 1 < p < ∞. It is also shown that certain weighted inequalities with monotone kernels are equivalent to inequalities in which one of the weights is monotone. Using this, a characterisation of those weights for which (1.1) holds with 0 < q < I = p is given. Results for (1.1), considered as an inequality over monotone functions f are presented.

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