What question did this study set out to answer?

To establish a necessary-and-sufficient two-weight criterion and optimal constant bounds for Hardy-type operators on arbitrary time scales in the quasi-Banach range 0 < q < 1 < p < ∞.

August 19, 2026AxiomsOpen Access

Weighted Hardy Inequalities on Time Scales in the Range 0 < q < 1 < p < ∞

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To establish a necessary-and-sufficient two-weight criterion and optimal constant bounds for Hardy-type operators on arbitrary time scales in the quasi-Banach range 0 < q < 1 < p < ∞.
Developed a gap-compatible weighted level-function construction utilizing interval averages evaluated over right-scattered gaps on arbitrary time scales.
Formulated conditions based on the finiteness of a mixed head–tail integral quantity combining forward-jump delta operators and weight functions.
Established explicit two-sided bounds for the optimal inequality constant via the finiteness of the mixed head–tail integral criterion.
Recovered classical continuous and discrete characterizations while extending results to dynamic averaging, Bennett–Copson-type, Hardy–Flett-type, and shifted quantum specializations.

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Implication

Mathematical analysis establishes necessary-and-sufficient two-weight criteria for Hardy inequalities on time scales, unifying continuous, discrete, and quantum specializations.

Cite This Study

Mahmoud et al. (2026) studied this question.

synapsesocial.com/papers/6a85639d03308d306e2d6d22 https://doi.org/10.3390/axioms15080613

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Implication

Mathematical analysis establishes necessary-and-sufficient two-weight criteria for Hardy inequalities on time scales, unifying continuous, discrete, and quantum specializations.

Weighted Hardy Inequalities on Time Scales in the Range 0 < q < 1 < p < ∞

Key Points

Discussion

Implication

Cite This Study

Weighted Hardy Inequalities on Time Scales in the Range 0 < q < 1 < p < ∞

Key Points

Discussion

Implication

Cite This Study

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