Weighted Hardy-type inequalities
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Date
2026
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SDU University
Abstract
This thesis is devoted to weighted Hardy-type inequalities and identities obtained by the factorization method. The main focus is on Hardy-type identities in the Baouendi–Grushin setting, where the Euclidean radial derivative is replaced by the radial derivative associated with the Grushin structure. More precisely, we consider the homogeneous distance function ρ and the Grushin gradient ∇γ, and study identities involving the radial Grushin derivative ∇γρ · ∇γu |∇γρ| . Using suitable first-order differential operators and their formal adjoints, we derive an improved weighted Hardy identity with an explicit non-negative remainder term. The result is formulated under the assumption that the weight functions V and W form a Bessel pair adapted to the homogeneous dimension Q. The obtained identity extends classical radial Hardy-type identities with Bessel pairs to the degenerate Baouendi–Grushin framework. Several special cases are also discussed, including power-type and logarithmic weights, which illustrate how the general theorem leads to concrete weighted Hardy-type inequalities.
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Keywords
Hardy identity, applications and special cases, Bessel pairs
Citation
Atina A / Weighted Hardy-type inequalities / SDU University / Department of Mathematics and Natural Sciences