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New equivalent conditions for Hardy-type inequalities

Alois Kufner, Komil Kuliev, Gulchehra Kulieva, Mohlaroyim Eshimova (2024)

Mathematica Bohemica

We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties.

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2012)

Open Mathematics

We obtain Hardy type inequalities ∫ 0 ∞ M ω r u r ρ r d r ⩽ C 1 ∫ 0 ∞ M u r ρ r d r + C 2 ∫ 0 ∞ M u ' r ρ r d r , and their Orlicz-norm counterparts ω u L M ( ℝ + , ρ ) ⩽ C ˜ 1 u L M ( ℝ + , ρ ) + C ˜ 2 u ' L M ( ℝ + , ρ ) , with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

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