Gagliardo-Nirenberg inequalities in weighted Orlicz spaces

Agnieszka Kałamajska; Katarzyna Pietruska-Pałuba

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On a Variant of the Gagliardo-Nirenberg Inequality Deduced from the Hardy Inequality

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We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.

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We obtain Hardy type inequalities $\int_{0}^{\infty} M (ω (r) |u (r)|) ρ (r) d r ⩽ C_{1} \int_{0}^{\infty} M (|u (r)|) ρ (r) d r + C_{2} \int_{0}^{\infty} M (|u^{'} (r)|) ρ (r) d r,$ and their Orlicz-norm counterparts ${∥ω u∥}_{L^{M} (ℝ_{+}, ρ)} ⩽ {\tilde{C}}_{1} {∥u∥}_{L^{M} (ℝ_{+}, ρ)} + {\tilde{C}}_{2} {∥u^{'}∥}_{L^{M} (ℝ_{+}, ρ)},$ 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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