Motivated and inspired by the improved Hardy inequalities studied in their well-known works by Brezis and Vázquez (Rev Mat Univ Complut Madrid 10:443–469, 1997) and Brezis and Marcus (Ann Scuola Norm Sup Pisa Cl Sci 25(1–2):217–237, 1997), we establish in this paper several identities that imply many sharpened forms of the Hardy type inequalities on upper half spaces xN>0\documentclass[12pt]{minimal}
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\begin{document}$$\left\{ x_{N}>0\right\} $$\end{document}. We set up these results for the distance to the origin, the distance to the boundary of any strip RN-1×0,R\documentclass[12pt]{minimal}
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\begin{document}$$ \mathbb {R} ^{N-1}\times \left( 0,R\right) $$\end{document} and the distance to the hyperplane xN=0\documentclass[12pt]{minimal}
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\begin{document}$$\left\{ x_{N}=0\right\} $$\end{document}, using both the usual full gradient and radial derivative (in the case of distance to the origin) or only the partial derivative ∂u∂xN\documentclass[12pt]{minimal}
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\begin{document}$$\frac{\partial u}{\partial x_{N}}$$\end{document} (in the case of distance to the boundary of the strip or hyperplane). One of the applications of our main results is that when Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is the strip RN-1×0,2R\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^{N-1}\times \left( 0,2R\right) $$\end{document}, the bound λΩ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \left( \Omega \right) $$\end{document} given by Brezis and Marcus in Brezis and Marcus (1997) can be improved to z02R2\documentclass[12pt]{minimal}
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\begin{document}$$\frac{z_{0}^{2}}{R^{2}}$$\end{document}, where z0=2.4048…\documentclass[12pt]{minimal}
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\begin{document}$$z_{0} =2.4048 \ldots $$\end{document} is the first zero of the Bessel function J0z\documentclass[12pt]{minimal}
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\begin{document}$$J_{0}\left( z\right) $$\end{document}. Our approach makes use of the notion of Bessel pairs introduced by Ghoussoub and Moradifam (Math Ann 349(1):1–57, 2011) and (Functional inequalities: new perspectives and new applications. Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2013) and the method of factorizations of differential operators. In particular, our identities and inequalities offer sharpened and more precise estimates of the second remainder term in the existing Hardy type inequalities on upper half spaces in the literature, including the Hardy-Sobolev-Maz’ya type inequalities.
