On the framework of L summations for functions
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Cited by 2 publications
References 53 publications
On Multiple $$L_p$$-Curvilinear-Brunn–Minkowski Inequalities
We construct the extension of the curvilinear summation for bounded Borel measurable sets to the $$L_p$$ L p space for multiple power parameter $$\bar{\alpha }=(\alpha _1, \ldots , \alpha _{n+1})$$ α ¯ = ( α 1 , … , α n + 1 ) when $$p>0$$ p > 0 . Based on this $$L_{p,\bar{\alpha }}$$ L p , α ¯ -curvilinear summation for sets and the concept of compression of sets, the $$L_{p,\bar{\alpha }}$$ L p , α ¯ -curvilinear-Brunn–Minkowski inequality for bounded Borel measurable sets and its normalized version are established. Furthermore, by utilizing the hypo-graphs for functions, we enact a brand new proof of $$L_{p,\bar{\alpha }}$$ L p , α ¯ Borell–Brascamp–Lieb inequality, as well as its normalized version, for functions containing the special case of the $$L_{p}$$ L p Borell–Brascamp–Lieb inequality through the $$L_{p,\bar{\alpha }}$$ L p , α ¯ -curvilinear-Brunn–Minkowski inequality for sets. Moreover, we propose the multiple power $$L_{p,\bar{\alpha }}$$ L p , α ¯ -supremal-convolution for two functions together with its properties. Last but not least, we introduce the definition of the surface area originated from the variation formula of measure in terms of the $$L_{p,\bar{\alpha }}$$ L p , α ¯ -curvilinear summation for sets as well as $$L_{p,\bar{\alpha }}$$ L p , α ¯ -supremal-convolution for functions together with their corresponding Minkowski type inequalities and isoperimetric inequalities for $$p\ge 1,$$ p ≥ 1 , etc.
On Multiple $$L_p$$-Curvilinear-Brunn–Minkowski Inequalities
We construct the extension of the curvilinear summation for bounded Borel measurable sets to the $$L_p$$ L p space for multiple power parameter $$\bar{\alpha }=(\alpha _1, \ldots , \alpha _{n+1})$$ α ¯ = ( α 1 , … , α n + 1 ) when $$p>0$$ p > 0 . Based on this $$L_{p,\bar{\alpha }}$$ L p , α ¯ -curvilinear summation for sets and the concept of compression of sets, the $$L_{p,\bar{\alpha }}$$ L p , α ¯ -curvilinear-Brunn–Minkowski inequality for bounded Borel measurable sets and its normalized version are established. Furthermore, by utilizing the hypo-graphs for functions, we enact a brand new proof of $$L_{p,\bar{\alpha }}$$ L p , α ¯ Borell–Brascamp–Lieb inequality, as well as its normalized version, for functions containing the special case of the $$L_{p}$$ L p Borell–Brascamp–Lieb inequality through the $$L_{p,\bar{\alpha }}$$ L p , α ¯ -curvilinear-Brunn–Minkowski inequality for sets. Moreover, we propose the multiple power $$L_{p,\bar{\alpha }}$$ L p , α ¯ -supremal-convolution for two functions together with its properties. Last but not least, we introduce the definition of the surface area originated from the variation formula of measure in terms of the $$L_{p,\bar{\alpha }}$$ L p , α ¯ -curvilinear summation for sets as well as $$L_{p,\bar{\alpha }}$$ L p , α ¯ -supremal-convolution for functions together with their corresponding Minkowski type inequalities and isoperimetric inequalities for $$p\ge 1,$$ p ≥ 1 , etc.
The Dual Hamilton–Jacobi Equation and the Poincaré Inequality
Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity shown by L. Gross, and applying the ideas and methods of the work by Bobkov, Gentil and Ledoux, we would like to establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations. In addition, Poincaré inequality is also recovered by the dual Hamilton–Jacobi equations.
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