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تسجيل مستخدم جديدA Variation on Holder-Brascamp-Lieb Inequalities
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Kevin ONeill
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The Holder-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Holder-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlying linear maps match those of the convolution inequality of Young.
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define the associated Tauberian functions $mathsf{C}_{mathsf{HL},w}(alpha)$ and $mathsf{C}_{mathsf{S},w}(alpha)$ on $(0,1)$ by [ mathsf{C}_{mathsf{HL},w}(alpha) :=sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M chi_E(x) > alpha}) ] and [ mathsf{C}_{mathsf{S},w}(alpha) := sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M _{mathsf S}chi_E(x) > alpha}). ] Utilizing weighted Solyanik estimates for $mathsf M$ and $mathsf M_{mathsf S}$, we show that the function $mathsf{C}_{mathsf{HL},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}})^{-1}}(0,1)$ and $mathsf{C}_{mathsf{S},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}^ast})^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.
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