Asymptotic constants in averaged Hölder inequalities
Let \(p>0\), \(B_p:=\{ x\in \mathbb{R}^n \mid \|x\|_p \le 1\}\) be the \(l^p\)-unit ball, \(V_n(p)\) be the volume of \(B_n(p)\), \(S_n(p) : = \{ x \in \mathbb{R}^n \mid \|x\|_p =1 \}\) and \(A_n(p)\) be the surface area of \(S_n(p)\) with respect to the regular surface measure \(\sigma_n\). Let \(\tau_{q,p} : = p^{p^{-1}} \left( \frac{ \Gamma((q+1)p^{-1})}{\Gamma(p^{-1}) }\right)^{q^{-1}}\) (\(p< \infty\)), \(\tau_{q,\infty} := (q+1)^{-q^{-1}}\) (\(q<\infty\) and \(\tau_{\infty,\infty}:= 1\). Using a Korovkin-type result for linear functions from [\textit{G. Herzog} and \textit{P. C. Kunstmann}, Am. Math. Mon. 126, No. 5, 449--454 (2019; Zbl 1428.26062)], the integration of monomials over \(B_n(p)\) and the asymptotic relation of surface measure and cone measure on \(S_n(p)\) from [\textit{A. Naor} and \textit{D. Romik}, Ann. Inst. Henri Poincaré, Probab. Stat. 39, No. 2, 241--261 (2003; Zbl 1012.60025)], the authors prove the following: \begin{itemize} \item[(1)] Let \(0<q\le p\le \infty\). If \(f: [0,1] \to \mathbb{R}\) is a bounded Borel measurable function, which is continuous at \(\tau_{q,p}\), then \[ \frac{1}{V_n(p)} \int_{B_n(p)} f\left((n^{p^{-1}-q^{-1}})\|x\|_q ) dx \to f(\tau_{q,p}\right) ~~ (n\to \infty). \] \item[(2)] Let \(0< p\le \infty\). If \(f: [0,1] \to \mathbb{R}\) is a bounded Borel measurable function, which is continuous at \(\tau_{0,p}\), then \[ \frac{1}{V_n(p)} \int_{B_n(p)} f\left(n^{p^{-1}}(\prod_{k=1}^n |x_k|)^{n^{-1}}\right) dx \to f(\tau_{0,p}) ~~ (n\to \infty). \] \item[(3)] Let \(1\le p\le \infty\) and \(0< q \le p\). If \(f: [0,1] \to \mathbb{R}\) is a bounded Borel measurable function, which is continuous at \(\tau_{q,p}\), then \[ \frac{1}{A_n(p)} \int_{S_n(p)} f\left((n^{p^{-1}-q^{-1}})\|x\|_q ) d \sigma_n (x) \to f(\tau_{q,p}\right) ~~ (n\to \infty). \] \end{itemize} The results are useful for researchers on Hölder inequalities.
scientific article; zbMATH DE number 7229056