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Asymptotic constants in averaged Hölder inequalities. (English) Zbl 1448.26030

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 [G. Herzog and 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 [A. Naor and D. Romik, Ann. Inst. Henri Poincaré, Probab. Stat. 39, No. 2, 241–261 (2003; Zbl 1012.60025)], the authors prove the following:
(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 \[ \frac{1}{V_n(p)} \int_{B_n(p)} f\left((n^{p^{-1}-q^{-1}})\|x\|_q ) dx \to f(\tau_{q,p}\right) ~~ (n\to \infty). \]
(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 \[ \frac{1}{V_n(p)} \int_{B_n(p)} f\left(n^{p^{-1}}(\prod_{k=1}^n |x_k|)^{n^{-1}}\right) dx \to f(\tau_{0,p}) ~~ (n\to \infty). \]
(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 \[ \frac{1}{A_n(p)} \int_{S_n(p)} f\left((n^{p^{-1}-q^{-1}})\|x\|_q ) d \sigma_n (x) \to f(\tau_{q,p}\right) ~~ (n\to \infty). \]

The results are useful for researchers on Hölder inequalities.

MSC:

26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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Full Text: Euclid

References:

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