Multivariate Hardy-type inequalities (Q2724920)

Let \(f(x) \geq 0\) for \(0 < x < \infty\) and write \(G(x) = \frac 1{x}\int^x_0 f(t) dt.\) If \(p>1\), Hardy's inequality reads NEWLINE\[NEWLINE\int^{\infty}_0 G^{p}(x) dx < \Big(\frac{p}{p-1}\Big)^p \int^{\infty} _0 f^{p}(x) dx.NEWLINE\]NEWLINE In this paper, the authors consider the multivariate inequalities of Hardy-type for the continuous and the discrete cases. They show that several results of the continuous case may be derived directly from the known univariate results. The results obtained extend many classical ones, in particular, those of \textit{K.-C. Lee} and \textit{G.-S. Yang} [Tamkang J. Math. 17, No. 4, 109-119 (1986; Zbl 0626.26009)], \textit{G.-S. Yang} and \textit{S.-P. Jean} [Tamkang J. Math. 24, No. 3, 341-354 (1993; Zbl 0801.26008)] and \textit{B. G. Pachpatte} [Anal. Ştiinţ. Univ. Al. I. Cuza Iaşi, Mat. 38, No. 3, 355-361 (1992; Zbl 0813.26007)]. The discrete multivariate Hardy-type inequalities are considered and a simpler proof of the main result of \textit{D.-Y. Hwang} [Tamkang J. Math. 27, No. 2, 125-132 (1996; Zbl 0874.26014)] is given.