Sinnamon, Gordon A weighted gradient inequality. (English) Zbl 0686.26004 Proc. R. Soc. Edinb., Sect. A 111, No. 3-4, 329-335 (1989). Hardy’s averaging operator and its dual given by \((P_ 1f)(x)=\frac{1}{x}\int^{x}_{0}f(t)dt\) and \((Q_ 1f)(x)=\int^{\infty}_{x}f(t)\frac{dt}{t}\) satisfy differential equations whose natural analogues for operators on functions on \(R^ n\) are \(x.\nabla (P_ nf)(x)+(P_ nf)(x)=f(x)\) and \(x.\nabla (Q_ nf)(x)+f(x)=0,\) respectively. The solutions \((P_ nf)(x)=\int^{1}_{0}f(\lambda x)d\lambda\) and \((Q_ nf)(x)=\int^{\infty}_{1}f(\lambda x)\frac{d\lambda}{\lambda}\) are not dual operators for \(n>1,\) but it is shown that weighted norm inequalities for \(Q_ n\) yield other ones for \(P_ n\). The author then concentrates on the inequality \[ (1)\quad (\int_{R^ n}| (Q_ nf)(x)|^ qv(x)dx)^{1/q}\leq c(\int_{R^ n}| f(x)|^ pu(x)dx)^{1/p}, \] where v, u are weights, i.e. non-negative measurable functions on \(R^ n\), and \(1\leq p<\infty,\quad 0<q<\infty.\) The change to polar coordinates enables to utilise the one-dimensional Hardy inequalities to obtain conditions equivalent to (1) in the cases \(1<p=q<\infty\) or \(0<q<p<\infty\) with \(p>1\) or \(1\leq p<q<\infty\) (particularly, in the last case, if \(n>1\) and u is locally integrable, then (1) holds for every f if and only if \(v=0).\) As a consequence, conditions are derived for the inequality \[ (\int_{R^ n}| g(x)|^ qv(x)dx)^{1/q}\leq c(\int_{R^ n}| x.\nabla g(x)|^ pu(x)dx)^{1/p} \] to hold for every \(g\in C_ 0^{\infty}(R^ n)\). Reviewer: J.Rákosník Cited in 2 ReviewsCited in 12 Documents MSC: 26D10 Inequalities involving derivatives and differential and integral operators Keywords:gradient inequality; n-dimensional analogue; Hardy’s averaging operator; weighted norm inequalities PDFBibTeX XMLCite \textit{G. Sinnamon}, Proc. R. Soc. Edinb., Sect. A, Math. 111, No. 3--4, 329--335 (1989; Zbl 0686.26004) Full Text: DOI References: [1] Muckenhoupt, Studia Math. 44 pp 31– (1972) [2] DOI: 10.4153/CMB-1978-071-7 · Zbl 0402.26006 · doi:10.4153/CMB-1978-071-7 [3] Maz’ja, Sobolev Spaces (1985) · doi:10.1007/978-3-662-09922-3 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.