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A weighted gradient inequality. (English) Zbl 0686.26004

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

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
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References:

[1] Muckenhoupt, Studia Math. 44 pp 31– (1972)
[2] DOI: 10.4153/CMB-1978-071-7 · Zbl 0402.26006
[3] Maz’ja, Sobolev Spaces (1985)
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