## 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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