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- W2803864694 abstract "Our goal in this paper is to find a characterization of $n$-dimensional bilinear Hardy inequalities begin{align*} bigg| ,int_{B(0,cdot)} f cdot int_{B(0,cdot)} g ,bigg|_{q,u,(0,infty)} & leq C , |f|_{p_1,v_1,{mathbb R}^n} , |g|_{p_2,v_2,{mathbb R}^n}, quad f,,g in {mathfrak M}^+ ({mathbb R}^n), end{align*} and begin{align*} bigg| ,int_{,^{^{mathsf{c}}}! B(0,cdot)} f cdot int_{,^{^{mathsf{c}}}! B(0,cdot)} g ,bigg|_{q,u,(0,infty)} &leq C , |f|_{p_1,v_1,{mathbb R}^n} , |g|_{p_2,v_2,{mathbb R}^n}, quad f,,g in {mathfrak M}^+ ({mathbb R}^n), end{align*} when $0 < q le infty$, $1 le p_1,,p_2 le infty$ and $u$ and $v_1,,v_2$ are weight functions on $(0,infty)$ and ${mathbb R}^n$, respectively. Since the solution of the first inequality can be obtained from the characterization of the second one by usual change of variables we concentrate our attention on characterization of the latter. The characterization of this inequality is easily obtained for the range of parameters when $p_1 le q$ using the characterizations of multidimensional weighted Hardy-type inequalites while in the case when $q < p_1$ the problem is reduced to the solution of multidimensional weighted iterated Hardy-type inequality. To achieve the goal, we characterize the validity of multidimensional weighted iterated Hardy-type inequality $$ left|left|int_{,^{^{mathsf{c}}}! B(0,cdot)}h(z)dzright|_{p,u,(0,t)}right|_{q,mu,(0,infty)}leq c |h|_{theta,v,(0,infty)},~ h in mathfrak{M}^+({mathbb R}^n) $$ where $0 < p,,q < +infty$, $1 leq theta le infty$, $uin {mathcal W}(0,infty)$, $v in {mathcal W}({mathbb R}^n)$ and $mu$ is a non-negative Borel measure on $(0,infty)$." @default.
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- W2803864694 date "2018-05-17" @default.
- W2803864694 modified "2023-10-14" @default.
- W2803864694 title "Multidimensional bilinear Hardy inequalities" @default.
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