Abstract:
We find the sharp constant in the inequality
$$
\|w(\cdot) x(\cdot)\|_{L_q(T)}\le K\|w_0(\cdot) x(\cdot)\|_{L_p(T)}^{\gamma}\biggl(\sum_{j=1}^d\|\varphi_j(\cdot) x(\cdot)\|_{L_r(T)}^r\biggr)^{(1-\gamma)/r},
$$
where $T$ is a cone in $\mathbb R^d$ and the weights $w(\cdot)$, $w_0(\cdot)$ and $\varphi_j(\cdot)$, $j=1,\dots,d$, are homogeneous measurable functions. We also obtain similar inequalities for some differential operators.
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