On some Carlson-type inequalities

We find the sharp constant in the inequality $$ \|w(\cdot) x(\cdot)\|_{L_q(T)}\le K\|w_0(\cdot) x(\cdot)\|_{L_p(T)}^{\gamma}(\sum_{j=1}^d\|\varphi_j(\cdot) x(\cdot)\|_{L_r(T)}^r)^{(1-\gamma)/r}, $$ where $T$ is a cone in $\mathbb R^d$ and the weights $w(\cdot)$, $w_0(\cdot)$ and $\varphi_j(\cdot)$, $j=1,…,d$, are homogeneous measurable functions. We also obtain similar inequalities for some differential operators. Bibliography: 7 titles.