Sharp Carlson Type Inequalities with Many Weights

The paper is concerned with sharp Carlson type inequalities of the form
$$\|w(\cdot) x(\cdot)\|_{L_q(T)}\le K\|w_0(\cdot) x(\cdot)\|_{L_p(T)}^{\gamma}\max_{1\le j\le n}\|w_j(\cdot) x(\cdot)\|_{L_r(T)}^{1-\gamma},$$
where $T$ is a cone in $\mathbb R^d$ and the weight functions $w_j(\cdot)$, $j=1,\ldots,n$, are homogeneous with some symmetry property.