Sharp Carlson Type Inequalities with Many Weights

The paper is concerned with sharp Carlson type inequalities of the form \(\|w(\cdot)x(\cdot)\|_{L_{q}(T)}\leq K\|w_{0}(\cdot)x(\cdot)\|_{L_{p}(T)}^{ \gamma}\max_{1\leq j\leq n}\|w_{j}(\cdot)x(\cdot)\|_{L_{r}(T)}^{1-\gamma},\) where \(T\) is a cone in \(\mathbb{R}^{d}\) and the weight functions \(w_{j}(\cdot)\), \(j=1,\mathinner{\ldotp\ldotp\ldotp},n\), are homogeneous with some symmetry property.