Carlson-type inequalities with many weights and optimal recovery

We find a sharp constant in the inequality $$ \|\psi(\cdot) x(\cdot)\|_{L_q(T,\mu)}\le C \max_{1\le j\le l}\|\varphi_j(\cdot) x(\cdot)\|_{L_p(T,\mu)}^\gamma \max_{l+1\le j\le n}\|\varphi_j(\cdot) x(\cdot)\|_{L_r(T,\mu)}^{1-\gamma}, $$ where $T$ is a cone in $\mathbb R^d$, and the weights $\psi( {\cdot} )$ and $\varphi_j( {\cdot} )$, $j=1,…,n$, are homogeneous measurable functions satisfying certain additional symmetry conditions. The sharp inequality is a consequence of a more general problem of optimal recovery in weighted $ L_q(T,\mu)$-spaces from incompletely given functions, whose solution is also presented. The results obtained are used for the optimal recovery of powers of the generalized Laplace operator and for the derivation of relevant sharp inequalities. Bibliography: 11 titles.