In this paper we will establish a new Pólya–Szegö type inequality for a weighted gradient of a function on \({\mathbb {R}}^2\) with respect to a weighted area. In order to do that we need to study an isoperimetric problem for the weighted area. We then apply the inequality to prove embedding theorems for weighted Sobolev spaces and to calculate the best constant in the Sobolev imbedding theorems. In our upcoming manuscript the obtained results in this note will be used to study boundary value problems for semilinear degenerate elliptic equations, see Luyen et al. (arXiv:2303.14661).
