Parabolic methods for ultraspherical interpolation inequalities (Q6160090)

Let \(\mathbb S^d\) be the \(d\)-sphere, \(n\) be a strictly positive parameter, and \(d\mu_n=Z^{-1}\rho^{n-d}d\mu\) be a suitable probability measure such that \(\rho(x)^2=1-(x\cdot\mathbf{e})^2\) with \(\mathbf{e}\in\mathbb S^d\), \(d\mu\) is the uniform probability measure on \(\mathbb S^d\), \(Z_n=\frac{\sqrt{\pi}\Gamma\left(\frac{n}{2}\right)}{\Gamma\left( \frac{n+1}{2}\right)},\) and \(H^1(\mathbb S^d,d\mu_n)\) be the classical Sobolev space w.r.t. \(d\mu_n\). The purpose of the article is to provide a parabolic approach of the following weighted interpolation inequality for \(u\in H^1(\mathbb S^d,d\mu_n).\) \[ \int_{\mathbb S^d}|\nabla u|^2d\mu_n \gtrsim_{n,p} \left(\left[\int_{\mathbb S^d}|u|^pd\mu_n\right]^{2/p}-\int_{\mathbb S^d}|u|^2d\mu_n\right). \]