Weighted Young inequalities for convolutions
Let \(1 < p, q < \infty\) and let \(u\) and \(v\) be weighted functions on \(\mathbb R^n\). The aim of the paper is to find sufficient conditions for the validity of the inequality \[ \Bigl(\int_{\mathbb R^n} (g \times f)^q (x)\, u (x) \, dx\Bigr)^{1/q} \leq C \| g\| _X \Bigl(\int_{\mathbb R^n} f (x)^p \, v (x) \, dx\Bigr)^{1/p} \] for all measurable and non-negative functions \(f, g\) on \(\mathbb R^n\) with \(g\) radial and satisfying a convenient growth condition. Here \(X\) is a Banach function space and \(C\) is a~ positive constant.
scientific article; zbMATH DE number 1884054