Sobolev's inequality for Riesz potentials of functions in non-doubling Morrey spaces
Let \(G\) be an open subset of a separable metric space \(X\), and let \(\mu\) be a diffuse nonnegative Radon measure on \(X\). The Riesz potential of order \(\alpha\) for a nonnegative measurable function \(f\) on \(G\) is defined by \[ U_\alpha f(x)= \int_G \frac{|x-y|^\alpha f(y)}{\mu(B(x,4|x-y|))} d\mu(y), \] where \(|x-y|\) is the distance of \(x\) and \(y\) and \(B(x,r)\) is the open ball centered at \(x\) of radius \(r>0\). The authors prove Sobolev's inequality and Trudinger exponential integrability for Riesz potentials of functions \(f\) in non-doubling Morey spaces, i.e., \[ \sup\limits_{x\in G, 0<r<d_G} \frac{r^\nu}{\mu(B(x,kr))} \int_{G\cap B(x,r)} |f(y)|^p d\mu(y) \;< \infty, \] where \(1<p<\infty, \nu>0,k>1\), and \(d_G\) denotes the diameter of \(G\).
scientific article; zbMATH DE number 5543719