Boundedness of Riesz potentials in nonhomogeneous spaces
The authors consider the operator defined by \[ T_\alpha f(x) = \int_{\mathbb R^d} k_\alpha(x, y)f (y)\,d\mu(y), \] where the measure \(\mu\) satisfies the growth condition, \(0 < n \leq d,\) \[ \mu(B(x, r)) \leq Cr^n \text{ for all } x \in\mathbb R^d \text{ and }r > 0, \] and the kernel \(k\alpha \in L^1_{loc}(\mathbb R^d\times \mathbb R^d\backslash\{x = y\})\) satisfies, for \(0 <\alpha < n,\) (i) \(|k_\alpha(x, y)| \leq C|x - y|^{-n+\alpha}\) for \(x \neq y,\) (ii) \( |k_\alpha(x, y)- k_\alpha(x', y)| + |k_\alpha(y, x) - k_\alpha(y, x')| \leq C\frac{|x-x'|^\delta}{|x - y|^{n-\alpha+\delta}}\) for some \(\delta\in (0, 1]\) and for all \(|x - y| \geq 2|x - x'|\) with \(x\neq y.\) They show that the following statements are equivalent: (1) \(T_\alpha\) is bounded from \(L^p(\mu)\) into \(L^q(\mu)\) for \(1 < p < n/\alpha\) and \(1/q = 1/p -\alpha/n\); (2) \(T_\alpha\) is bounded from \(L^1(\mu)\) into \(L^{n/(n-\alpha),\infty}(\mu)\); (3) \(\|T_\alpha f\|_{RBMO(\mu)}\leq C\|f\|_{L^{n/\alpha}(\mu)}\) for all \(f \in L^{n/\alpha}(\mu)\) with \(T_\alpha f\) being finite almost everywhere; (4) \(T_\alpha\) is bounded from \(H^1(\mu)\) into \(L^{n/(n-\alpha)}(\mu)\); (5) \(T_\alpha\) is bounded from \(H^1(\mu)\) into \(L^{n/(n-\alpha),\infty}(\mu)\).
scientific article; zbMATH DE number 5524048