Inequalities for fractional integral operators on different metric measure spaces.
The author extends the classical Hardy--Littlewood--Sobolev Inequality and the Hedberg Inequality for two types of fractional integral operators defined on metric measure spaces. More precisely, the following is shown. Let \((X, \rho, \mu)\) be a homogeneous measurable metric space and let \(p,q,\alpha\in \mathbb{R}\) be such that \(0 < \alpha < 1\), \(1\leq p < 1/\alpha\), and \(1/q = 1/p - \alpha\). Then the fractional integral operator \(T_\alpha\) defined by \[ T_\alpha f := \int_X V(\cdot, y)^{\alpha - 1} f(y) d\mu(y) \] maps \(L^p (X)\) into \(L^q (X)\) for \(p > 1\) and \(L^1(X)\) into \(L^{q,\infty}(X)\) for \(p = 1\). Let \((X, \rho, \mu)\) be a Vitali measurable metric space such that for all \(x\in X\), the volume \(V(x,r)\) of its balls \(B(x,r)\) has polynomial growth in \(r\) in the sense that \[ \forall\,r < 1:\,V(x, r) \leq c r^d\quad\text{and}\quad \forall\,r \geq 1:\,V(x, r) \leq c r^D, \] for some positive constant \(c\), where \(a,n\in \mathbb{R}\) and \(d\leq n\leq D\). Suppose that \(k_a:X\times X\to \mathbb{R}\) is a measurable function such that there exists a positive constant \(c\) so that \[ \text{\(\forall x,y\in X\) with \(\rho(x,y) < 1\)}: \;k_a (x,y) \leq c \rho(x,y)^{a - d} \] and \[ \text{\(\forall x,y\in X\) with \(\rho(x,y) \geq 1\)}: k_a (x,y) \leq c \rho(x,y)^{a - D}. \] Let \(I_a\) be the fractional integral operator whose associated kernel is \(k_a\). Then \(I_a\) maps \(L^p (X)\) into \(L^q(X)\) for \(p > 1\) and \(L^1(X)\) into \(L^{q,\infty}(X)\) for \(p = 1\). Here \(p\) and \(q\) satisfy \(1\leq p < n/a\) and \(1/q = 1/p - a/n\).
scientific article; zbMATH DE number 5989455
scientific article