Weak estimates for commutators of fractional integral operators
The authors consider the commutators \([b,I_\alpha]f(x) = b(x)I_\alpha f(x)-I_\alpha(bf)(x)\) generated by the fractional integral operator \(I_\alpha\) defined by \[ I_\alpha f(x)\int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n-\alpha}} dy\quad (0<\alpha<n), \] and a BMO function \(b(\mathbb{R}^n)\). It is proved that for \(q = n/(n-\alpha)\), \([b,I_\alpha]\) satisfies the following weak type \(L\log L\) estimates: Let \(b\in \text{BMO}(\mathbb{R}^n)\), \(0 <\alpha < n\), \(\Phi(t) = t(1+\log^+t)\). Then there exists a positive constant \(C\) such that for any \(\lambda > 0\) \[ |\{x\in\mathbb{R}^n:|[b,I_\alpha]f(x)|>\lambda\}|^{1/q}\leq C\Phi(\Phi(\|b\|_*))\|\phi(|f(\cdot)|/\lambda)\|_1\{1+\tfrac{\alpha}{n} \log^+\|\Phi(|f(\cdot)|/\lambda\|_1\}; \] where \(\|b\|_*\) is the BMO norm of \(b\). In addition, for the higher order commutators the weak type \(L \log L\) estimates are shown by using the technique of sharp functions.
scientific article; zbMATH DE number 1782133
scientific article