6. Fractional integrals
The fractional singular integrals are defined as convolution operators of the form $$ I_{\alpha} f(x) = \int_{\mathbb{R}^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy, $$ where $0 < \alpha \leq d$.-
Separated bump conjecture
Problem 6.1.
Determine if the conditions $$ \begin{cases} |Q|^{\alpha /d} \langle u \rangle_Q^{1/p} \| \sigma^{1/p'} \|_{\Phi , Q} \leq C, & \Phi \in B_p, \\ |Q|^{\alpha / d} \| u^{1/p} \|_{\Psi , Q} \langle \sigma \rangle_Q^{1/p'} \leq C, & \Psi \in B_{p'}, \end{cases} $$ imply that $I_{\alpha}$ is bounded from $L^p(\sigma)$ to $L^p(u)$.
Cite this as: AimPL: Sparse domination of singular integral operators, available at http://aimpl.org/sparsedomop.