
## 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 $0 < \alpha \leq d$.
1. ### 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.