
## 1. Rough singular integrals

Given $\Omega \in L^{\infty}(\mathbb{S}^{d-1})$ with mean zero. Define the rough homogeneous singular integral $T_{\Omega}$ by $$T_{\Omega} f (x) = p.v. \int_{\mathbb{R}^d} f(x-y) \frac{ \Omega(y / |y|)}{ |y|^n } \, dy.$$
1. ### $A_2$ conjecture for rough singular integrals.

#### Problem 1.1.

Prove or disprove: $$\| T_{\Omega} \|_{ L^2(w) \rightarrow L^2(w) } \leq C_d \| \Omega \|_{L^{\infty}} [w]_{A_2}.$$

1. Is it possible to get linear dependence on $[w]_{A_2}$, by allowing a worse power for $\| \Omega \|_{L^{\infty}}$?

2. Related to the previous point, we can look at an upper bound of the type $C_{d,\Omega} [w]_{A_2}$.

3. Any positive result for a $2 - \varepsilon$ power of $[w]_{A_2}$ would be an improvement.

• ### Linear bound for the Beurling operator

#### Problem 1.2.

Prove or disprove: $$\| B^m \|_{ L^2(w) \rightarrow L^2(w) } \leq C \cdot m \cdot [w]_{A_2}.$$

1. The unweighted results are consistent with this conjecture.

2. This is a consequence of Problem 1.1.

• ### $A_2$ for Commutators

#### Problem 1.3.

Consider the commutator defined by $$[b , T_{\Omega}](f) = bT_{\Omega} f - T_{\Omega}(bf).$$ Then we have $$\| [b , T_{\Omega} ] \| _{L^2(w) \rightarrow L^2(w)} \leq C_{d , b, \Omega} [w]_{A_2}^2, \quad \forall \Omega \in L^{\infty}(\mathbb{S}^{d-1}).$$

1. The bound is known with $[w]_{A_2}^3$.

2. This is also a consequence of 1.1.

• ### Boundedness of $T_{\Omega}$ on $L^p$

#### Problem 1.4.

Prove or disprove $$\| T_{\Omega} \|_{L^p(w) \rightarrow L^p(w)} \leq C_{d, \Omega} p p' [w]_{A_1}.$$

1. This would imply quadratic dependence in 1.1.

2. This is easier than 1.1, but not a consequence.

• ### Weak-type estimates

#### Problem 1.5.

For all the problems 1.1 to 1.4, it would be interesting to find a sparse approach to obtain sharp weak-type estimates.
• ### A generalized related problem

#### Problem 1.6.

Determine, by using sparse operators, if the following inequality is true for $r \in [ 1, 2 )$. $$\left( \frac{1}{|B|^2} \int_B \int_B \left| T_{\Omega}(f \mathbf{1}_{(2B)^c})(x) - T_{\Omega}(f \mathbf{1}_{(2B)^c})(y) \right|^2 \, dx \, dy \right)^{1/2} \lesssim_{\Omega} \sup{B' \supset B} \left( \frac{1}{|B'|} \int_{B'} |f|^r \right)^{1/r}.$$