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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.


      Add remark

      Problem 1.1.

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

      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


          Add remark

          Problem 1.2.

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

          1. The unweighted results are consistent with this conjecture.

          2. This is a consequence of Problem 1.1.

            • $A_2$ for Commutators


              Add remark

              Problem 1.3.

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

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

              2. This is also a consequence of 1.1.

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


                  Add remark

                  Problem 1.4.

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

                  1. This would imply quadratic dependence in 1.1.

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

                    • Weak-type estimates


                      Add remark

                      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


                          Add remark

                          Problem 1.6.

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

                          Understanding this problem could give an idea on how to solve 1.1.

                              Cite this as: AimPL: Sparse domination of singular integral operators, available at http://aimpl.org/sparsedomop.