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7. Non homogeneous setting

The usual sparse techniques usually rely strongly in the doubling conditions of the measures involved. New sparse theory for non-homogeneous setting has to be developed.

We consider operators $$ Tf(x) = \int_{\mathbb{R}^d} K(x,y) f(y) d \mu (y), $$ where $T$ is $L^2(\mu)$-bounded; the measure $\mu$ satisfies: $\mu(B(x,r)) \lesssim r^d$ and $\mu(B(x,r)) \lesssim \lambda(x,r) \lesssim \lambda(x, r/2)$, and the Kernel satisfies the usual Calderón-Zygmund decay and smoothness condition.
    1. $A_2$ conjecture in non-homogeneous setting

          We introduce the following variants of the $A_2$ weight condition:

      1. $\displaystyle [w]_{{A_2}^0(\mu)} = \sup_{B} \frac{w(B)}{\mu(B)} \frac{w^{-1}(B)}{\mu(B)}$.

      2. $\displaystyle [w]_{{A_2}^{\eta}(\mu)} = \sup_{B} \frac{w(B)}{\mu(B)^{1-\eta} r(B)^{n \eta }} \frac{w^{-1}(B)}{\mu(B)^{1-\eta} r(B)^{n \eta }}$.

      In both cases, the supremum is taken over balls $B$.

      Problem 7.1.

      Find a sparse domination result, to prove (one of) the following conjectures:

      1. $\displaystyle \| T \|_{L^2(w) \rightarrow L^2(w)} \lesssim [w]_{A_2^0(\mu)}$

      2. $\displaystyle \| T \|_{L^2(w) \rightarrow L^2(w)} \lesssim [w]_{A_2^{\eta}(\mu)}$

        • Control of a maximal function

              We look at the following maximal function, considered by Tolsa (2007), $$ \widetilde{\mathcal{M}} f(x) = \sup_{0 < r < R < \infty} \frac{1}{1 + \| \phi_{x,r,R} \|_{L^1(\mu)}} \int | \phi_{x,r,R} * f | \, d\mu, $$ where the function $\phi_{x,r,R}$ is given by $$ \phi_{x,n,R}(y) = \begin{cases} \frac{1}{rd}, & |x-y| < r, \\ \frac{1}{|x-y|^d}, & r \leq |x-y| \leq R, \\ 0, & |x-y|> R. \end{cases} $$ Consider also $Z_p(\mu)$ to be the set of weights $w$ such that $\widetilde{\mathcal{M}}$ is bounded on $L^p(\mu)$.

          It was proven by Tolsa (2007) that the boundedness of this maximal function on $L^p(w)$ is equivalent to the boundedness of all Calderón-Zygmund operators with respect to $\mu$, over $L^p(w)$.

          Problem 7.2.

          What is the relation between $Z_p(\mu)$ and $A_2^{\eta}(\mu)$ defined in problem 7.1.

          It was proven by Tolsa that $A_p^0(\mu) \subsetneq Z_p(\mu)$.
            • Sparse domination of $\widetilde{\mathcal{M}}$

              Problem 7.3.

              Can we find a sparse operator that controls the maximal function $\widetilde{\mathcal{M}}$ related to the Tolsa $Z_p(\mu)$ condition.
                • Representative class

                  Problem 7.4.

                  Find a representative class for Tolsa’s theorem. That is, find a subfamily of operators (for example, Riesz transforms) such that the boundedness on $L^p(w)$ of the elements of this class of operators, implies such boundedness for all Calderón-Zygmund operators.
                    • Structure theory

                      Problem 7.5.

                      Develop some structure theory for $Z_p$ weights. This includes, explore extrapolation for such class of weights.
                        • Different characterizations of $\widetilde{\mathcal{M}}$

                          Problem 7.6.

                          Find another characterization for the boundedness of $\widetilde{\mathcal{M}}$ on $L^p(w)$ similar to the work of Jawerth, in terms of an $A_p$ condition and boundedness of two weighted maximal operators.
                            • $T1$ theorem

                              Problem 7.7.

                              Prove some version of the $T1$ theorem for the maximal function $\widetilde{\mathcal{M}}$ or for the Calderón-Zygmund operators.

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