7. Non homogeneous setting
The usual sparse techniques usually rely strongly in the doubling conditions of the measures involved. New sparse theory for nonhomogeneous 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 CalderónZygmund decay and smoothness condition.

$A_2$ conjecture in nonhomogeneous setting
We introduce the following variants of the $A_2$ weight condition: $\displaystyle [w]_{{A_2}^0(\mu)} = \sup_{B} \frac{w(B)}{\mu(B)} \frac{w^{1}(B)}{\mu(B)}$.
 $\displaystyle [w]_{{A_2}^{\eta}(\mu)} = \sup_{B} \frac{w(B)}{\mu(B)^{1\eta} r(B)^{n \eta }} \frac{w^{1}(B)}{\mu(B)^{1\eta} r(B)^{n \eta }}$.
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: $\displaystyle \ T \_{L^2(w) \rightarrow L^2(w)} \lesssim [w]_{A_2^0(\mu)}$
 $\displaystyle \ T \_{L^2(w) \rightarrow L^2(w)} \lesssim [w]_{A_2^{\eta}(\mu)}$
 $\displaystyle [w]_{{A_2}^0(\mu)} = \sup_{B} \frac{w(B)}{\mu(B)} \frac{w^{1}(B)}{\mu(B)}$.

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 $$ \phi_{x,n,R}(y) = \begin{cases} \frac{1}{rd}, & xy < r, \\ \frac{1}{xy^d}, & r \leq xy \leq R, \\ 0, & xy> R. \end{cases} $$ 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 CalderónZygmund 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 CalderónZygmund 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 CalderónZygmund operators.
Cite this as: AimPL: Sparse domination of singular integral operators, available at http://aimpl.org/sparsedomop.