
## 2. Discrete operators

We consider operators in a discrete setting, that is, defined for the integers. There are some preliminary results of sparse domination for discrete operators, but many questions remain open.
1. ### Vector valued bounds for maximal operator

Consider the operator $$Af(m) = \sup_{N} \left| \frac{1}{N} \sum_{n = 1}^N f(m + n^2) \right|$$

#### Problem 2.1.

Is the operator $A$ bounded on $\ell^p (\ell^r)$ for $1 < p, r < \infty$?
•     Consider the operator $$Af(m) = \sup_{N} \left| \frac{1}{N} \sum_{n = 1}^N f(m + n^2) \right|$$

#### Problem 2.2.

Find a sparse control over the operator $A$, that is, find $s \geq 1$, $0 < \varepsilon \leq 1$ and a sparse colection $\mathcal{S}$ such that $$\langle Af, g \rangle \lesssim \sum_{S \in \mathcal{S}} \langle f \rangle_{S, 2-\varepsilon} \langle g \rangle_{S, 2-\varepsilon} |S|.$$