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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 $$ \langle Af, g \rangle \lesssim \sum_{S \in \mathcal{S}} \langle f \rangle_{S, 2-\varepsilon} \langle g \rangle_{S, 2-\varepsilon} |S|. $$

          The scalar result is equivalent to the vector valued result.

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