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5. Multi-parameter theory

The domination by sparse operators has been done so far over a one parameter settings. Nothing exists for the case of two or more parameters.
    1. Sparse operators

      Problem 5.1.

      It is necessary to find the correct notion of a sparse operator in a two or more parameters setting.

      1. In particular, one can try to develop a sparse theory that gives domination for a strong maximal function.

      2. It is worth exploring an alternative version to the approach by A. Barron and J. Pipher.

      3. We can look for quantitative estimates for the strong maximal function, bi-parameter square functions and weak bounds.

        • Exponential results

              Let $H_1$ and $H_2$ be the Hilbert transform acting over the first and second parameter respectively. Similarly, define the maximal functions $\mathcal{M}_1$ and $\mathcal{M}_1$.

          Problem 5.2.

          Determine if we have the following $$ \iint_{\mathbb{T} \times \mathbb{T}} \exp \left( \frac{|H_1 H_2 f|}{\mathcal{M}_1 \mathcal{M}_2 f} \right)^{1/2} \, dx \, dy < \infty. $$

          1. In one parameter the result is known.

          2. In two parameters there are partial results in a subset of the Torus.

            • Zygmund dilations

                  Consider the operator $$ T_{\varepsilon} f = \sum_{ \substack{I,J,K \in \mathcal{D} \\ \ell(I) = \ell(J)\ell(K) } } \varepsilon_{IJK} \langle f, h_I \otimes h_{JK} \rangle h_I \otimes h_{JK}. $$

              Problem 5.3.

              Is there a notion of sparse domination for this kind of operators?
                • Representation theorem

                      The analysis of Calderón-Zygmund operators can be reduced to analyzing the Haar functions via a representation theorem in terms of dyadic shifts.

                  Problem 5.4.

                  Find a representation theorem for Calderón-Zygmund operators with Zygmund dilations that allows to produce an sparse bound by dominating the Haar shifts.
                    • $L^p$ boundedness for strong maximal function

                      Problem 5.5.

                      Considering the weighted strong maximal function $\mathcal{M}_s^{w}$. Is it possible to obtain by means of sparse domination, a sharp dependence of $\| \mathcal{M}_s^{w} \|_{L^p(w) \rightarrow L^p(w)}$ on $[w]_{A_{\infty}}$ or $[w]_{A_p}$ ?

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