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4. Matrix weights

We consider weighted inequalities with vector valued functions, where the weight is given by a matrix $W$, which is assumed to be positive semi-definite and with locally integrable entries. The weighted $L^2(W)$ space is defined by the norm $$ \| f \|_{L^2}^2 = \int_{\mathbb{R}^d} \langle W(x)f(x), f(x) \rangle_{\mathbb{C}^d} \, dx. $$
    1. Lower bound for the square function

          Let $\mathcal{D}$ be a dyadic filtration, $h_I$ represent the Haar function, and $W$ a matrix weight, the square function can be defined as $$ S^2 f(x) = \sum_{I \in \mathcal{D}} \| \langle f(x), h_I(x) \rangle \|_{\mathbb{C}^d}^2 \frac{\mathbf{1}_{I}(x)}{|I|}. $$

      Problem 4.1.

      Find a sparse domination proof for the lower bound $$ \| f \|_{L^2(W)} \lesssim [W]_{A_2}^{1/2} \| S f \|_{L^2(W)}. $$

      1. It is necessary to find a proof first for the scalar case.

      2. The inequality fails in the scalar case if the space is not homogeneous.

      3. The lower bound implies the upper bound for the square function.

      4. This opens up the question about sparse techniques to prove lower bounds.

      5. In the scalar case, the problem can be turned into an upper bound problem.

        • Matrix $A_2$ conjecture

          Problem 4.2.

          Prove or disprove: $$ \| T \|_{L^2(W) \rightarrow L^2(W)} \lesssim [W]_{A_2}. $$

          1. Find a notion of sparse that incorporates the weight.

          2. Start by analyzing particular instances of $T$, like Haar shifts and Hilbert transforms before moving to general CalderoĢn-Zygmund operators.

            • Extrapolation results

              Problem 4.3.

              Is it possible to develop some extrapolation theory for matrix weights?

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