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3. Spherical averages and Radon transforms

This section includes some problems with a more geometric flavor, such as averaging operators over spheres, and special types of radial multipliers.

    1. Variation of spherical averages

          Consider the operator $$ V f(x) =\sup \left( \sum_{i} \left| \int_{\mathbb{S}^1} f(x + t_{i} \sigma ) \, d \sigma - \int_{\mathbb{S}^1} f(x + t_{i-1} \sigma ) \, d \sigma \right|^r \right)^{1/r}. $$ Where the supremum is taken over increasing sequences $\{ t_{i} \}$.

      Add remark

      Problem 3.1.

      Find $\varepsilon > 0$ and a sparse colection $\mathcal{S}$ such that $$ \langle Vf , g \rangle \lesssim \sum_{S \in \mathcal{S}} \langle f \rangle_{S, 2+\varepsilon} \langle g \rangle_{S, \frac{4}{3}+\varepsilon} |S|. $$

        • Generalized spherical averages

              Consider the operator $$ A_{t} f(x) = \int_{B(0,1)} (1 - |y|^2)^{\beta} f (x - ty) \, dy $$

          Add remark

          Problem 3.2.

          Find $r,s$ and a sparse collection $\mathcal{S}$ such that $$ \langle \sup_{t \geq 0} A_t f , g \rangle \lesssim \sum_{S \in \mathcal{S}} \langle f \rangle_{r,S} \langle g \rangle_{r,S} |S|. $$

            • Bochner-Riesz operator

                  Let $K_j $ be the Kakeya type maximal function defined by $$ K_j f(x) = \sup{R: ec(R) \geq 2^{-j}} \langle f \rangle_R \mathbf{1}_{R}(x). $$ Here, $ec(R)$ represents the eccentricity of the rectangle $R$, that is, the ration between the lengths of the longer and shorter side. Let $\chi$ be a smooth bump in a neighborhood of the origin, then define $$ \widehat{S_j f}{\xi} = \chi( 2^j (|\xi| - 1) ) \hat{f} (\xi). $$

              Add remark

              Problem 3.3.

              Prove the existence of an $\varepsilon > 0$ such that the following weighted inequality holds $$ \int |S_j f (x)|^2 w(x) \, dx \leq \int |f(x)|^2 \left( K_{j/2} |w(x)|^{2-\varepsilon} \right)^{\frac{1}{2- \varepsilon}} \, dx $$

                • End point sparse bounds

                      Consider the spherical average $$ A_t f(x) = \int_{\mathbb{S}^{d-1}} f(x - ty) \, d \sigma (y). $$

                  Add remark

                  Problem 3.4.

                  Find domination by sparse operators at the end points for the lacunary spherical maximal average $$ \mathcal{M}_{lac} f = \left| \sup_{k} A_{2^k} f \right|. $$

                    • Sparse adapted to the geometry


                      Add remark

                      Problem 3.5.

                      Find sparse bounds for the lacunary spherical maximal average, using sparse forms given by sets better adapted to the geometry of the problem.
                          Comments:

                      It was suggested to look at annular regions or ellipsoids on a sphere.

                        • Radial weights


                          Add remark

                          Problem 3.6.

                          Find sparse bounds adapted to estimates with respect to radial weights and/or radial functions.
                              Comments:

                          For example, Bochner-Riesz, and other Riesz means.

                            • Bilinear spherical maximal operator


                              Add remark

                              Problem 3.7.

                              Find a correct way to define a sparse bound for the bilinear spherical maximal averaging operator.

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