
## 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 $\{ t_{i} \}$.

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

#### 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 $$\widehat{S_j f}{\xi} = \chi( 2^j (|\xi| - 1) ) \hat{f} (\xi).$$

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

#### 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

#### 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.

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