Maximal subspace averages

We study maximal operators associated to singular averages along finite subsets $\Sigma$ of the Grassmannian $\mathrm{Gr}(d,n)$ of $d$-dimensional subspaces of $\mathbb R^n$. The well studied $d=1$ case corresponds to the the directional maximal function with respect to arbitrary finite subsets of $\mathrm{Gr}(1,n)=\mathbb S^{n-1}$. We provide a systematic study of all cases $1\leq d<n$ and prove essentially sharp $L^2(\mathbb R^n)$ bounds for the maximal subspace averaging operator in terms of the cardinality of $\Sigma$, with no assumption on the structure of $\Sigma$. In the codimension $1$ case, that is $n=d+1$, we prove the precise critical weak $(2,2)$-bound. Drawing on the analogy between maximal subspace averages and $(d,n)$-Nikodym maximal averages, we also formulate the appropriate maximal Nikodym conjecture for general $1<d<n$ by providing examples that determine the critical $L^p$-space for the $(d,n)$-Nikodym problem. Unlike the $d=1$ case, the maximal Kakeya and Nikodym problems are shown not to be equivalent when $d>1$. In this context, we prove the best possible $L^2(\mathbb R^n)$-bound for the $(d,n)$-Nikodym maximal function for all combinations of dimension and codimension. Our estimates rely on Fourier analytic almost orthogonality principles, combined with polynomial partitioning, but we also use spatial analysis based on the precise calculation of intersections of $d$-dimensional plates in $\mathbb R^n$.Comment: 40 pages, 1 figure, submitted for publicatio

Linear and multilinear spherical maximal functions

In dimensions n [greater than or equal to] 2 we obtain Lp1(Rn) x ... x Lpm(Rn) to Lp(Rn) boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results. Moreover, we obtain weak type and Lorentz space estimates as well as counterexamples in the endpoint cases. We also study a family of maximal operators that provides a continuous link connecting the Hardy-Littlewood maximal function to the spherical maximal function. Our theorems are proved in the multilinear setting but may contain new results even in the linear case. For this family of operators we obtain bounds between Lebesgue spaces in the optimal range of exponents. Moreover, we provide multidimensional versions of the Kakeya, Nikodym, and Besicovitch constructions associated with a fixed rectifiable set. These yield counterexamples indicating that maximal operators given by translations of spherical averages are unbounded on all Lp(Rn) for p [less than] [infinity]. For lower-dimensional sets of translations, we obtain Lp boundedness for the associated maximally translated spherical averages and for the uncentered spherical maximal functions for a certain range of p that depends on the upper Minkowski dimension of the set of translations. This implies that the Nikodym sets associated with spheres have full Hausdorff dimension.Includes bibliographical references (pages 91-96)

The polynomial method over varieties

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2019, Director: Martín Sombra[en] In 2010, Guth and Katz introduced the polynomial partitioning theorem as a tool in incidence geometry and in additive combinatorics. This allowed the application of results from algebraic geometry (mainly on intersection theory and on the topology of real algebraic varieties) to the solution of long standing problems, including the celebrated Erdős distinct distances problem. Recently, Walsh has extended the polynomial partitioning method to an arbitrary subvariety. This result opens the way to the application of this method to control the point-hypersurface incidences and, more generally, of variety-variety incidences, in spaces of arbitrary dimension. This final project consists in studying Walsh’s paper, to explain its contents and explore its applications to t his kind of incidence problems