Related papers: $L^p$ improving bounds for averages along curves

A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which…

A method is described for predicting extremes values beyond the span of historical data. The method - based on extending a curve fitted to a location- and scale-invariant variation of the double-logarithmic QQ-plot - is simple and…

We prove sharp $\ell^{p}L^{p}$ decoupling inequalities for $2$ quadratic forms in $4$ variables. We also recover several previous results (arXiv:1409.1634, arXiv:1501.07224, arXiv:1609.02022, arXiv:1609.04107) in a unified way.

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.

We prove sharp L^2 boundary decay estimates for the eigenfunctions of certain second order elliptic operators acting in a bounded region, and of their first order space derivatives, using only the Hardy inequality. We then deduce bounds on…

Measuring the similarity of curves is a fundamental problem arising in many application fields. There has been considerable interest in several such measures, both in Euclidean space and in more general setting such as curves on Riemannian…

We undertake a systematic study of the mapping properties of forms based on distance graphs in $\mathbb{Z}^{d}$ to see how the structure of a graph, $G$, affects the $\ell^{p}$ improving estimates of the form, $\Lambda_{G}$, based on $G$.…

Let $ \lambda ^2 \in \mathbb N $, and in dimensions $ d\geq 5$, let $ A_{\lambda } f (x)$ denote the average of $ f \;:\; \mathbb Z ^{d} \to \mathbb R $ over the lattice points on the sphere of radius $\lambda$ centered at $x$. We prove $…

We establish sharp (H^1, L^{1,q}) and local (L \log^r L, L^{1,q}) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map H^1 to L^{1,\infty} and L…

We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random…

In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally…

We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal…

Let $X$ be a semistable curve and $L$ a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of $X$. We establish an upper bound for $h^0(X,L)$, which generalizes the…

We prove the local boundedness for solutions to a class of obstacle problems with non-standard growth conditions. The novelty here is that we are able to establish the local boundedness under a sharp bound on the gap between the growth…

We establish some new $L^p$-improving bounds for the $k$-simplex averaging operators $S^k$ that hold in dimensions $d \geq k$. As a consequence of these $L^p$-improving bounds we obtain nontrivial bounds $S^k\colon L^{p_1}\times\cdots\times…

Consider the wave equation associated with the Kohn Laplacian on groups of Heisenberg type. We construct parametrices using oscillatory integral representations and use them to prove sharp $L^p$ and Hardy space regularity results.

In this paper, we obtain local smoothing estimates for the averages over nondegenerate surfaces of codimension $2$ in $\mathbb R^4$. We make use of multilinear restriction estimates and decoupling inequalities for a hypersurface in $\mathbb…

In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…

We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below, in the spirit of the classical bound on the distances between conjugates points in surfaces…