Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDecoupling inequalities for quadratic forms
135
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove sharp $ell^q L^p$ decoupling inequalities for arbitrary tuples of quadratic forms. Our argument is based on scale-dependent Brascamp-Lieb inequalities.
rate research
Read More
We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola ${(t, t^2) : 0 leq t leq 1}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeters decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to $[0, 1]$. Our ideas are inspired by the recent work on ellipsephic sets by Biggs using nested efficient congruencing.
We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $f colon mathbb{R}^{12} to mathbb{R}$ is an integrable function that is not identically zero. Normalize its Fourier transform $widehat{f}$ by $widehat{f}(xi) = int_{mathbb{R}^d} f(x)e^{-2pi i langle x, xirangle}, dx$, and suppose $widehat{f}$ is real-valued and integrable. We show that if $f(0) le 0$, $widehat{f}(0) le 0$, $f(x) ge 0$ for $|x| ge r_1$, and $widehat{f}(xi) ge 0$ for $|xi| ge r_2$, then $r_1r_2 ge 2$, and this bound is sharp. The construction of a function attaining the bound is based on Viazovskas modular form techniques, and its optimality follows from the existence of the Eisenstein series $E_6$. No sharp bound is known, or even conjectured, in any other dimension. We also develop a connection with the linear programming bound of Cohn and Elkies, which lets us generalize the sign pattern of $f$ and $widehat{f}$ to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.
Equivalencies of many basic elementary inequalities are given
We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincare series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenstein series has to be extended to include also iterated integrals of holomorphic cusp forms to find expressions for all modular invariant functions of depth two. The coefficients of these cusp forms are identified as ratios of their L-values inside and outside the critical strip.
This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as (sin x)/x and x/(sinh x) are proved.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
