Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Variation on Holder-Brascamp-Lieb Inequalities
111
0
0.0
(
0
)
Authors
Kevin ONeill
Ask ChatGPT about the research
No Arabic abstract
The Holder-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Holder-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlying linear maps match those of the convolution inequality of Young.
rate research
Read More
In this paper, we derive sharp nonlinear dimension-free Brascamp-Lieb inequalities (including hypercontractivity inequalities) for distributions on Polish spaces, which strengthen the classic Brascamp-Lieb inequalities. Applications include the extension of Mr. and Mrs. Gerbers lemmas to the cases of Renyi divergences and distributions on Polish spaces, the strengthening of small-set expansion theorems, and the characterization of the exponent of $q$-stability of Boolean functions. Our proofs in this paper are based on information-theoretic and coupling techniques.
A symmetrization inequality of Rogers and of Brascamp-Lieb-Luttinger states that for a certain class of multilinear integral expressions, among tuples of sets of prescribed Lebesgue measures, tuples of balls centered at the origin are among the maximizers. Under natural hypotheses, we characterize all maximizing tuples for these inequalities for dimensions strictly greater than 1. We establish a sharpened form of the inequality.
We study the infimum of the best constant in a functional inequality, the Brascamp-Lieb-like inequality, over auxiliary measures within a neighborhood of a product distribution. In the finite alphabet and the Gaussian cases, such an infimum converges to the best constant in a mutual information inequality. Implications for strong converse properties of two common randomness (CR) generation problems are discussed. In particular, we prove the strong converse property of the rate region for the omniscient helper CR generation problem in the discrete and the Gaussian cases. The latter case is perhaps the first instance of a strong converse for a continuous source when the rate region involves auxiliary random variables.
We present reverse Holder inequalities for Muckenhoupt weights in $mathbb{R}^n$ with an asymptotically sharp behavior for flat weights, namely $A_infty$ weights with Fujii-Wilson constant $(w)_{A_infty}to 1^+$. That is, the local integrability exponent in the reverse Holder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Holder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Holder inequalities for one-sided and two sided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Holder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.
Let $mathsf M$ and $mathsf M _{mathsf S}$ respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $mathbb{R}^n$, and let $w$ be a nonnegative locally integrable function on $mathbb{R}^n$. We define the associated Tauberian functions $mathsf{C}_{mathsf{HL},w}(alpha)$ and $mathsf{C}_{mathsf{S},w}(alpha)$ on $(0,1)$ by [ mathsf{C}_{mathsf{HL},w}(alpha) :=sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M chi_E(x) > alpha}) ] and [ mathsf{C}_{mathsf{S},w}(alpha) := sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M _{mathsf S}chi_E(x) > alpha}). ] Utilizing weighted Solyanik estimates for $mathsf M$ and $mathsf M_{mathsf S}$, we show that the function $mathsf{C}_{mathsf{HL},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}})^{-1}}(0,1)$ and $mathsf{C}_{mathsf{S},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}^ast})^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
