Local dimension-free estimates for volumes of sublevel sets of analytic functions

Abstract

§1. The resultIn what follows, we denote complex balls {z ∈ C n : |z − w| < r} by B c (w, r) and real balls {x ∈ R n : |x − u| < r} by B(u, r). For any real ball B, we denote by

Discrete Schrödinger Operators with Random Alloy-type Potential

“…For these results, one needs to apply the Cartan estimate to functions depending on many variables, and thus needs an estimate that is well behaved in the number of variables. Such an estimate was first proven by Nazarov, Sodin, and Volberg [NSV03]. Unfortunately, they work on balls and not on polydisks as necessary for our applications.…”

Dilation Type Inequalities for Strongly-Convex Sets in Weighted Riemannian Manifolds

“…As we will see in Subsection 3.3, we will also discuss Kahane-Khintchine inequalities with positive and negative exponents for symmetric quasi-convex functions via Theorem 1.2 (and Theorem 2.4 and Proposition 2.3), and discuss deviation inequalities as their application. Similar inequalities for general functions have been already investigated in [30,5,15,32] where we need to assume the Remez type inequality. On the other hand, we can obtain Kahane-Khintchine inequalities and deviation inequalities without the Remez type inequality.…”

“…On the other hand, Nazarov-Sodin-Volberg [30] showed a new sharp isoperimetric-type inequality for a log-concave probability measure on R n , which we call the dilation inequality in this paper. This inequality is originally given by Borell [9] and investigated by many researchers in [25,17,30,4,5,8,15,20,32] where the sharpness and generalization of the dilation inequality are discussed. Here a measure µ on R n is log-concave if for any compact subsets A, B ⊂ R n , it holds µ((1 − t)A + tB) ≥ µ(A) 1−t µ(B) t , ∀t ∈ (0, 1),…”

“…and define a dilation area of A by Then Nazarov-Sodin-Volberg [30] (see also [32]) showed that every log-concave probability measure µ on R n satisfies µ * (A) ≥ −(1 − µ(A)) log(1 − µ(A)) (1.4) for any Borel subset A ⊂ R n . We note that it is natural to consider (1.4) as a counterpart of Cheeger's isoperimetric inequality, namely…”

A sufficient condition for mobile sampling in terms of surface density

“…For the proof of Claim 4.5 we shall appeal to a Remez inequality. The following inequality is a well-known simple special case of much more general results, for instance [Br,NSV2], but we give a concise proof in an appendix for the benefit of the reader.…”