Best constants for a family of Carleson sequences

We consider a general family of Carleson sequences associated with dyadic $A_2$ weights and find sharp -- or, in one case, simply best known -- upper and lower bounds for their Carleson norms in terms of the $A_2$-characteristic of the weight. The results obtained make precise and significantly generalize earlier estimates by Wittwer, Vasyunin, Beznosova, and others. We also record several corollaries, one of which is a range of new characterizations of dyadic $A_2.$ Particular emphasis is placed on the relationship between sharp constants and optimizing sequences of weights; in most cases explicit optimizers are constructed. Our main estimates arise as consequences of the exact expressions, or explicit bounds, for the Bellman functions for the problem, and the paper contains a measure of Bellman-function innovation.Comment: 29 pages, 2 figure

IBEX, SWCX and a Consistent Model for the Local ISM

The Local Interstellar Medium (LISM) makes its presence felt in the heliosphere in a number of ways including inflowing neutral atoms and dust and shaping of the heliosphere via its ram pressure and magnetic field. Modelers of the heliosphere need to know the ISM density and magnetic field as boundary conditions while ISM modelers would like to use the data and models of the heliosphere to constrain the nature of the LISM. An important data set on the LISM is the diffuse soft X-ray background (SXRB), which is thought to originate in hot gas that surrounds the local interstellar cloud (LIC) in which the heliosphere resides. However, in the past decade or so it has become clear that there is a significant X-ray foreground due to emission within the heliosphere generated when solar wind ions charge exchange with inflowing neutrals. The existence of this SWCX emission complicates the interpretation of the SXRB. We discuss how data from IBEX and models for the Ribbon in particular provide the possibility of tying together heliosphere models with models for the LISM, providing a consistent picture for the pressure in the LISM, the ionization in the LIC and the size and shape of the heliosphere.Comment: 6 pages, 2 figures. To be published in the proceedings of the 12th AIAC Conference "Outstanding problems in Heliosphysics: from coronal heating to the edge of the heliosphere" (ASP Conference Series

Cincinnati lectures on Bellman functions

In January-March 2011, the Department of Mathematical Science at the University of Cincinnati held a Taft Research Seminar "Bellman function method in harmonic analysis." The seminar was made possible by a generous grant from the Taft Foundation. The principal speaker at the seminar was Vasily Vasyunin. The local host and convener of the seminar was Leonid Slavin. The seminar was in effect a 10-week lecture- and discussion-based course. This manuscript represents a slightly revised content of those lectures. In particular, it includes some technical details that were omitted in class due to time constraints.Comment: 33 pages, 4 figure

The absence of the selfaveraging property of the entanglement entropy of disordered free fermions

We consider the macroscopic system of free lattice fermions in one dimension assuming that the one-body Hamiltonian of the system is the one dimensional discrete Schr\"odinger operator with independent identically distributed random potential. We show analytically and numerically that the variance of the entanglement entropy of the segment $[-M,M]$ of the system is bounded away from zero as $M\rightarrow \infty$. This manifests the absence of the selfaveraging property of the entanglement entropy in our model, meaning that in the one-dimensional case the complete description of the entanglement entropy is provided by its whole probability distribution. This also may be contrasted the case of dimension two or more, where the variance of the entanglement entropy per unit surface area vanishes as $$ \cite{El-Co:17}, thereby guaranteeing the representativity of its mean for large $M$ in the multidimensional case.Comment: arXiv admin note: substantial text overlap with arXiv:1703.0605

On the Area Law for Disordered Free Fermions

We study theoretically and numerically the entanglement entropy of the $d$-dimensional free fermions whose one body Hamiltonian is the Anderson model. Using basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $\langle S_\Lambda \rangle$ of the $d$ dimension cube $\Lambda$ of side length $l$ admits the area law scaling $\langle S_\Lambda \rangle \sim l^{(d-1)}, \ l \gg 1$ even in the gapless case, thereby manifesting the area law in the mean for our model. For $d=1$ and $l\gg 1$ we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not selfaveraging, i.e., has non vanishing random fluctuations even if $l \gg 1$.Comment: 4 pages, 1 figur

Bellman Function and the H1−BMOH^1-BMO Duality

A Bellman function approach to Fefferman's $H^1-BMO$ duality theorem is presented. One Bellman-type argument is used to handle two different one-dimensional cases, dyadic and continuous. An explicit estimate for the constant of embedding $BMO\subset (H^1)^*$ is given in the dyadic case. The same Bellman function is then used to establish a multi-dimensional analog.Comment: 14 pages, 2 figures, final versio

Sharp L^p estimates on BMO

We construct the upper and lower Bellman functions for the $L^p$ (quasi)-norms of BMO functions. These appear as solutions to a series of Monge--Amp\`ere boundary value problems on a non-convex plane domain. The knowledge of the Bellman functions leads to sharp constants in inequalities relating average oscillations of BMO functions and various BMO norms.Comment: 42 pages, 12 figure

The John--Nirenberg constant of BMOp{\rm BMO}^p, p>2p>2

This paper is a continuation of earlier work by the first author who determined the John--Nirenberg constant of ${\rm BMO}^p\big((0,1)\big)$ for the range $1\le p\le 2.$ Here, we compute that constant for $p>2.$ As before, the main results rely on Bellman functions for the $L^p$ norms of logarithms of $A_\infty$ weights, but for $p>2$ these functions turn out to have a significantly more complicated structure than for $1\le p\le 2.$Comment: 16 pages, 3 figures. To appear in St. Petersburg Math Journa

Inequalities for BMO on α\alpha-trees

We develop technical tools that enable the use of Bellman functions for BMO defined on $\alpha$-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John--Nirenberg inequality and an inequality relating $L^1$- and $L^2$-oscillations for BMO on $\alpha$-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in $\mathbb{R}^n,$ the inequalities proved are sharp. We also reformulate the John--Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be used for any function class that corresponds to a non-convex Bellman domain.Comment: 17 pages, 1 figur

Sharp results in the integral-form John--Nirenberg inequality

We consider the strong form of the John-Nirenberg inequality for the $L^2$-based BMO. We construct explicit Bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant as well as the precise bound on the inequality's range of validity, both previously unknown. The results for the two cases are substantially different. The paper not only gives another instance in the short list of such explicit calculations, but also presents the Bellman function method as a sequence of clear steps, adaptable to a wide variety of applications.Comment: 37 pages, 8 figures, final version; Trans. Amer. Math. Soc., Vol. 363, No. 8 (2011