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This paper is a companion to our prior paper arXiv:0705.4619 on the `Small Ball Inequality in All Dimensions. In it, we address a more restrictive inequality, and obtain a non-trivial, explicit bound, using a single essential estimate from our prior paper. The prior bound was not explicit and much more involved.
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The Small Ball Inequality is a conjectural lower bound on sums the L-infinity norm of sums of Haar functions supported on dyadic rectangles of a fixed volume in the unit cube. The conjecture is fundamental to questions in discrepancy theory, approximation theory and probability theory. In this article, we concentrate on a special case of the conjecture, and give the best known lower bound in dimension 3, using a conditional expectation argument.
Let h_R denote an L ^{infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{infty} norm of the `hyperbolic sums $$ sum _{|R|=2 ^{-n}} alpha(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^{eta} for some positive eta, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.
We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers.
For the weight function $W_mu(x) = (1-|x|^2)^mu$, $mu > -1$, $lambda > 0$ and $b_mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ la f,g ra = {b_mu [int_{BB^d} f(x) g(x) W_mu(x) dx + lambda int_{BB^d} abla f(x) cdot abla g(x) W_mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $la cdot,cdotra$, can be generated through a recursive formula.
We give in this paper some equivalent definitions of the so called $rho$-Carleson measures when $rho(t)=(log(4/t))^p(loglog(e^4/t))^q$, $0le p,q<infty$. As applications, we characterize the pointwise multipliers on $LMOA(mathbb S^n)$ and from this space to $BMOA(mathbb S^n)$. Boundedness of the Ces`aro type integral operators on $LMOA(mathbb S^n)$ and from $LMOA(mathbb S^n)$ to $BMOA(mathbb S^n)$ is considered as well.
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