This note is a companion paper to arXiv:1608.01630 [math.CA]. Here we generalize some of the geometric results of arXiv:1608.01630 [math.CA] to the case of a $3\times 3$ matrix function $A(x)\approx \mathrm{diag}\{1,f(x_1), g(x_1)\}$. More precisely, we make explicit calculations of the geodesics in the Carnot-Carathéodory space associated to $A$, and provide estimates on the Lebesgue measures of metric balls centered at the origin in that space.
We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (arXiv:math/9907127 [math.RT]). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are \emph{not} determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the $4\times4$ problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size $n\ge4$ which appear new for $n\ge 5$. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (arXiv:math/9804027 [math.CA]), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.
This paper is the third in an investigation begun in arXiv:1906.05602 and arXiv:1907.07571 of extending the T1 theorem of David and Journé, and optimal cancellation conditions, to more general weight pairs. The main result here is that the familiar T1 testing conditions over indicators of cubes, together with the one-tailed A2 conditions, imply polynomial testing. Analogous results for fractional singular integrals hold as well. Applications include a T1 theorem for fractional CZO's T in the case of doubling measures when one of the weights is A infinity, and then to optimal cancellation conditions for such CZO's in similar situations.
We extend some one parameter theorems on fractional integrals due to Sawyer and Wheeden to the two parameter setting using a recent iteration method of Tanaka and Yabuta.
We show that the energy conditions are not necessary for boundedness of fractional Riesz transforms in dimension at least 2. We also give a weak converse, namely that the energy conditions are necessary for boundedness of families of twisted localizations of fractional singular integrals having the positive gradient property.
In this paper we construct a wavelet basis in weighted L^2 of Euclidean space possessing vanishing moments of a fixed order for a general locally finite positive Borel measure. The approach is based on a clever construction of Alpert in the case of Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calderón-Zygmund operator on the real line and show that under suitable natural conditions, including a weaker energy condition, the operator is bounded from one weighted L^2 space to another if certain stronger testing conditions hold on polynomials. An example is provided showing that this result is logically different than existing results in the literature.
Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero
We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T on the real line, and any pair of locally finite positive Borel measures on the line. This includes the Hilbert transform and in a sense improves on the T1 theorem by the authors and M. Lacey.
Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero
This paper is a sequel to our paper Rev. Mat. Iberoam. 32 (2016), no. 1, 79-174. Let T be a standard fractional Calderon Zygmund operator. Assume appropriate Muckenhoupt and quasienergy side conditions. Then we show that T is bounded from one weighted space to another if the quasicube testing conditions hold for T and its dual, and if the quasiweak boundedness property holds for T. Conversely, if T is bounded, then the quasitesting conditions hold, and the quasiweak boundedness condition holds. If the vector of fractional Riesz transforms (or more generally a strongly elliptic vector of transforms) is bounded, then the appropriate Muckenhoupt conditions hold. We do not know if our quasienergy conditions are necessary in higher dimensions, except for certain situations in which one of the measures is one-dimensional as in arXiv:1310.4820 and arXiv:1505.07822v4, and for certain side conditions placed on the measures such as doubling and k-energy dispersed, which when k=n-1 is similar to the condition of uniformly full dimension in Lacey and Wick arXiv:1312.6163v3.
Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero
Using our T1 theorem with an energy side condition allowing common point masses, we extend our previous work in arXiv:1310.4484v3 on one measure supported on a line, to include regular C(1,delta) curves and to permit common point masses. In the special case of the Cauchy transform with one measure supported on the circle, this gives a slightly different conclusion than that in arXiv:1310.4820v4.
Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero
For alpha in [0,n) we demonstrate the failure of energy reversal for the vector of alpha-fractional Riesz transforms, and more generally for any vector of alpha-fractional convolution singular integrals having a kernel with vanishing integral on every great circle of the sphere.
This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and seven chapters that represent part of my work influenced by Apéry's proof from 1978 of the irrationality of $ζ(2)$ and $ζ(3)$, the values of Riemann's zeta function. Chapter 1 is about "at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ and $ζ(11)$ is irrational" (based in part on arXiv:math.NT/0206176). Chapter 2 explains a connection between the generalized multiple integrals introduced by Beukers in his proof of Apéry's result and the very-well-poised hypergeometric series; it is based on arXiv:math.CA/0206177. Chapter 3 surveys some arithmetic and hypergeometric $q$-analogies and establishes the irrationality measure $μ(ζ_q(2))<3.518876$ for a $q$-analogue of $ζ(2)$; it closely follows the text in Sb. Math. 193 (2002), 1151--1172, but also incorporates the sharper analysis of the hypergeometric construction by Smet and Van Assche (arXiv:0809.2501 [math.CA]) to produce the improvement upon the 2002 result. Chapter 4 is devoted to the measure $μ(ζ(2))<5.095412$ and is based on arXiv:1310.1526 [math.NT]; Chapter 5 is establishing the estimate $||(3/2)^k||>0.5803^k$ for the distance from $(3/2)^k$ to the nearest integer, with the English version published in J. Théor. Nombres Bordeaux 19 (2007), 313--325. Chapter 6 reproduces the solution (from arXiv:math.CA/0311195) to the problem of Asmus Schmidt about generalized Apéry's numbers. Finally, Chapter 7 is about expressing the special $L$-values as periods (in the sense of Kontsevich and Zagier), in particular, as values of hypergeometric functions; it is based on the publication in Springer Proc. Math. Stat. 43 (2013), 381--395.
Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero
We prove a two weight theorem for fractional singular integrals in higher dimensions assuming energy side conditions on the weights. The testing conditions are taken over quasicubes, namely globally biLipschitz images of cubes.
Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero
We prove a two weight theorem for alpha-fractional singular integrals in higher dimensions, assuming energy side conditions. We also show that reversal of the Energy Lemma fails for the vector Riesz transforms in the plane, as well as other collections of convolution Calderon-Zygmund operators in the plane, and when alpha = 1, even for the infinite vector of all classical 1-fractional Calderon-Zygmund operators.