474 research outputs found

### On the Hilbert transform and lacunary directions in the plane

Let $H_k$ be the one dimensional Hilbert transform computed in the direction
$(1,2^k)$ in the plane. We show that the maximal operator $\sup_k |H_kf|$ maps
$L^p$ of the plane into itself for $1<p<\infty$.
The same result with the Hilbert transform replaced by the one dimensional
maximal function was proved by Nagel, Stein and Wainger in 1978.Comment: 11 pages, 1 fiure. This is version to appear in Illinois J. Mat

### On the Discrepancy Function in Arbitary Dimension, Close to L ^{1}

Let $\mathcal A_N$ to be $N$ points in the unit cube in dimension $d$, and
consider the Discrepency function
D_N(\vec x) \coloneqq \sharp \mathcal A_N \cap [\vec 0,\vec x)-N \abs{[\vec
0,\vec x)}
Here, $\vec x= (x_1 ,...c, x_d)$ and $[ 0,\vec x)=\prod_{t=1} ^{d} [0,x_t)$.
We show that necessarily
\norm D_N. L ^{1} (\log L) ^{(d-2)/2}. \gtrsim (\log N) ^{d/2} .
In dimension $d=2$, the `$\log L$' term has power zero, which corresponds to
a Theorem due to \cite{MR637361}.Comment: 17 pages. To appear in Analysis Mathematica. Many changes, and an
additional section on Hardy space and the Discrepancy functio

### Commutators with Reisz Potentials in One and Several Parameters

Let $M_b$ be the operator of pointwise multiplication by $b$, that is
$\operatorname M_b f=bf$. Set $[ A,B]={} AB- BA$. The Reisz potentials are the
operators
R_\alpha f(x)=\int f(x-y)\frac{dy}{\abs y ^{\alpha}},\qquad 0<\alpha<1.
They map $L^p\mapsto L^q$, for $1-\alpha+\frac1q=\frac1p$, a fact we shall take
for granted in this paper. A Theorem of Chanillo \cite{MR84j:42027} states that
one has the equivalence \norm [ M_b, R_\alpha].p\to q.\simeq \norm
b.\operatorname{BMO}. with the later norm being that of the space of
functions of bounded mean oscillation. We discuss a new proof of this result in
a discrete setting, and extend part of the equivalence above to the higher
parameter setting.Comment: To appear in Hokkaido Math J. This is the final version of the paper.
Several typos correcte

### The Linear Bound in A_2 for Calder\'on-Zygmund Operators: A Survey

For an L ^2-bounded Calderon-Zygmund Operator T, and a weight w \in A_2, the
norm of T on L ^2 (w) is dominated by A_2 characteristic of the weight. The
recent theorem completes a line of investigation initiated by
Hunt-Muckenhoupt-Wheeden in 1973, has been established in different levels of
generality by a number of authors over the last few years. It has a subtle
proof, whose full implications will unfold over the next few years. This sharp
estimate requires that the A_2 character of the weight can be exactly once in
the proof. Accordingly, a large part of the proof uses two-weight techniques,
is based on novel decomposition methods for operators and weights, and yields
new insights into the Calder\'on-Zygmund theory. We survey the proof of this
Theorem in this paper.Comment: 19 pages. Submitted to the proceedings of the Jozef Marcinkiewicz
Centenary Conferenc

### Sparse Bounds for Spherical Maximal Functions

We consider the averages of a function $f$ on $\mathbb R ^{n}$ over spheres
of radius $0< r< \infty$ given by $A_{r} f (x) = \int_{\mathbb S ^{n-1}} f
(x-r y) \; d \sigma (y)$, where $\sigma$ is the normalized rotation invariant
measure on $\mathbb S ^{n-1}$. We prove a sharp range of sparse bounds for two
maximal functions, the first the lacunary spherical maximal function, and the
second the full maximal function. $M_{{lac}} f = \sup_{j\in \mathbb Z }
A_{2^j} f , \qquad M_{{full}} f = \sup_{ r>0 } A_{r} f .$ The sparse bounds
are very precise variants of the known $L^p$ bounds for these maximal
functions. They are derived from known $L ^{p}$-improving estimates for the
localized versions of these maximal functions, and the indices in our sparse
bound are sharp. We derive novel weighted inequalities for weights in the
intersection of certain Muckenhoupt and reverse H\"older classes.Comment: 20 pages, 7 figures. To appear in J D'Analyse Mat

### On the Separated Bumps Conjecture for Calderon-Zygmund Operators

We study the `separated bump conjecture' of Cruz-Uribe & Perez, and
Cruz-Uribe & Reznikov & Volberg. In the L^p setting, we formulate a stronger
version of this conjecture, and show that under it, a two weight inequality
holds for all CZOs. When p=2, this is the result of Nazarov & Reznikov &
Volberg (1306.2653). Our argument is based on stopping time arguments and the
extra hypothesis is used in a clear-cut and seemingly essential way. This
argument could be of some help in searching for a counterexample to the
conjecture.Comment: 15 pages. v4: to appear in Hokkaido Math

### Small Ball and Discrepancy Inequalities

This is a comprehensive set of notes on the ArXiV paper math.CA/0609815 by
Dmitry Bilyk and the author.
The focus of that paper is a new inequality for sums of hyperbolic Haar
functions in three variables, extending a famous result of J Beck from 1987.
This is an improvement on what is known as the Small Ball Conjecture. In this
paper, that result is proved, in a more leisurely fashion and additional
remarks.
In addition, background material is gathered together, including a complete
proof of the necessary Harmonic Analysis; a summary of known results on the
Small Ball inequality; Irregularities of Distribution; the relationship with
conjectures in Approximation Theory and Probability Theory.Comment: 73 page

### Issues related to Rubio de Francia's Littlewood--Paley Inequality: A Survey

Rubio de Francia's Littlewood Paley inequality is an extension of the
classical Littlewood Paley inequality to one that holds for a decomposition of
frequency space into arbitrary disjoint intervals. We survey this inequality,
its higher dimensional analogs, and the implications for multipliers.Comment: 43 pages, 43 references. Additional material and references added. A
final polishing of the manuscript. To appear in NYJ Mat

### Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II

A conjecture of Nazarov--Treil--Volberg on the two weight inequality for the
Hilbert transform is verified. Given two non-negative Borel measures u and w on
the real line, the Hilbert transform $H_u$ maps $L^2(u)$ to $L^2(w)$ if and
only if the pair of measures of satisfy a Poisson $A_2$ condition, and dual
collections of testing conditions, uniformly over all intervals. This
strengthens a prior characterization of Lacey-Sawyer-Shen-Uriate-Tuero
arxiv:1201.4319. The latter paper includes a `Global to Local' reduction. This
article solves the Local problem.Comment: Final Version, to appear in Duk

### An A_p --A_infty inequality for the Hilbert Transform

Continuing a theme of Lerner and Hytonen-Perez, we establish an L^p(w)
inequality for a Haar shift operator of bounded complexity, that quantifies the
contribution of the A_infty characteristic of the weight to the L^p norm. Here,
1<p<\infty. The Hytonen-Perez inequality is only for p=2, and we improve an
inequality of the author and 6 other collaborators. As a corollary, the same
inequality holds for all Calderon-Zygmund operators in the convex hull of Haar
shifts of a bounded complexity, of which the canonical example is the Hilbert
transform. We conjecture that the same inequality holds for all
Calderon-Zygmund operators.Comment: 15 pages. (v2) Proof at end of section 4 is reworded. (v3) final
version, accepted to Houston J Mat

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