6,556 research outputs found
On the Discrepancy Function in Arbitary Dimension, Close to L ^{1}
Let to be points in the unit cube in dimension , 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, and .
We show that necessarily
\norm D_N. L ^{1} (\log L) ^{(d-2)/2}. \gtrsim (\log N) ^{d/2} .
In dimension , the `' 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 be the operator of pointwise multiplication by , that is
. Set . 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 , for , 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
On the Hilbert transform and lacunary directions in the plane
Let be the one dimensional Hilbert transform computed in the direction
in the plane. We show that the maximal operator maps
of the plane into itself for .
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
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
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
Sparse Bounds for Spherical Maximal Functions
We consider the averages of a function on over spheres
of radius given by , where is the normalized rotation invariant
measure on . 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. The sparse bounds
are very precise variants of the known bounds for these maximal
functions. They are derived from known -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
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 maps to if and
only if the pair of measures of satisfy a Poisson 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
The bilinear maximal functions map into L^p for 2/3 < p <= 1
The bilinear maximal operator defined below maps into
provided 1
, and . Mfg(x)=\sup_{t>0}\frac1{2t}\int_{-t}^t\abs{f(x+y)g(x-y)} dy. In particular is integrable\thinspace if and are square integrable, answering a conjecture posed by Alberto Calder\'on.Comment: 23 page
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