1. CJM Online first
 Chen, Xianghong; Seeger, Andreas

Convolution powers of Salem measures with applications
We study the regularity of convolution powers for measures supported
on
Salem sets, and prove related results on Fourier restriction
and Fourier multipliers. In particular we show
that for $\alpha$ of the form
${d}/{n}$, $n=2,3,\dots$ there exist $\alpha$Salem measures
for which the $L^2$ Fourier restriction theorem holds in the
range $p\le \frac{2d}{2d\alpha}$.
The results rely on ideas of KÃ¶rner.
We extend some of his constructions to obtain upper regular $\alpha$Salem
measures, with sharp regularity results for $n$fold convolutions
for all $n\in \mathbb{N}$.
Keywords:convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of BochnerRiesz type Categories:42A85, 42B99, 42B15, 42A61 

2. CJM Online first
3. CJM Online first
 Guo, Xiaoli; Hu, Guoen

On the commutators of singular integral operators with rough convolution kernels
Let $T_{\Omega}$ be the singular integral operator with kernel
$\frac{\Omega(x)}{x^n}$, where $\Omega$ is homogeneous of degree
zero, has mean value zero and belongs to $L^q(S^{n1})$ for
some
$q\in (1,\,\infty]$. In this paper, the authors establish the
compactness on weighted $L^p$ spaces, and the Morrey spaces,
for the commutator generated by $\operatorname{CMO}(\mathbb{R}^n)$ function
and $T_{\Omega}$. The associated maximal operator and the discrete
maximal operator are also considered.
Keywords:commutator, singular integral operator, compact operator, completely continuous operator, maximal operator, Morrey space Categories:42B20, 47B07 

4. CJM Online first
 Yattselev, Maxim L.

Strong Asymptotics of HermitePadÃ© Approximants for Angelesco Systems
In this work type II HermitePadÃ© approximants for a vector
of Cauchy transforms of smooth Jacobitype densities are considered.
It is assumed that densities are supported on mutually disjoint
intervals (an Angelesco system with complex weights). The formulae
of strong asymptotics are derived for any ray sequence of multiindices.
Keywords:HermitePadÃ© approximation, multiple orthogonal polynomials, nonHermitian orthogonality, strong asymptotics, matrix RiemannHilbert approach Categories:42C05, 41A20, 41A21 

5. CJM 2015 (vol 67 pp. 1161)
 Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun

Nontangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights
or in the class of $QC(\mathbb{R}^n)$ weights, and
$L_w:=w^{1}\mathop{\mathrm{div}}(A\nabla)$
the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$,
$n\ge 2$. In this article, the authors establish the nontangential
maximal function characterization
of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for
$p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and
$w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$,
the authors prove that the associated Riesz transform $\nabla L_w^{1/2}$
is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical
Hardy space $H_w^p(\mathbb{R}^n)$.
Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform Categories:42B30, 42B35, 35J70 

6. CJM 2014 (vol 66 pp. 1358)
 Osėkowski, Adam

Sharp Localized Inequalities for Fourier Multipliers
In the paper we study sharp localized $L^q\colon L^p$ estimates for
Fourier multipliers resulting from modulation of the jumps of
LÃ©vy
processes.
The proofs of these estimates rest on probabilistic methods and
exploit related sharp bounds for differentially subordinated
martingales, which are of independent interest. The lower bounds
for
the constants involve the analysis of laminates, a family of
certain
special probability measures on $2\times 2$ matrices. As an
application, we obtain new sharp bounds for the real and imaginary
parts of the BeurlingAhlfors operator .
Keywords:Fourier multiplier, martingale, laminate Categories:42B15, 60G44, 42B20 

7. CJM 2013 (vol 66 pp. 284)
 Eikrem, Kjersti Solberg

Random Harmonic Functions in Growth Spaces and Blochtype Spaces
Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces
of harmonic functions in the unit disk and multidimensional unit
ball
which admit a twosided radial majorant $v(r)$.
We consider functions $v $ that fulfill a doubling condition. In the
twodimensional case let $u (re^{i\theta},\xi) = \sum_{j=0}^\infty
(a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$
where $\xi =\{\xi_{ji}\}$ is a sequence of random
subnormal variables and $a_{ji}$ are
real; in higher dimensions we consider series of spherical harmonics.
We will obtain conditions on the coefficients $a_{ji} $ which imply
that $u$ is in $h^\infty_v(\mathbf B)$ almost surely.
Our estimate improves previous results by Bennett, Stegenga and
Timoney, and we prove that the estimate is sharp.
The results for growth spaces can easily be applied to Blochtype
spaces, and we obtain a similar characterization for these spaces,
which generalizes results by Anderson, Clunie and Pommerenke and by
Guo and Liu.
Keywords:harmonic functions, random series, growth space, Blochtype space Categories:30B20, 31B05, 30H30, 42B05 

8. CJM 2013 (vol 66 pp. 1382)
 Wu, Xinfeng

Weighted Carleson Measure Spaces Associated with Different Homogeneities
In this paper, we introduce weighted Carleson measure spaces associated
with different homogeneities and prove that these spaces are the dual spaces
of weighted Hardy spaces studied in a forthcoming paper.
As an application, we establish
the boundedness of composition of two CalderÃ³nZygmund operators with
different homogeneities on the weighted Carleson measure spaces; this,
in particular, provides the weighted endpoint estimates for the operators
studied by PhongStein.
Keywords:composition of operators, weighted Carleson measure spaces, duality Categories:42B20, 42B35 

9. CJM 2013 (vol 65 pp. 1217)
 Cruz, Victor; Mateu, Joan; Orobitg, Joan

Beltrami Equation with Coefficient in Sobolev and Besov Spaces
Our goal in this work is to present some function spaces on the
complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of
the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f
(z)$, have first derivatives locally in $X(\mathbb C)$, provided that the
Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.
Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³nZygmund operators Categories:30C62, 35J99, 42B20 

10. CJM 2012 (vol 65 pp. 510)
 Blasco de la Cruz, Oscar; Villarroya Alvarez, Paco

Transference of vectorvalued multipliers on weighted $L^p$spaces
We prove
restriction and extension of multipliers between
weighted Lebesgue spaces with
two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability which can be
below one.
We also develop some adhoc methods which apply to weights
defined by the product of periodic weights with functions of power type.
Our vectorvalued approach allow us to extend results
to transference of maximal multipliers and provide transference of LittlewoodPaley inequalities.
Keywords:Fourier multipliers, restriction theorems, weighted spaces Categories:42B15, 42B35 

11. CJM 2012 (vol 66 pp. 102)
 Birth, Lidia; Glöckner, Helge

Continuity of convolution of test functions on Lie groups
For a Lie group $G$, we show that the map
$C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$,
$(\gamma,\eta)\mapsto \gamma*\eta$
taking a pair of
test functions to their convolution is continuous if and only if $G$ is $\sigma$compact.
More generally, consider $r,s,t
\in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$
and a continuous bilinear map $b\colon E_1\times E_2\to F$
to a complete locally convex space $F$.
Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$,
$(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map.
The main result is a characterization of those $(G,r,s,t,b)$
for which $\beta$ is continuous.
Convolution
of compactly supported continuous functions on a locally compact group
is also discussed, as well as convolution of compactly supported $L^1$functions
and convolution of compactly supported Radon measures.
Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigmacompactness, convolution, continuity, seminorm, product estimates Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25 

12. CJM 2012 (vol 65 pp. 299)
 Grafakos, Loukas; Miyachi, Akihiko; Tomita, Naohito

On Multilinear Fourier Multipliers of Limited Smoothness
In this paper,
we prove certain $L^2$estimate
for multilinear Fourier multiplier operators
with multipliers of limited smoothness.
As a result,
we extend the result of CalderÃ³n and Torchinsky
in the linear theory to the multilinear case.
The sharpness of our results and some
related estimates in Hardy spaces
are also discussed.
Keywords:multilinear Fourier multipliers, HÃ¶rmander multiplier theorem, Hardy spaces Categories:42B15, 42B20 

13. CJM 2012 (vol 65 pp. 600)
 Kroó, A.; Lubinsky, D. S.

Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials
We establish asymptotics for Christoffel functions associated with
multivariate orthogonal polynomials. The underlying measures are assumed to
be regular on a suitable domain  in particular this is true if they are
positive a.e. on a compact set that admits analytic parametrization. As a
consequence, we obtain asymptotics for Christoffel functions for measures on
the ball and simplex, under far more general conditions than previously
known. As another consequence, we establish universality type limits in the
bulk in a variety of settings.
Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions Categories:42C05, 42C99, 42B05, 60B20 

14. CJM 2011 (vol 64 pp. 1201)
 Aistleitner, Christoph; Elsholtz, Christian

The Central Limit Theorem for Subsequences in Probabilistic Number Theory
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying
\begin{equation}
\tag{1}
f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad
\operatorname{Var_{[0,1]}}
f \lt \infty.
\end{equation}
If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$
the distribution of
\begin{equation}
\tag{2}
\frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}}
\end{equation}
converges to a Gaussian distribution. In the case
$$
1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty
$$
there is a complex interplay between the analytic properties of the
function $f$, the numbertheoretic properties of $(n_k)_{k \geq 1}$,
and the limit distribution of (2).
In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying
$\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial
subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying
(1) the distribution of
$$
\frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}}
$$
converges to a Gaussian distribution. This result is best possible: for any
$\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to
\infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial
subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution
of (2) does not converge to a Gaussian distribution for some $f$.
Our result can be viewed as a Ramsey type result: a sufficiently dense
increasing integer sequence contains a subsequence having a certain
requested numbertheoretic property.
Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem Categories:60F05, 42A55, 11D04, 05C55, 11K06 

15. CJM 2011 (vol 64 pp. 1036)
 Koh, Doowon; Shen, ChunYen

Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields
In this paper we study the extension problem, the
averaging problem, and the generalized ErdÅsFalconer distance
problem associated with arbitrary homogeneous varieties in three
dimensional vector spaces over finite fields. In the case when the
varieties do not contain any plane passing through the origin, we
obtain the best possible results on the aforementioned three problems. In
particular, our result on the extension problem modestly generalizes
the result by Mockenhaupt and Tao who studied the particular conical
extension problem. In addition, investigating the Fourier decay on
homogeneous varieties enables us to give complete mapping properties
of averaging operators. Moreover, we improve the size condition on a
set such that the cardinality of its distance set is nontrivial.
Keywords:extension problems, averaging operator, finite fields, ErdÅsFalconer distance problems, homogeneous polynomial Categories:42B05, 11T24, 52C17 

16. CJM 2011 (vol 64 pp. 892)
 Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong

Boundedness of CalderÃ³nZygmund Operators on Nonhomogeneous Metric Measure Spaces
Let $({\mathcal X}, d, \mu)$ be a
separable metric measure space satisfying the known upper
doubling condition, the geometrical doubling condition, and the
nonatomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$.
In this paper, we show that the boundedness of a CalderÃ³nZygmund
operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on
$L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$
to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a
CalderÃ³nZygmund operator bounded on $L^2(\mu)$,
then its maximal operator is bounded on $L^p(\mu)$
for all $p\in (1, \infty)$ and from
the space of all complexvalued Borel measures on
${\mathcal X}$ to $L^{1,\,\infty}(\mu)$.
All these results generalize the corresponding results of Nazarov et al.
on metric spaces with
measures satisfying the socalled polynomial growth condition.
Keywords:upper doubling, geometrical doubling, dominating function, weak type $(1,1)$ estimate, CalderÃ³nZygmund operator, maximal operator Categories:42B20, 42B25, 30L99 

17. CJM 2011 (vol 64 pp. 257)
 Chen, Yanping; Ding, Yong; Wang, Xinxia

Compactness of Commutators for Singular Integrals on Morrey Spaces
In this paper we characterize the
compactness of the commutator $[b,T]$ for the singular integral
operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More
precisely, we prove that if
$b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO}
(\mathbb R^n)$closure of $C_c^\infty(\mathbb R^n)$,
then $[b,T]$ is a compact operator on the
Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and
$0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and
$[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$
for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$.
Moreover, the boundedness of a rough singular integral operator $T$
and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also
given. We obtain a sufficient condition for a
subset in Morrey space to be a strongly precompact set,
which has interest in its own right.
Keywords:singular integral, commutators, compactness, VMO, Morrey space Categories:42B20, 42B99 

18. CJM 2011 (vol 63 pp. 798)
 Daws, Matthew

Representing Multipliers of the Fourier Algebra on NonCommutative $L^p$ Spaces
We show that the multiplier algebra of the Fourier algebra on a
locally compact group $G$ can be isometrically represented on a direct
sum on noncommutative $L^p$ spaces associated with the right von
Neumann algebra of $G$. The resulting image is the idealiser of the
image of the Fourier algebra. If these spaces are given their
canonical operator space structure, then we get a completely isometric
representation of the completely bounded multiplier algebra. We make
a careful study of the noncommutative $L^p$ spaces we construct and
show that they are completely isometric to those considered recently
by Forrest, Lee, and Samei. We improve a result of theirs about module
homomorphisms. We suggest a definition of a FigaTalamancaHerz
algebra built out of these noncommutative $L^p$ spaces, say
$A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to
$L^1(G)$, generalising the abelian situation.
Keywords:multiplier, Fourier algebra, noncommutative $L^p$ space, complex interpolation Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52 

19. CJM 2011 (vol 63 pp. 689)
 Olphert, Sean; Power, Stephen C.

Higher Rank Wavelets
A theory of higher rank multiresolution analysis is given in the
setting of abelian multiscalings. This theory enables the
construction, from a higher rank MRA, of finite wavelet sets
whose multidilations have translates forming an orthonormal basis
in $L^2(\mathbb R^d)$. While tensor products of uniscaled MRAs provide
simple examples we construct many nonseparable higher rank
wavelets. In particular we construct \emph{Latin square
wavelets} as rank~$2$ variants of Haar wavelets. Also we construct
nonseparable scaling functions for rank $2$ variants of Meyer
wavelet scaling functions, and we construct the associated
nonseparable wavelets with compactly supported Fourier transforms.
On the other hand we show that compactly supported scaling
functions for biscaled MRAs are necessarily separable.
Keywords: wavelet, multiscaling, higher rank, multiresolution, Latin squares Categories:42C40, 42A65, 42A16, 43A65 

20. CJM 2010 (vol 63 pp. 181)
 Ismail, Mourad E. H.; Obermaier, Josef

Characterizations of Continuous and Discrete $q$Ultraspherical Polynomials
We characterize the continuous $q$ultraspherical polynomials in
terms of the special form of the coefficients in the expansion
$\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$
being the AskeyWilson divided difference operator. The polynomials
are assumed to be symmetric, and the connection coefficients
are multiples of the reciprocal of the square of the $L^2$ norm of
the polynomials. A similar characterization is given for the discrete
$q$ultraspherical polynomials. A new proof of the evaluation of
the connection coefficients for big $q$Jacobi polynomials is given.
Keywords:continuous $q$ultraspherical polynomials, big $q$Jacobi polynomials, discrete $q$ultra\spherical polynomials, AskeyWilson operator, $q$difference operator, recursion coefficients Categories:33D45, 42C05 

21. CJM 2010 (vol 62 pp. 1419)
 Yang, Dachun; Yang, Dongyong

BMOEstimates for Maximal Operators via Approximations of the Identity with NonDoubling Measures
Let $\mu$ be a nonnegative Radon measure
on $\mathbb{R}^d$ that satisfies the growth condition that there exist
constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and
$r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball
centered at $x$ and having radius $r$. In this paper, the authors prove
that if $f$ belongs to the $\textrm {BMO}$type space $\textrm{RBMO}(\mu)$ of Tolsa, then
the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an
initial cube) and the inhomogeneous maximal function
$\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube)
associated with a given approximation of the identity $S$ of Tolsa are
either infinite everywhere or finite almost everywhere,
and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from
$\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$type
space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous
maximal operator $\mathcal{M}_S$ is bounded from the local
$\textrm {BMO}$type space $\textrm{rbmo}(\mu)$
to the local $\textrm {BLO}$type space $\textrm{rblo}(\mu)$.
Keywords:Nondoubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu) Categories:42B25, 42B30, 47A30, 43A99 

22. CJM 2010 (vol 62 pp. 1182)
 Yue, Hong

A Fractal Function Related to the JohnNirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces
is defined as a Haar wavelet decomposition, with the coefficients
depending on a fixed parameter $\beta>0$. On its support $I_0=[0,
1]^n$, $f(x)$ can be expressed by the binary expansions of the
coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb
R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for
$\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a
JohnNirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a selfaffine function. It is continuous almost
everywhere and discontinuous at all dyadic points inside $I_0$. In
addition, it is not monotone along any coordinate direction in any
small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from
$I_0$ to $[\frac{1}{12^{\beta}}, \frac{1}{12^{\beta}}]$, and the
graph of $f$ has a noninteger fractal dimension $n+1\beta$.
Keywords:Haar wavelets, Q spaces, JohnNirenberg inequality, Greedy expansion, selfaffine, fractal, Box dimension Categories:42B35, 42C10, 30D50, 28A80 

23. CJM 2010 (vol 62 pp. 827)
 Ouyang, Caiheng; Xu, Quanhua

BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces
This paper studies the relationship between vectorvalued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1z)^{q1}\\nabla f(z)\^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\f(z)f(z_0)\^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$uniformly convex, where $$P_{z_0}(z)=\frac{1z_0^2}{1\bar{z_0}z^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$uniformly smooth norm.
Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces Categories:46E40, 42B25, 46B20 

24. CJM 2009 (vol 61 pp. 807)
25. CJM 2009 (vol 61 pp. 141)
 Green, Ben; Konyagin, Sergei

On the Littlewood Problem Modulo a Prime
Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow
\mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f}
\Vert_1 \leq 1$. Then
$\min_{x \in \Zp} f(x) = O(\log p)^{1/3 + \epsilon}$.
One should think of $f$ as being ``approximately continuous''; our
result is then an ``approximate intermediate value theorem''.
As an immediate consequence we show that if $A \subseteq \Zp$ is a
set of cardinality $\lfloor p/2\rfloor$, then
$\sum_r \widehat{1_A}(r) \gg (\log p)^{1/3  \epsilon}$. This
gives a result on a ``mod $p$'' analogue of Littlewood's wellknown
problem concerning the smallest possible $L^1$norm of the Fourier
transform of a set of $n$ integers.
Another application is to answer a question of Gowers. If $A
\subseteq \Zp$ is a set of size $\lfloor p/2 \rfloor$, then there is
some $x \in \Zp$ such that
\[  A \cap (A + x)  p/4  = o(p).\]
Categories:42A99, 11B99 
