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\begin{document}

\title{Vertical variation of  harmonic functions in upper half spaces.}
\author{Michael D. O'Neill}



\address{ Department of Mathematics, UTEP\\El Paso, TX 79968}




\email{michael@math.utep.edu}
\begin{abstract}
Some results of Bourgain on the radial variation of harmonic functions
in the disk are extended to the setting of harmonic functions in upper
half
spaces.
\end{abstract}





\newtheorem{dfn}{Definition}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{claim}{Claim}
\newtheorem{theorem}{Theorem}
\newtheorem{Theorem}{Theorem 1}
\newtheorem*{TheoremA}{Theorem A}
\newtheorem*{TheoremB}{Theorem B}
\newtheorem{ex}{Example}
\newtheorem{rmk}{Remark}
\renewcommand{\theTheorem}{}
\renewcommand{\thetheorem}{}
\newtheorem{prop}{Proposition}[section]
\def\pfclm{\noindent {\em Proof of claim}\quad}
\def\proof{\noindent{\em Proof.}\quad}
\def\pflm{\noindent {\em Proof of lemma.}\quad}
\def\pfthm{\noindent {\em Proof of theorem.}\quad}
\def\rse#1{\mbox{$^{\raisebox{.3ex}{$\scriptstyle{#1}$}}$}}

\maketitle
  
1991 Mathematics subject classification: 31B05, 31B25, 42B15

\section{Introduction}
Let $f$ be a bounded analytic function in the unit disk $\mathbb D$
and let 
$$
V(f, \theta) = \int \limits _0 ^1 |f'( \rho e^{i\theta}) | \, d\rho.
$$

It was shown in \cite{Rudin} that the set 
$$FRV(f)\equiv \{ \theta \in \mathbb T: V(f, \theta) < \infty \}
$$
may have Lebesgue measure zero and may be of first category.

Answering a question from \cite{Rudin}, Bourgain showed in
\cite{Bourgain1}
that $FRV(f)$ is always of Hausdorff dimension 1 when $f$ is bounded
and analytic or even when $f$ is a bounded real harmonic function. By
similar
arguments, it was shown in \cite{Bourgain2} that the same conclusion
holds if $f$ is a real positive harmonic function.

Here we will show that the technique and results in \cite{Bourgain1}
and \cite{Bourgain2} can be generalized to real valued harmonic
functions
in the upper half space $\mathbb R^{n+1}$.
To be specific, we will replace the notion of radial variation with
variation on vertical lines and prove the following theorems.
\begin{TheoremA}
If $u $ is a bounded real valued harmonic function in the upper half space 
$\mathbb R^{(n+1)}$ then there is a set $E \subset \mathbb R^n$
having Hausdorff dimension $n$  such that
$$
\int \limits _0 ^1 |\nabla u (x,y)| \,dy < + \infty
$$
for each $x\in E$.
\end{TheoremA} 
\begin{TheoremB}
If $u $ is a positive  harmonic function in the upper half space 
$\mathbb R^{(n+1)}$ then there is a set $E \subset \mathbb R^n$
having Hausdorff dimension $n$  such that
$$
\int \limits _0 ^1 |\nabla u (x,y)| \,dy < + \infty
$$
for each $x\in E$.
\end{TheoremB}


The line of reasoning for proving the theorems
 follows \cite {Bourgain1} but the results appear not to have been known.
 The generalization to higher dimensions requires  some manipulation 
of convolution kernels and the  asymptotic estimation of certain oscillatory
integrals by a  technique of Wong, \cite{Wong1}, \cite{Wong2}.
The lemmas containing the required estimates are given in section 3
and their proofs in section 7.

 

A construction of P.W. Jones, \cite{Jones}, shows that there are
bounded analytic functions $f_1$ and $f_2$ in $\mathbb D$ such that 
$$
\int \limits _{\gamma} (| f_1'| + |f_ 2'|)\, ds = + \infty
$$
for any curve $\gamma$ in $\mathbb D$ joining $0$ to $\partial \mathbb
D$.
Using Jones' idea, it is easy to generate examples of  bounded complex
valued or vector valued harmonic functions in the upper half space
which have infinite vertical variation at every point.
Simple extensions of  examples of the type in
\cite{Rudin} show that in the real valued  case, the set of finite vertical
variation can be of measure zero and of first category. In the above
two senses then the results are the best possible.

In \cite{JonesMueller}, Bourgain's
result in the disk is used in conjunction with a stopping time
construction of Lipschitz sub-domains to prove that for any Bloch
function $b$ defined in the disk there is a countable set of points $E$ in
$\partial\mathbb D$ such that
$$
\liminf \limits _{r\to 1} \frac{\mbox{ Re }b(r\xi)}{\int \limits
_0^r
|b'(\rho \xi)| d\rho} >0
$$
for each $\xi \in E$. The proof depends upon the Riemann mapping
theorem to pull Bourgain's result back to the Lipschitz sub-domains.
It is an open question whether an analog of the Jones-Mueller result
holds for harmonic Bloch functions in upper half spaces and in light
of Theorems A and B the question can be reduced to finding a
substitute for the Riemann mapping part of the Jones-Mueller argument.
 


\section{Poisson kernels}

Let
$$
p_y (x) = \int \limits _ {\mathbb R^n} e^{-2 \pi i t\cdot x} e^{-2 \pi
|t| y} \, dt = \frac{ c_n y}{(|x|^2 + y^2 )^{\frac{n+1}2}}
$$
be the usual $n$-dimensional  Poisson kernel and let $P_N =
p_{\frac1{N}}$.

We will use the following properties of $P_N$.
\begin{enumerate}
\item $P_N \ge 0$
\item $\int P_N = 1 $
\item  $|\nabla P_N| \le C N P_N $ pointwise.
\item If $N_1 > N$ and $t\in \mathbb R ^n$
$$
\int \limits _ { \mathbb R^n} | P_N (t + \tau) - P_N(t)| P_{N_1}(\tau)
\, d\tau \le \frac{N}{N_1} \log (2 + \frac{N}{N_1} ) P_N(t)
$$
\item If $N_1 > N$ and $t\in \mathbb R^n$ 
$$
|P_N * P_{N_1} - P_N | \le (\frac{N}{N_1})^{\frac12} P_N(t)
$$
\end{enumerate}

The proofs, being exactly as in section 1 of \cite{Bourgain1} are omitted.

\section{Two lemmas on convolution kernels}
To prove theorems A and B we will need two lemmas which give pointwise
control of certain convolution kernels.  The lemmas are stated here
for reference and proved in section 7.

Let $\omega$ denote a fixed unit vector in $\mathbb R^n$.

\lemma
For each positive integer $j$ there is a function $Y_j$ such that
$$
\widehat{Y_j}(\xi) =\bigl (\widehat{P}_{2^{k-2}}(\xi)
\widehat{P}_{2^{k-1}}(\xi - 2^{k-1} \omega)\bigr)^{-1} 
$$
on 
$$ 
\frac14 2^{j-1} \le |\xi| \le \frac32 2^{j-1}
$$
and
$$
|Y_j(t) | \le \frac{c}{ 2^j(|t|^2 + 4 ^{-j} ) ^{\frac{n+1}2}}
$$
where $c$ is some numerical constant.
\em

For the second lemma, let $\sigma _j$ denote radial multipliers
such that
\begin{enumerate}
\item $0\le \sigma _ j (r) \le 1$
\item $\mbox{supp }\sigma _ j(r) \subset [\frac14 2^{j-1}, \frac32
2^{j-1} ]$
\item $\sum \limits _ j \sigma_j \equiv 1 $
\item $|\sigma_j^{(s)}| \le c 2^{-js}, \qquad s= 1,2,3,\dots$
\item $\sigma _j(r)$ vanishes to all orders at the endpoints of its
support.
\item $\sigma_j (r)= Q(2^{-(j+1)}r) - Q(2^{-j} r)$ for some fixed
 $C^{\infty}$ function $Q$ with compact support.
\end{enumerate}
\lemma
For any positive integer $j$ and positive real $y$, the kernels
$$
Z_{0,j,y}(x)= \int\limits _{\mathbb R^n} \frac{|s|}{2^j} e^{-2\pi |s|
y} e^{2\pi \frac{2^j}{10} y} \sigma _ j(s) e^{-2\pi i s\cdot x} \, ds
$$
and
$$
Z_{k,j,y}(x)= \int\limits _{\mathbb R^n} \frac{s_k}{2^j} e^{-2\pi |s|
y} e^{2\pi \frac{2^j}{10} y} \sigma _ j(s) e^{-2\pi i s\cdot x} \, ds
$$
for $k=1,2,\dots,n$, satisfy
$$
|Z_{\ell,j,y}(x)| \le \frac{c}{2^j(|x|^2 + 4^{-j} )^{\frac{n+1}2}}
$$
for $\ell = 0,1,\dots,n$. The constant is independent of $y$ and $j$.
\em
\section{Littlewood -Paley estimates for vertical variation}
Let $f$ denote the boundary values of the bounded harmonic function
$u$ in the upper half space $\mathbb R^{n+1}$. Let the multipliers
$\sigma_j$ be as in lemma 3.2.
 Considering the tempered distribution $\widehat{f}$ we estimate
\begin{eqnarray}
|\frac{\partial u}{\partial x_k}|& \le &
2\pi \left| \int s_k e^{-2\pi |s| y } e^{- 2 \pi i s \cdot x} \widehat{f}
(s) \, ds \right| \nonumber\\
&\le& 2 \pi \sum \limits _ j 2^j e^{-2\pi \frac{2^j}{10} y } \left| f * \int
\frac{s_k}{2^j} e^{-2 \pi | s| y } e^{2 \pi \frac{2^j}{10} y } \sigma
_ j (s) e^{- 2 \pi i s \cdot x} \, ds \right| \nonumber
\end{eqnarray} 
and similarly
\begin{eqnarray}
|\frac{\partial u}{\partial y}| \le 
 2 \pi \sum \limits _ j 2^j e^{-2\pi \frac{2^j}{10} y } \left| f * \int
\frac{|s|}{2^j} e^{-2 \pi | s| y } e^{2 \pi \frac{2^j}{10} y } \sigma
_ j (s) e^{- 2 \pi i s \cdot x} \, ds \right|. \nonumber
\end{eqnarray}

Since 
$$
\int \limits _ 0 ^ 1 2^j e^{-2\pi \frac{2^j}{10} y } \, dy \le \frac{10}{2\pi},
$$
we find that
$$
\int \limits _0 ^1 |\nabla u (x,y)| \, dy 
\le C  \sum \limits _{\ell = 0}^n \sum \limits _j |f*Z_{\ell,j,y}(x)|.
$$
Recalling lemmas 3.1 and 3.2, write
$$
X_j =P_{2^{j-2}} *(e^{2\pi i 2^{j-1}x\cdot \omega}P_{2^{j-1}})
$$
then
\begin{eqnarray}
|f*Z_{\ell,j,y}|& =& |f * X_j * Y_j* Z_{\ell,j,y}|\nonumber\\
&\le&|f*X_j|*|Y_j| *|Z_{\ell,j,y}| \nonumber \\
&\le&  C|f*X_j|*P_{2^{j-1}}. \nonumber
\end{eqnarray}
So we have 
$$
\int \limits _0 ^1 |\nabla u (x,y)| \, dy 
\le C \sum \limits _j |f*X_j|*P_{2^{j-1}}(x).
$$

To prove theorems A and B we  show  in the next two sections
that these last expressions on
the right are finite  on a non- empty closed set.
Then we estimate the dimension of these sets in section 7.

\section{Proof of Theorem A at a point}


As before, let $f$ denote the boundary values of the bounded harmonic function
$u$ in the upper half space $\mathbb R^{n+1}$. Let $\omega$ denote a fixed
unit
vector in $\mathbb R^{n}$.
 By the semi-group property of the Poisson kernel we can define 
$$
f_k = f*P_{2^{k-2}} = f * P_{2^k}*P_{2^k}*P_{2^{k+1}}*
P_{2^{k+1}}*\dots
$$

For each $x \in \mathbb R^n$ and for each angle $\psi$
\begin{eqnarray}
\int \limits _ { \mathbb R^n}f_{k+1}(\eta) P_{2^k} (x - \eta) (1 + \cos
(2 \pi 2^k (x- \eta)\cdot \omega  + \psi) \, d\eta \nonumber \\
= f_{k+1} * P_{2^k} (x) + \mbox{Re }[e^{i\psi} f_{k+1} *(e^{2\pi i 2^k
\omega \cdot  x } P_{2^k}) (x)] \nonumber
\end{eqnarray}
So we may choose $\psi = \psi _ x$ such that
\begin{eqnarray}
\int \limits _ { \mathbb R^n}|f_{k+1}(\eta)| P_{2^k} (x - \eta) (1 + \cos
(2 \pi 2^k (x- \eta)\cdot \omega  + \psi_x) \, d\eta \nonumber \\
\ge  |f_{k+1} * P_{2^k} (x)| + | f_{k+1} *(e^{2\pi i 2^k
\omega \cdot  x } P_{2^k}) (x)| \nonumber
\end{eqnarray}

Now with
$$
L(x^k, \eta) = \int P_{2^k}(x^k - x) P_{2^k}(x - \eta) (1 + \cos(2 \pi
2^k (x- \eta)\cdot \omega  + \psi_x) \, dx
$$ 
we have
\begin{eqnarray}
&{}&\int |f_{k+1}(\eta)| L(x^k, \eta) \, d \eta  \\
&\ge& | f_{k}(x^k)| + | f_{k+1}*(e^{2\pi i 2^k
\omega \cdot  x } P_{2^k})|*P_{2^k}(x^k). \nonumber
\end{eqnarray}

Suppose we have found points $x^1,\dots, x^k \in \mathbb R^n$ and 
constants
$c_j ^{\ell} >c>0$
 for $0\le j\le \ell$ and $0\le \ell \le k$  such that
\begin{equation}
|f_k(x^k)| \ge \sum \limits _{j=0} ^k c_j^k(|f_j * (e^{2\pi 2^{j-1}
x\cdot\omega} P_{2^{j-1}})|*P_{2^{j-1}})(x^k) - C 2^{-\frac{j}2} .
\end{equation}
Writing
$$
g_j = f_j*(e^{2\pi 2^{j-1}
x\cdot\omega} P_{2^{j-1}})
$$
and using the properties in section 1 we have 
\begin{eqnarray}
&{}&\iint |g_j(x)| | P_{2^{j-1}}(\eta - x) - P_{2^{j-1}}(x^k - \eta)|
L(x^k, \eta)\, dx\,d\eta \nonumber \\
&\le& C2^{-\frac{k-j}2}(|g_j|*P_{2^{j-1}})(x^k), \nonumber  
\end{eqnarray}
which implies
\begin{eqnarray}
\int(|g_j|*P_{2^{j-1}})(\eta) L(x^k, \eta) \, d\eta 
\le (1+ C2^{-\frac{k-j}2})(|g_j|*P_{2^{j-1}})(x^k).
\end{eqnarray}
Putting  together $(1),(2),(3)$  we get
\begin{eqnarray}
&{}&\int |f_{k+1}(\eta)| L(x_k, \eta) \, d \eta \nonumber \\
&\ge&\int \left(\sum \limits _{j=0} ^k (1+ C 2^{-\frac{k-j}2})^{-1} c_j^k(|g_j|*P_{2^{j-1}})(\eta) 
+ \frac1{C}(|g_{k+1}|*P_{2^k})(\eta)      \right)L(x^k, \eta) \,d\eta.
\nonumber 
\end{eqnarray}

As $L$ is positive and
$$
\int \limits _{|\eta - x^k| > 2^{-\frac{k}2}} L(x^k,\eta) \, d\eta \le
C 2^{-\frac {k}2},
$$
 we find a point $x^{k+1}$ such that $|x^k - x^{k+1}|
<2^{-\frac{k}2}$ and   satisfying  $(2)$ with
$k$ increased to $k+1$. By induction, there is a constant $C>0$ and a
bounded  closed interval $I$ centered at $x^0$ such
that  the decreasing sequence of
closed sets 
$$
\{x\in \mathbb R^n : \sum \limits_{j=0}^k (|g_j|*P_{2^{j-1}}(x) \le C
\}\cap I
$$
has non-empty intersection. So there is a point  $x\in R^n$ such that
$$
\sum \limits_{j=0}^{\infty} (|g_j|*P_{2^{j-1}})(x) \le C
$$
and by section 4 such a point satisfies
$$
\int \limits _0 ^1 |\nabla u(x,y)| \, dy \le C'<+\infty.
$$





\section{Proof of Theorem B at a point}

The argument from the previous section only requires a few changes.
Recall that any positive harmonic function $u$ in the upper half space
$\mathbb R^{n+1}$ has the form
$$
\int \limits _{\mathbb R^n} p_y(x-t) \, d\mu(t) + cy, \quad c\ge 0
$$
where $\mu$ is a positive Borel measure satisfying
$$
\int \limits _{\mathbb R^n} \frac{d\mu}{(1+ |t|^2)^{\frac{n+1}2}}.
$$

We may assume that $c=0$ and now let
$$
f_k = \mu * P_{2^k}*P_{2^k} * P_{2^{k+1}}* P_{2^{k+1}}*\dots
$$
Choose $\psi_x$ so that
\begin{eqnarray}
\int \limits _ { \mathbb R^n}f_{k+1}(\eta) P_{2^k} (x - \eta) (1 + \cos
(2 \pi 2^k (x- \eta)\cdot \omega  + \psi_x) \, d\eta \nonumber \\
=  f_{k+1} * P_{2^k} (x) - | f_{k+1} *(e^{2\pi i 2^k
\omega \cdot  x } P_{2^k}) (x)|. \nonumber
\end{eqnarray}
Forming the kernel $L$ as before and assuming inductively that
\begin{equation}
f_k(x^k) \le f_0(x^0) - \sum \limits _{j=0} ^k c_j^k(|f_j * (e^{2\pi 2^{j-1}
x\cdot\omega} P_{2^{j-1}})|*P_{2^{j-1}})(x^k) .
\end{equation}
for points $x^k$ and constants $c_j^k >c>0$, we find in the same way a
point $x^{k+1}$ and constants $c_j^{k+1} >c>0$ such that $(4)$ holds
with $k$ increased to $k+1$. To get a decreasing sequence of compact
sets, replace the kernels $L$ by $L\cdot \chi_{\{|x-\eta| <
2^{-\frac{k}2}\}}$ and change the induction assumption to 
\begin{equation}
f_k(x^k) \le \prod \limits _{j\le k} (1-c 2^{-\frac{j}2})^{-1} f_0(x^0) - \sum \limits _{j=0} ^k c_j^k(|f_j * (e^{2\pi 2^{j-1}
x\cdot\omega} P_{2^{j-1}})|*P_{2^{j-1}})(x^k) .
\end{equation}
We can do this because all the $f_k$ are positive.
 
Now we find a point $x^{k+1}$ such that $|x^k - x^{k+1}| <
2^{-\frac{k}2}$
and such that $(5)$ holds with $k$ increased to $k+1$.

The rest of the proof is finished as before, replacing $f$ with $\mu$
and $\hat{f}$ with $\hat{\mu}$ in section 4.
 \section{Proofs of the lemmas}
\subsection{Proof of lemma 3.1}
It will be  enough to consider 
$$
(\hat{P}_{2^{j-2}}(\xi))^{-1} = e^{2 \pi |\xi| 2^{-(j-2)}}.
$$
We will define a multiplier which is equal to this function on
a sufficiently large ball so that the translated multiplier will also
have the desired properties.
 Then the
lemma will follow because the convolution of two functions satisfying
the required  pointwise estimate will be the desired function $Y_j$.

Let $\hat{S}_ {2^{k-2}}$ be a $C^{\infty}$ function with derivatives
satisfying the pointwise bound
$$
|\hat{S}^{(m)} | \le C 2^{-m |k|}
$$
 and such that
$$
\hat{S}_ {2^{k-2}}(\xi)= 
\begin{cases}
e^{2\pi |\xi| 2^{-(k-2)}}& \text{if $0\le |\xi| \le \frac32
2^{(k+1)}$},\\
e^{48 \pi}e^{-2 \pi |\xi| 2^{-(k-2)}}& \text{if $ |\xi| \ge \frac74
2^{(k+1)}$}.
\end{cases}
$$
Let 
$$
\hat{T}_{2^{k-2}}= e^{48 \pi}e^{-2 \pi |\xi| 2^{-(k-2)}} -
\hat{S}_{2^{k-2}}.
$$
Then
$$
\hat{T}_{2^{k-2}} = 2 e^{24 \pi} \sinh(24\pi - 2\pi |\xi| 2^{-(k-2)})
$$
on the set $[-\frac32 2^{k+1},\frac32 2^{k+1}]$, 
  vanishes to
infinite
order at $\pm \frac 74 2^{k+1}$ and has 
$$
|\hat{T} _ {2^{k-2}}^{(m)}| \le C 2^{-m|k|}
$$
pointwise for each $m$. Since $\hat{T}$ differs from $\hat{S}$ by
a Poisson multiplier, it is enough to estimate $T_{2^{k-2}}$.

Noting that the function $\hat{T}_{2^{k-2}}$ is radial and abusing notation
we have
\begin{eqnarray}
T_1(x)&=& \int \limits _ {\mathbb R^n}\hat{ T}_1(\xi) e^{-2\pi i x
\cdot \xi} \, d\xi \nonumber \\
&=& 2\pi |x|^{-(\frac{n-2}2)} \int \limits _ 0 ^{\infty}
\hat{T}_1 (r) J_{(\frac{n-2}2)} (2\pi |x| r )\, r^{\frac{n}2} \,
dr \nonumber
\end{eqnarray}
where $J_{\nu}$ denotes the Bessel function of order $\nu$, see for
example \cite{SteinWeiss} pg. 155.

By the result of Wong  in \cite{Wong1}, \cite{Wong2}, this is 
\begin{equation}
= \frac{C}{|x|^{n+1}} + O(\frac1{|x|^{n+2}}), \qquad |x| >M_0
\end{equation}
for some $M_0 >0$.
 Now by the  properties of Fourier transforms when composed with
dilations
$$
T_{2^{k-2}}(x) = \frac{C2^{-(k-2)}}{|x|^{n+1}} +\dots \qquad 2^{k-2}|x|
> M_0.
$$
Since each of  the $T_{2^j}$ is bounded by $C2^{nj}$, the required
pointwise estimate follows.

For the reader's convenience we briefly explain Wong's proof of (6).

For $x>0$, $\eta \in \mathbb R$ and $\mbox{Re }(\mu + \alpha) >0$,
$$
\int \limits _ 0 ^{\infty} e^{-\eta^2 t^2} J_{\alpha}(x t) t^{\mu -1}
\, dt = \frac{x^{\alpha} \Gamma(\frac{\mu}2 + \frac{\alpha}2)
e^{-\frac{x^2}{4\eta^2}}}{ 2^{\alpha+1} \eta^{\alpha + \mu}
\Gamma(\alpha + 1)}\, { _{1}F_1}(\frac{\alpha}2 - \frac{\mu}2 + 1; \alpha +
1 ; \frac{x^2}{4\eta^2})
$$
where $_{1}F_1$ denotes Kummer's hypergeometric function. (See
\cite{EMOT} pg. 50
or compute the integral on Mathematica.)

Kummer's function satisfies
$$
_{1}F_1 (0;c;z)\equiv 1 , \qquad z>0
$$
and 
$$
_{1}F_1 (a;c;z) \sim \frac{\Gamma(c)}{\Gamma (a)} e^z z^{a-c} \qquad
z\to \infty.
$$
So if $\frac{\alpha}2 - \frac{\mu}2 +1 =0$ then
$$
\lim _{\eta \to 0} \int \limits_0 ^{\infty} e^{-\eta^2 t^2} J_{\alpha}(x t) t^{\mu -1}
\, dt = 0
$$
and otherwise
$$
= \frac{\Gamma(\frac{\mu}2 + \frac{\alpha}2) 2^{\mu
-1}}{\Gamma(\frac{\alpha}2-
\frac{\mu}2 + 1) x^{\mu}}.
$$
Writing
$$
\hat{T}_1(r) = c_0 + c_1 r + c_2 r^2 + \epsilon(r)
$$
where $\epsilon(r) = O(r^3)$ as $r\to 0^+$, it follows that
$$
T_1(x) = \frac C{|x|^{n+1}} + \frac{ C'}{|x|^{n+2}} + \delta(|x|)
$$
where
\begin{equation} 
\delta(|x|)= 2\pi |x|^{-(\frac{n-2}2)} \lim _{\eta \to 0} \int \limits
_ 0 ^{\infty} e^{-\eta^2 r^2} J_{(\frac{n-2}2)}(x r) \epsilon(r)
r^{\frac n2}
\, dr.
\end{equation}

We require the following well known properties of Bessel functions
\begin{enumerate}
\item $\frac{d}{dt}[t^{\nu + 1} J_{\nu +1} (t) ] = t^{\nu + 1}
J_{\nu}(t)$
\item $J_{\alpha} (t) \sim \frac{ t^{\alpha}}{2^{\alpha} \Gamma(\alpha
+ 1)} \qquad t\to 0$
\item $J_{\alpha} (t) = \sqrt{\frac2{\pi t}}\cos(t-\frac{\alpha \pi}2
- \frac{\pi}4 ) + O(t^{-\frac32}) \qquad t\to \infty$
\end{enumerate}

Define inductively
\begin{eqnarray}
\epsilon_0 (r)& =& \epsilon(r) r^{\frac{n}2} \nonumber \\
\epsilon _ {j+1} &=& \epsilon_j'(r) - (\frac{n-2}{2} + j + 1) \epsilon
_j(r) \frac1r. \nonumber
\end{eqnarray}

Using the properties $1-3$ and integrating the integral in (7)
repeatedly by parts gives
$$
 |x|^{\frac{n-2}2} \delta(|x|)= \frac{2\pi}{|x|^m} \lim _{\eta \to 0} \int \limits _ 0 ^
\infty e^{\eta^2 r^2} \epsilon _m(r) J_{(\frac{n}2 + m-1 )}(2 \pi |x| r )
\, dr
$$
where $m$ is an integer such that $\frac{n+5}2 <m < \frac{n+8}2 $ .
Using $1-3$ again and the definition of $T$ shows that
$$
\delta(|x|) = o(\frac1{|x|^{n+2}})
$$
and completes the proof of $(6)$.
\subsection{Proof of lemma 3.2}
A change of variables shows that 
$$
Z_{0,j,y}(x) = 2^{nj} Z_{0,1,2^j y} (2^j x).
$$
So it is enough to show that
$$
|Z_{0,1,y}| \le \frac{C}{|x|^{n+1}}
$$
for a constant $C$ which is independent of $y$.

But this follows by integrating by parts $m= [\frac{n}2 + 2]$ times
using the above property 1 of Bessel functions, the definition of
$\sigma$ and the fact that $y^m e^{-2 \pi r y } e^{2 \pi \frac{y} {
10}}$
is bounded independently of $y$ when $r\in [\frac18,\frac34]$.
 For $k>0$, notice that
$$
2^j Z_{k,j,y} (x) = \frac{\partial}{\partial x_k}
  \int\limits_{\mathbb R^n}
  e^{-2\pi |s|
y} e^{2\pi \frac{2^j}{10} y} \sigma _ j(s) e^{-2\pi i s\cdot x} \, ds.
$$
The pointwise estimate follows again from repeated integration by parts.
\section{Dimension estimates}
Recall {\em Polya's criterion} that a real nonnegative even  function
$\phi(t)$ on $\mathbb R$ with $\phi(0) = 1$ and which is convex on
$(0,\infty)$ can be represented as
$$
\phi(t) = \int \limits _ 0 ^{\infty} \bigl(1- \frac{|t|}{|s|}\bigr)^+
\,\nu(ds)
$$ 
where $\nu$ is a probability measure on $(0,\infty]$. Since
$(1-|t|)^+$ is the Fourier transform of a positive function,
so is any $\phi$ satisfying Polya's criterion. See for example
\cite{Durrett}
pg.87.

Let $\phi$ be a function satisfying Polya's criterion which is  $C^{\infty}$ in $(0,\infty)$,
supported in $[-1,1]$, vanishes to
all orders at 1 and is greater than $\frac1{10}$ on
$(\frac18,\frac34)$.
We may require further that 
$$
\phi = 1 - ct \qquad t\in (0,\frac1{16})
$$
for some $c>1$.
 
Let $\hat{\Phi}$ denote the radial Fourier  multiplier in $\mathbb R^n$ such that 
$\hat{\Phi}(\xi)= \hat{\Phi}(|\xi|)= \phi(|\xi|)$ and let
$\Phi_N (t)$ denote the function whose Fourier transform is
 $\hat{\Phi}(\frac{\xi}{N})$.

By Polya's criterion and Wong's technique from the proof of lemma 3.1
$\Phi_N$ is a positive function satisfying the pointwise bound
$$
\Phi_N(t) \le \frac{C} { N (N^{-2} + |t| ^2)^{\frac{n+1}2}}.
$$
Now define $K_N= P_N*\Phi_N$. Then $K_N$ satisfies the properties 1-5
in section 2 and has its Fourier transform supported in the ball of
radius $N$. Since $\phi $  is linear in $(0,\frac1{16})$, there is a
constant $c'>0$ such that  the Fourier transform of 
$$
K_{2^k}*K_{2^k}*K_{2^{k+1}}*K_{2^{k+1}}\dots
$$
is bounded below by $c'$ on $[\frac14 2^{(k-1)}, \frac32 2^{(k-1)}]$.

All the previous arguments can now be made with $K$ replacing $P$.
In particular, lemma 3.1 is proved in a similar way.

Following Bourgain, we note that in each of the inductive arguments for theorems A and B, the average
of
$$
\sum \limits _ {j\le k} | f_j * (e^{2\pi i 2^{j-1} x\cdot\omega}
K_{2^{j-1}})|* K_{2^{j-1}}
$$
with respect to the probability density $\Omega_k$ defined by
\begin{eqnarray}
\Omega_k(\eta) = \int L_1(x_1,x_2) L_2(x_2,x_3)\dots
L_{k-1}(x_{k-1},\eta)
\,dx_1dx_2\dots dx_{k-1} \nonumber\\
L_k(x,y)= \int K_{2^k}(x-\eta) K_{2^k}(\eta - y) (1+ \cos(2\pi 2^k(\eta
- y) + \psi_{\eta} \, d\eta \nonumber
\end{eqnarray}
is  uniformly bounded in $k$.

Given $\epsilon >0$ we alter the construction as follows:
Replace $L_k$ with 
$$
L_k(x,y)= \int K_{2^k}(x-\eta) K_{2^k}(\eta - y) (1+
\epsilon\cos(2\pi(2^k + [\frac{2^k}{10}]) (\eta
- y) + \psi_{\eta}) \, d\eta 
$$
and replace $\Omega_k$ with the densities obtained from the new $L$.

Replace the Littlewood-Paley expressions which control the vertical
variation with
$$
\sum \limits _ {j\le k} | f_j * (e^{2\pi i (2^{j-1} +
[\frac{2^{j-1}}{10}])
 x\cdot\omega}
K_{2^{j-1}})|* K_{2^{j-1}}
$$
Neither of these changes effects the argument but now
because of the shift by $[\frac{2^{j-1}}{10}]$ the densities
$\Omega_k$ have a $\mbox{weak-}*$ limit which is a probability
measure $\mu$ on $\mathbb R^n$. This can be seen by integration of the
$\Omega_k$ against a fixed function with compactly supported Fourier transform.

The integral of 
$$
\sum \limits _0^ {\infty} | f_j * (e^{2\pi i (2^{j-1} + [\frac{2^{j-1}}{10}]) x\cdot\omega}
K_{2^{j-1}})|* K_{2^{j-1}}
$$
with respect to $\mu$ is finite, and if $\gamma$ is a function with
 $\hat{\gamma}$ supported in $[\frac{-2^k}{20}, \frac{2^k}{20}]^n$ then
\begin{eqnarray}
|\int \gamma \, d\mu | &\le& C \int \Omega _k (\eta) [|\gamma| *(K_{2^k}*K_{2^k}*K_{2^{k+1}}*K_{2^{k+1}}\dots]
 (\eta) \, d\eta \nonumber \\
&\le& C(1+\epsilon)^k \|\gamma\|_{L^1} \nonumber
\end{eqnarray}

Approximating the characteristic function of a cube $Q$ with
sidelength $|Q|$ in $L^1$ norm by functions with compactly supported
Fourier transform now shows that
$$
\mu (Q) \le |Q|^{n-\frac{\log(1+\epsilon)}{\log 2}}
$$
which easily implies that the Hausdorff dimension of the support of
$\mu$ is greater than $n-\frac{\log(1+\epsilon)}{\log 2}$.
By section 5, this shows that theorems A and B hold on a set with 
Hausdorff dimension $n$.




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\end{document}
