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\def\rightheadline{EJDE--1999/37\hfil Dini-Campanato spaces
\hfil\folio}
\def\leftheadline{\folio\hfil Jay Kovats 
 \hfil EJDE--1999/37}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol. {\eightbf 1999}(1999), No.~37, pp.~1--20.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break 
ftp ejde.math.swt.edu (login: ftp)\bigskip} }

\topmatter
\title
Dini-Campanato spaces and applications to nonlinear elliptic equations
\endtitle

\thanks 
{\it 1991 Mathematics Subject Classifications:} 35B65, 41A10. \hfil\break\indent
{\it Key words and phrases:} 
Fully nonlinear elliptic equations, polynomial approximation, \hfil\break\indent 
Dini condition. \hfil\break\indent
\copyright 1999 Southwest Texas State University  and
University of North Texas.\hfil\break\indent
Submitted January 6, 1999. Revised July 19, 1999. Published September 25, 1999.
\endthanks
\author  Jay Kovats  \endauthor
\address Jay Kovats \hfil\break 
Department of Mathematical Sciences \hfil\break
Florida Institute of Technology \hfil\break
Melbourne, FL 32901, USA
\endaddress
\email jkovats\@zach.fit.edu
\endemail

\abstract
We generalize a result due to Campanato [C] and use this to obtain regularity 
results for classical solutions of fully nonlinear elliptic equations. 
We demonstrate this technique in two settings. 
First, in the simplest setting of Poisson's equation $\Delta u=f$ in $B$, 
where $f$ is Dini continuous in $B$, we obtain known estimates on the modulus 
of continuity of second derivatives $D^2u$ in a way that does not depend on 
either differentiating the equation or appealing to integral representations 
of solutions. 
Second, we use this result in the concave, fully nonlinear 
setting $F(D^2u,x)=f(x)$ to obtain estimates on the modulus of continuity of 
$D^2u$ when the $L^n$ averages of $f$ satisfy the Dini condition. 
\endabstract
\endtopmatter

\document
\head 0. Introduction \endhead

Let $1\le q\le \infty$ and let $\Omega$ be a domain in $\Bbb R^n$. For any 
Dini modulus of continuity $\omega(t)$ and $u \in L^q(\Omega)$, we define 
the seminorm
$$[u]'_{k,\omega}=[u]'_{q,k,\omega;\Omega}=\sup\Sb x_0 \in \overline{\Omega}
\\ 0<r \leq d(\Omega)\endSb \bigg[ {1 \over {r^{kq+n} \omega(r)^q}}\, 
\inf_{P \in {\Cal P}_k} \int\limits_{\Omega_r(x_0)} |u(x)-P(x)|^q dx 
\bigg]^{1/q},$$
where $\Omega_r(x_0)=\overline B_r(x_0) \cap \Omega$ and ${\Cal
P}_k$ denotes the spaces of polynomials of degree less than or equal to $k$. 
We define the Dini-Campanato space ${\Cal M}_q^{k,\omega}(\Omega)$ as the 
space of functions 
$${\Cal M}_q^{k,\omega}(\Omega)=\left\{u \in  L^q(\Omega):[u]'_{q,k,\omega;
\Omega} < + \infty\right\}.$$

Following Campanato's original proof (in [C]) of the inclusion
${\Cal L}_k^{(q,\lambda)}(\Omega)\subset C^{k,\alpha}(\Omega)$, for 
$\alpha =(\lambda-n-kq)/q$, we obtain the regularity
result ${\Cal M}_q^{k,\omega}(\Omega) \subset C^{k,\omega_1}(\Omega)$,
under the assumption that $\omega(t)$ is a Dini modulus of
continuity and $\omega_1(t)=\int_0^t\frac{\omega(r)}r \,dr$. If $\omega(t)$
is a modulus of continuity, the space
$C^{k,\omega}(\Omega)$ is defined in obvious generalization of the H\"older
spaces, namely the space of all $u \in C^k(\Omega)$ with seminorm 
$$[u]_{k,\omega;\Omega}=\sup\Sb x,y \in \Omega\\ |\beta|=k \endSb
\frac {|D^\beta u(x)-D^\beta u(y)|}{\omega(|x-y|)} <+\infty\,.$$
Our present result is more general
than Campanato's original inclusion. Indeed, if $u \in {\Cal
L}_k^{(q,\lambda)}(\Omega)$, then $\omega(r)\sim
r^\beta$ for some $\beta \in (0,1]$ and $\lambda=kq+n+\beta
q$. Yet for $\omega(r)=r^\beta, \omega_1(t)=\int_0^t r^{\beta
-1},dr \sim t^\beta$ and so by our present result $u \in
C^{k,\omega_1}=C^{k,\beta}$, whereas by Campanato's original inclusion,
$u \in C^{k,\alpha}$ for
$\alpha=\frac{\lambda-n-kq}q=\frac{(kq+n+\beta q)-n-kq}q=\beta$.
On the other hand, there are examples of $u \in {\Cal
M}_q^{k,\omega}$, where $u \notin C^{k,\alpha}$ for any $\alpha >0$. 
The special case $k=0,\, q=1$ was proved by Spanne [Sp]. 

In [C1],[CC], L. Caffarelli uses polynomial approximation to obtain 
{\it pointwise} H\"older estimates for derivatives of viscosity solutions to 
fully nonlinear elliptic equations. In the special case 
$\omega (t) \sim t^\alpha,\, C^{1,\alpha}$ estimates involve approximation by 
affine functions ($q=\infty, k=1$), while  $C^{2,\alpha}$ estimates involve 
approximation by paraboloids ($q=\infty,k=2$).  Using a generalization of the
argument in Chapter 8 of [CC], we use the Dini Campanato inclusion to obtain 
regularity results for solutions of fully nonlinear elliptic equations. 
We illustrate this technique in two settings. In Chapter 2, in the simplest 
setting of Poisson's equation $\Delta u=f$ in $B$, where $f$ is Dini continuous 
in $B$, we obtain known estimates on the modulus of continuity of second 
derivatives $D^2u$ in a way that does not depend on either differentiating the 
equation or appealing to integral representations of solutions. In Chapter 3, 
we use this technique in the concave, fully nonlinear setting $F(D^2u,x)=f(x)$ 
to obtain estimates on the modulus of continuity of $D^2u$, when $f$ and the 
oscillations of $F$ in $x$ are Dini continuous. Here, Dini continuity is 
measured in the weaker setting of $L^n$ averages instead of the usual 
$L^\infty$ norm. This condition was proposed by Wang (see his closing remark 
of Section 1.1) in [W].

Finally, we remark that even in the simplest setting of Poisson's equation, 
second derivatives of $C^2$ solutions will not, in general, be Dini continuous 
even when $f$ is. For example, direct calculation shows that the function
$$u(x)=u(x_1,x_2)=x_1x_2\left(\ln\frac1{|x|}\right)^{-1},\quad x \in B
=B_{1/2}(0)$$
satisfies $$\Delta u(x)=\frac{x_1x_2\left( \ln\frac1{|x|}
\right)^{-2}}{|x|^2} \left( \frac{2}{\ln\frac1{|x|}}
+4\right):=f(x)$$
in $B$, where $f(x)=O\bigg(\big(\ln\frac1{|x|}\big)^{-2}\bigg)$ is Dini 
continuous in $B$ with Dini modulus of continuity 
$\omega(t)\sim \left(\ln\frac1{t}\right)^{-2}$. However direct calculation 
shows that 
$$D_{12} u(x)=\frac{\partial^2u(x)}{\partial x_1 \partial x_2}
=O\bigg(\big( \ln\frac1{|x|}\big)^{-1}\bigg)$$ 
has modulus of continuity 
$\sim \left(\ln\frac1{t}\right)^{-1}$, which fails the Dini condition, since 
for any $\varepsilon >0$
$$\int_0^\varepsilon \frac{\left(\ln\frac1{r}\right)^{-1}}r\,dr=
\int_{\ln\frac1{\varepsilon}}^{\infty} u^{-1}\,du= +\infty\,.$$
It is well known, (see [GT] Chapter 4) that if $u \in C^2(B_r)$ satisfies 
$\Delta u = f$ in $B_r$, where $f \in C^\alpha(B_r)$, then 
$D^2u \in C^\alpha(B_{r/2})$. This ``reproducing'' regularity occurs not 
only for $\omega(t)=t^\alpha$ but more generally, for 
$\omega(t)=t^\alpha\left(\ln\frac1{t}\right)^{\beta},\alpha \in (0,1)$. 
This can be seen by noting that both integrals in (13) are 
$\sim t^\alpha\left(\ln\frac1{t}\right)^{\beta}$, when 
$\omega(t)=t^\alpha\left(\ln\frac1{t}\right)^{\beta}$. See also [B],[K].

We recall that any modulus of continuity
$\omega(t)$ is non-decreasing, subadditive, continuous and
satisfies $\omega(0)=0.$ Hence any modulus of continuity $\omega(t)$ satisfies
$$ \frac{\omega(r)}r\leq 2\frac{\omega(h)}h, \qquad 0<h<r.$$
Indeed, by subadditivity, for $m\in \Bbb N$ and $h>0$, we have
$\omega(mh)\le\ m \omega(h)$. Thus for $0<h<r$, 
$\omega(r)=\omega(\frac rh h)\le \omega(\left\lceil \frac rh
\right\rceil h)\le \left\lceil\frac rh \right\rceil\omega(h)\le 2\frac
rh \omega(h)$,
where $\lceil a\rceil$ denotes the smallest integer $\ge a$. In particular, it 
immediately follows that $\omega(t)\le 2 \omega_1(t)$, since for $t>0,\omega_1(t)
=\int_0^t\frac{\omega(r)}r \,dr \ge \frac{\omega(t)}{2t}\int_0^t\,dr
=\frac{\omega(t)}{2}$.


\head 1. The Dini-Campanato Inclusion 
${\Cal M}_q^{k,\omega} \subset C^{k,\omega_1}$ \endhead

We restrict ourselves to domains $\Omega \subset \Bbb R^n$ which satisfy the 
following property (this includes Lipschitz domains). 

\proclaim{Definition 1.0} We say that $\Omega$ satisfies property (I) if there
exists a constant $A>0$ such that $\forall x_0 \in \Omega,\forall
r\in [0,d(\Omega)]$, the Lebesque measure of
$\Omega_r(x_0),\,|\Omega_r(x_0)|$ satisfies
$$|\Omega_r(x_0)|\ge A r^n.$$ 
\endproclaim 

\proclaim{Main Theorem} Let $1\le q \le \infty$. If $u\in {\Cal M}_q^{k,\omega}
(\Omega)$, where
$\Omega$ satisfies property (I), then $u \in C^{k,\omega_1}(\Omega)$,
where $\omega_1(t)=\int_0^t\frac{\omega(r)}r \,dr$. That is, the $k$th
order derivatives of $u$ satisfy 
$$|D^ku(x)-D^ku(y)|\le C(n,k,q,A) \omega_1(|x-y|)\qquad \forall x,y 
\in \Omega.$$
\endproclaim
We begin the  proof of the main theorem for the case $1\le q <
\infty$ with a lemma due to De Giorgi.

\proclaim{Lemma 1.1 (De Giorgi)} If $P(x) \in {\Cal P}_k$, $q\ge 1$ and $E$ is 
any measurable subset of $\overline B_r(x_0)$ satisfying 
$$ |E|\ge A r^n,$$ 
then $\exists$ constant $c_1(k,q,n,A)$ such that $\forall$ n-tuple $p$ of 
non-negative integers, we have
$$\left|\left[D^p P(x)\right]_{x=x_0}\right|^q \leq {c_1 \over {r^{n+|p|q}}} 
\int_E |P(x)|^q\,dx.$$
\endproclaim
If $u \in {\Cal M}_q^{k,\omega}(\Omega)$, one can show that $\forall x_0
\in \Omega,\forall r \in [0,d(\Omega)],\exists P_k(x,x_0,r,u) 
\in {\Cal P}_k$ such that 
$$\int\limits_{\Omega_r(x_0)}|u(x)-P_k(x,x_0,r,u)|^q\,dx =\inf_{P 
\in {\Cal P}_k} \int\limits_{\Omega_r(x_0)}|u(x)-P(x)|^q\,dx.\eqno (1)$$
If $P(x)$ is an arbitrary polynomial in ${\Cal P}_k$, then for
convenience we write it in the form 
$$P(x)=\sum_{|p|\leq k}{{a_p} \over {p!}} (x-x_0)^p$$
and henceforth put $P_k(x,x_0,r)$ for $P_k(x,x_0,r,u)$ and set $$a_p(x_0,r)
={[D^p P_k(x,x_0,r)]}_{x=x_0}.\eqno(2)$$
 
\proclaim{Lemma 1.2} If $u \in {\Cal M}_q^{k,\omega}(\Omega)$, then 
$\forall x_0 \in \overline \Omega, \forall \,0<r \leq d(\Omega)$ and integer 
$h\geq 0,$ we have
$$\int_{\Omega_{\frac{r}{2^{h+1}}}(x_0)} \tsize\left|P_k
\left(x,x_0,{r \over {2^h}}\right)-P_k\left(x,x_0,{r \over {2^{h+1}}}\right)
\right|^q\,dx \leq 2^{q+1}  {[u]'}^q \omega\left({r \over {2^h}}\right) 
{\left({r \over {2^h}}\right)}^{kq+n}.\eqno(3)$$
\endproclaim

\demo {Proof} $\forall x_0 \in \overline \Omega, \, 0<r \leq
d(\Omega)$, integer $h\geq 0, \,x \in \Omega_{r \over{2^{h+1}}}(x_0)$, by (1) 
and the definition of ${\Cal M}_q^{k,\omega}(\Omega)$ we have
$$\eqalign{
&\int\limits_{\Omega_{r \over {2^{h+1}}}(x_0)}\tsize\left|P_k\left(x,x_0,
{r \over{2^h}}\right)-P_k\left(x,x_0,{r \over {2^{h+1}}}\right)\right|^q\,dx \cr 
&\leq 2^q\Big\{ \int\limits_{\Omega_{r \over {2^{h}}}(x_0)}\tsize\left|P_k
\left(x,x_0,{r \over {2^h}}\right)-u(x)\right|^q\,dx 
+\dsize\int\limits_{\Omega_{r \over {2^{h+1}}}(x_0)}\tsize\left|u(x)-P_k
\left(x,x_0,{r \over {2^{h+1}}}\right)\right|^q\,dx\Big\} \cr
&\tsize\leq 2^q\left\{  {[u]'}^q \omega\left({r \over {2^h}}\right)^q \, 
{\left({r
\over {2^h}}\right)}^{kq+n} + {[u]'}^q \omega\left({r \over {2^{h+1}}}
\right)^q  {\left({r \over {2^{h+1}}}\right)}^{kq+n}\right\} \cr
&\leq 2^{q+1} {[u]'}^q \omega\left({r \over {2^h}}\right)^q\, 
{\left({r \over {2^h}}\right)}^{kq+n}.\qed \cr
}$$
\enddemo

\proclaim{Lemma 1.3} If $\Omega$ has property (I) and 
$u \in {\Cal M}_q^{k,\omega}(\Omega)$, then $\forall$ two points 
$x_0,y_0 \in \overline \Omega$ and any n-tuple $p$ of integers with 
$|p|=k, \exists \,c_2=c_2(k,n,q,A)$ such that
$$|a_p(x_0, 2|x_0-y_0|)-a_p(y_0,2|x_0-y_0|)|^q \leq c_2 \,{[u]'}^q\, 
\omega(|x_0-y_0|)^q.\eqno(4)$$ 
\endproclaim
\demo{Proof} Say $x_0,y_0 \in \overline \Omega$ and put $r=|x_0-y_0|,
\,I_r=\Omega(x_0,2r)\cap \Omega(y_0,2r).$ Then $\forall x \in
\Omega(x_0,r)\subset I_r$, we have, again by (1) and the fact that
$\omega(t)$ is a modulus of continuity  
$$\eqalign{
&\int_{\Omega_{r}(x_0)}|P_k(x,x_0,2r)-P_k(x,y_0,2r)|^q\,dx \cr 
&\leq 2^q \Big\{\int_{\Omega_{2r}(x_0)}
|P_k(x,x_0,2r)-u(x)|^q \,dx + \int_{\Omega_{2r}(y_0)}|u(x)
-P_k(x,y_0,2r)|^q \,dx\Big\}  \cr 
&\leq 2^q\left\{2 {[u]'}^q \omega(2r)^q {(2r)}^{kq+n} \right\}\cr
& \le 2^{2q+1+kq+n} {[u]'}^q\, r^{kq+n} \omega(r)^q. \cr
}$$
Applying Lemma 1.1 to the polynomial $P(x)=P_k(x,x_0,2r)-P_k(x,y_0,2r)$, 
and observing that the $k$-th derivatives of a polynomial of degree $k$ are 
constant and hence can be evaluated independent of any particular point, 
we see that
$$\eqalign{
|a_p(x_0, 2r)-a_p(y_0,2r)|^q &\leq {c_1 \over {r^{n+kq}}}
\int_{\Omega_r(x_0)} |P_k(x,x_0,2r)-P_k(x,y_0,2r)|^q\, dx \cr
&\leq {c_1 \over {r^{n+kq}}} \, 2^{2q+1+kq+n} {[u]'}^q\, r^{kq+n}
\omega(r)^q \cr
&=c_2 {[u]'}^q\omega(r)^q. \qed \cr
}$$
\enddemo
\proclaim{Lemma 1.4} If $\Omega$ has property (I) and 
$u \in {\Cal M}_q^{k,\omega} (\Omega)$, then $\exists$ constant $c_3(q,k,n,A)$ 
such that $\forall x_0 \in \overline \Omega, 0<r \leq d(\Omega)$ and integer 
$i\geq 0, \,|p|\leq k$, we have
$$\left|a_p(x_0,r)-a_p\left(x_0,{r \over {2^i}}\right)\right| 
\leq c_3  {[u]'}\ \sum_{h=0}^{i-1} \left(\frac r{2^h}\right)^{k-|p|}  
\omega\left({r \over {2^h}}\right).\eqno (5)$$
\endproclaim

\demo{Proof} With $x_0,r,\,|p|\leq k$ as in the hypotheses, note that by (2) and (3)
$$\eqalign{
& \left|a_p(x_0,r)-a_p\left(y_0,{r \over {2^i}}\right)\right| \leq
\sum_{h=0}^{i-1}\tsize\left|a_p\left(x_0,{r \over {2^h}}\right)-a_p\left(x_0,{r \over
{2^{h+1}}}\right)\right| \cr
&=\sum_{h=0}^{i-1}\Bigl|D^p \tsize\left[P_k\left(x,x_0,{r \over {2^h}}\right)
-P_k\left(x,x_0,{r \over {2^{h+1}}}\right)\right]_{x=x_0}\Bigr| \cr
&\leq \sum_{h=0}^{i-1} \frac{c_1^{1/q}}{(\frac r{2^{h+1}})
^{\frac{n+|p|q}{q}}}  \Bigg\{ \int_{\Omega_{r \over {2^{h+1}}}(x_0)}\tsize
\left|P_k\left(x,x_0,{r \over {2^h}}\right)-P_k\left(x,x_0,
{r \over {2^{h+1}}}\right)\right|^q\,dx \Bigg\}^{1/q} \cr 
&\le \sum_{h=0}^{i-1}\tsize \frac{c_1^{1/q}}{(\frac
r{2^{h+1}})^{\frac{n+|p|q}{q}}}2^{\frac{q+1}q}
\left(\frac{r}{2^h}\right)^{\frac{n+kq}{q}}\omega\left(\frac r{2^h}\right) [u]' \cr
&\le c_1^{1/q}\, 2^{\frac{n+(k+1)q +1}{q}} [u]' 
\sum_{h=0}^{i-1} \tsize\left(\frac r{2^h}\right)^{k-|p|}  \omega\left({r \over
{2^h}}\right).\qed \cr
}$$
\enddemo

\proclaim{Lemma 1.5} Let $\Omega$ have property (I) and $u \in {\Cal
M}_q^{k,\omega} (\Omega)$, where  $\omega(t)$ is a Dini modulus of
continuity. Then for every $l$ with $0\leq l\leq k$, there exists a system of
functions $\{v_p(x_0)\}_{|p|\leq l}$, defined in $\overline \Omega$
such that $\forall \,0<r\leq d(\overline \Omega), x_0 \in \overline
\Omega$ and $|p|\leq l$, we have, for some constant $c_5=c_5(q,k,n,A)$
$$|a_p(x_0,r)-v_p(x_0)|\leq c_5[u]'\, r^{k-|p|}\omega_1(r),\quad \hbox{
where  }\quad  \omega_1(t)=\int_0^t\frac{\omega(r)}r \,dr.    \eqno(6)$$ 
$$\hskip -.75in \hbox{Consequently }\forall x_0 \in \overline \Omega, \,
\lim_{r\to 0} a_p(x_0,r)=v_p(x_0) \,\hbox{ uniformly} .\eqno(7)$$
\endproclaim

\demo{Proof} Fix $x_0 \in \overline\Omega, \,0<r\leq d(\Omega),\,|p|\leq l$. 
We will show that the sequence $\{a_p(x_0,{r \over {2^i}} )\}$ converges as 
$i\to \infty$. Indeed, if $i,j$ are non-negative integers with $j>i$, then by 
Lemma 1.4 we have 
$$\left|a_p\left(x_0,{r \over {2^j}}\right)-a_p\left(x_0,{r \over {2^i}}\right)
\right| \leq c_3 
[u]'  \sum_{h=i}^{j-1} \omega\left({r \over {2^h}}\right)\left({r \over {2^h}}
\right)^{k-|p|}.$$

But since $\omega(t)$ is a Dini modulus of continuity, the integral test, 
applied to the non-negative, non-increasing sequence $\{\omega({r \over {2^h}})
\}_{h=0}^{\infty}$ yields that the series $\sum_{h=0}^{\infty} 
\omega({r \over {2^h}})$ converges. Indeed, by the integral test
$$\sum_{h=0}^{\infty} \omega\left({r \over {2^h}}\right) \le \omega(r)
+\int_1^{\infty}\omega\left({r \over {2^{x-1}}}\right)\,dx=\omega(r) 
+ {1 \over {\ln2}}\int_0^r {{\omega(t)} \over t}\,dt\leq 
\Bigl(2+{1 \over {\ln2}}\Bigr)\omega_1(r).$$
Thus, $\{a_p(x_0,{r \over {2^i}} )\}$ is a Cauchy sequence, and hence 
convergent. Moreover the limit will be independent of our choice of 
$r \in  [0,d(\Omega)].$ Indeed, if $r_1, r_2$ satisfy 
$0<r_1\leq r_2\leq d(\Omega)$, then by Lemma 1.1 and the definition of 
${\Cal M}_q^{k,\omega}(\Omega)$, we get that
$$\eqalign{
&\tsize\left|a_p\left(x_0,{r_1 \over {2^i}}\right)-a_p\left(x_0,{r_2 \over {2^i}
}\right)\right|^q \leq \Bigl|D^p \left[P_k\left(x,x_0,{r_1 \over {2^i}}\right)
-P_k\left(x,x_0,{r_2 \over {2^i}}\right)\right]_{x=x_0}\Bigr|^q \cr 
&\leq {c_1 \over ({r_1 \over {2^i}})^{n+|p|q}} \int\limits_{
\Omega_{r_1 \over {2^i}}(x_0)} \tsize\left|P_k\left(x,x_0,{r_1
\over {2^i}}\right)-P_k\left(x,x_0,{r_2 \over {2^i}}\right)\right|^q \,dx \cr
&\le{{c_1 2^q} \over ({r_1 \over {2^i}})^{n+|p|q}} \Big\{
\int\limits_{\Omega_{r_1 \over {2^i}}(x_0)}\left|P_k\left(x,x_0,\tsize{r_1 
\over {2^i}}\right)-u(x)\right|^q\,dx
+\int\limits_{\Omega_{r_2 \over {2^i}}(x_0)} \left|u(x)-P_k\left(x,x_0,
\tsize{r_2
\over {2^i}}\right)\right|^q \,dx \Big\}\cr
&\le {{c_1 2^q} \over ({r_1 \over {2^i}})^{n+|p|q}} \left\{ {[u]'}^q
\omega\left({r_1 \over {2^i}}\right)^q  {\left({r_1 \over {2^i}}\right)}^{kq+n}
 +{[u]'}^q \omega\left({r_2 \over {2^i}}\right)^q  {\left({r_2 \over {2^i}}
 \right)}^{kq+n} \right\}\cr
&\leq {{c_1 2^{q+1} {[u]'}^q} \over ({r_1 \over {2^i}})^{n+|p|q}}
\omega\left({r_2 \over {2^i}}\right) {\left({{r_2}\over {2^i}}\right)}^{kq+n}\cr
&= c_1 2^{q+1} {[u]'}^q \omega\left({r_2 \over {2^i}}\right)
\frac{r_2^{kq+n}}{r_1^{|p|q+n}}\, 2^{iq(|p|-k)} \quad \to 0 \hbox{
as }i\to \infty \quad \hbox{ (even if } |p|=k).\cr
}$$
Thus, $\{a_p(x_0,{r \over {2^i}} )\}$ converges independent of our choice of 
$r \in  [0,d(\Omega)]$ and $\forall x_0 \in \overline \Omega,\, 
0<r\leq d(\Omega), \,|p| \leq l\leq k,\lim\limits_{i\to \infty}a_p(x_0,
{r\over {2^i}})=v_p(x_0).$ Furthermore, from (5) in Lemma 1.4 we have
$$\left|a_p(x_0,r)-a_p\left(x_0,{r \over {2^i}}\right)\right| \leq c_3 \, [u]'
\, r^{k-|p|}\sum_{h=0}^{i-1} \omega\left({r \over {2^h}}\right).$$
Letting $i \to \infty$, we get
$$|a_p(x_0,r)-v_p(x_0)| \leq c_3 \, [u]'\, r^{k-|p|} 
\sum_{h=0}^{\infty} \omega\left({r \over {2^h}}\right)\leq c_5 \, [u]'
\, r^{k-|p|} \omega_1(r).\qed \eqno(8)$$
\enddemo

\proclaim{Theorem 1.6} If $\Omega$ has property (I) and $u \in {\Cal
M}_q^{k,\omega} (\Omega)$, where  $\omega(t)$ is a Dini modulus of
continuity, then the functions $v_p(x)$ with $|p|=k$ have modulus of
continuity $\omega_1(t)$ in $\overline \Omega$ and $\forall x,y \in
\overline \Omega$, we have, with $\omega_1(t)=\int_0^t\frac{\omega(r)}r \,dr$ 
$$|v_p(x)-v_p(y)|\leq c_6 \, [u]'\omega_1(|x-y|),\eqno(9)$$
for some constant $c_6=c_6(k,q,n,A)$.
\endproclaim

\demo{Proof} Fix an n-tuple of nonnegative integers $p=(p_1,...,p_n)$
with $|p|=k$ and let $x,y \in \overline \Omega$. Since $\Omega$ is
connected, we may assume that $r=|x-y|\leq {d(\Omega)\over 2}$. By
Lemma 1.5 (with $2r$ in place of $r$), Lemma 1.3 and the fact that
$\omega_1(mr)\le m\omega_1(r)\forall m \in \Bbb N$, we have
$$\eqalign{
|v_p(x)-v_p(y)| &\leq |v_p(x)-a_p(x,2r)| +|a_p(x,2r)-a_p(y,2r)| +|a_p(y,2r)
-v_p(y)|\cr
&\leq  c_5  [u]'\,  \omega_1(2r)+ c_2^{1/q} [u]'\,  \omega(r) + c_5  [u]'\omega_1(2r) \cr
&\leq c_6  [u]'  \omega_1(|x-y|).\qed \cr
}$$
\enddemo

\proclaim{Theorem 1.7} If $\Omega$ has property (I) and 
$u \in {\Cal M}_q^{k,\omega}(\Omega)$, where  $\omega(t)$ is a Dini modulus 
of continuity, then the functions $v_p(x)$ with $|p|\leq k-1$ have first order 
partial derivatives in $\Omega$ and $\forall x \in \Omega$, we have
$${\partial v_p(x) \over {\partial x_i}}=v_{(p+e_i)}(x) \quad 
(i=1,2,...,n).\eqno(10)$$
\endproclaim

\demo{Proof} By Theorem 1.6, the $v_p(x)$ with $|p|=k$ are continuous in 
$\overline \Omega$. Our theorem will be proved by induction under the 
additional assumption that the $v_{(p+\delta e_i)}(x)$ are continuous in 
$\overline \Omega$ for $\delta=1,2,...,k-|p|.$ So let 
$|p|\leq k-1, \,1\leq i\leq n,\, x_0 \in \Omega$ and choose $r$ so small that 
$\overline B_{|r|}(x_0)\subset\Omega$. We have
$$\eqalign{
{{a_p(x_0+e_ir,2|r|)-a_p(x_0,2|r|)}\over r}&={ {D^p[P_k(x,x_0+e_ir,2|r|)
-P_k(x,x_0,2|r|)]_{x=x_0}} \over r}\cr
&-\sum_{\delta=1}^{k-|p|}{ {(-1)^{\delta}} \over {\delta !}} r^{\delta
-1}a_{(p+\delta e_i)}(x_0+e_ir,2|r|).\cr
}\eqno(11)$$
First, since $k-|p|-1\geq 0$, applying Lemma 1.1, we see that
$$\eqalign{
&\biggl|{{D^p[P_k(x,x_0+e_ir,2|r|)-P_k(x,x_0,2|r|)]_{x=x_0}} \over
r}\biggr|^q \cr
&\leq {c_1 \over {(2|r|)^{n+|p|q}}} \int\limits_{\Omega_{|r|}(x_0)}|P_k(x,x_0
+e_ir,2|r|)-P_k(x,x_0,2|r|)|^q\,dx \cr
&\le {c_1 \over {(2|r|)^{n+|p|q}}}\, 2^{2q+1+kq+n} \, |r|^{kq+n}  {[u]'}^q
\omega(|r|)\cr
&=c  {[u]'}^q \omega(|r|)\, |r|^{q(k-|p|)} \quad \to 0 \hbox{ as } r\to 0. \cr
}$$
Second, $\forall \delta \hbox{ with } 1\leq \delta \leq k-|p|$, by (8) we have
$$\eqalign{
&|a_{(p+\delta e_i)}(x_0+e_ir,2|r|)-v_{(p+\delta e_i)}(x_0)| \cr
&\leq |a_{(p+\delta e_i)}(x_0+e_ir,2|r|)-v_{(p+\delta e_i)}(x_0+e_ir)|
+|v_{(p+\delta e_i)}(x_0+e_ir)-v_{(p+\delta e_i)}(x_0)| \cr
&\leq c_5 [u]' (2|r|)^{k-|p|}  \omega_1(2|r|) + |v_{(p+\delta e_i)}(x_0+e_ir)
-v_{(p+\delta e_i)}(x_0)| \cr
}$$
But since the $v_{(p+\delta e_i)}(x)$ are continuous for
$\delta=1,2,\dots ,k-|p|$, we immediately get 
$$\lim_{r\to 0} a_{(p+\delta e_i)}(x_0+e_ir,2|r|)=v_{(p+\delta e_i)}(x_0)
\quad\delta=1,2,...,k-|p|.$$

Hence by (11), we get (uniformly with respect to $x_0$)
$$\lim_{r\to 0}{{a_p(x_0+e_ir,2|r|)-a_p(x_0,2|r|)}\over r}\biggl( 
=\lim_{r\to 0} a_{(p+e_i)}(x_0+e_ir,2|r|)\biggr)=v_{(p+e_i)}(x_0).$$

It remains only to verify that 
$$\lim_{r\to 0}{{v_p(x_0+e_ir)-v_p(x_0)} \over r}=\lim_{r\to 0}{{a_p(x_0
+e_ir,2|r|)-a_p(x_0,2|r|)}\over r}.$$

Recalling that $|p|\leq k-1$, we write 
$$\eqalign{
{{v_p(x_0+e_ir)-v_p(x_0)} \over r}
=&{{v_p(x_0+e_ir)-a_p(x_0 +e_ir,2|r|)} \over r} \cr
&+{{a_p(x_0+e_ir,2|r|)-a_p(x_0,2|r|)}\over r} 
+{{a_p(x_0,2|r|)-v_p(x_0)}\over r} \cr
}$$
But by the first inequality in (8), we see that the first and third
summands $\to 0$ as $r\to 0$, proving the theorem. \qed
\enddemo

\proclaim{Theorem 1.8} If $\Omega$ has property (I) and 
$u \in {\Cal M}_q^{k,\omega}(\Omega)$, where  $\omega(t)$ is a Dini modulus of 
continuity, then the function $v_{(0)}(x) \in C^{k,\omega_1}(\overline \Omega)$
 and $\forall x \in \Omega, \,|p|\leq k,$
$$D^p v_{(0)}(x)= v_p(x).$$
\endproclaim

\demo{Proof} Immediate corollary of Theorems 1.7 and 1.8.
\enddemo

\proclaim{Main Theorem} If $\Omega$ has property (I) and 
$u \in {\Cal M}_q^{k,\omega}(\Omega)$, where $\omega(t)$ is a Dini modulus of
continuity, then $u \in C^{k,\omega_1}(\Omega)$  
$$[u]_{k,\omega_1;\Omega}\leq c_6 [u]'_{q,k,\omega;\Omega}.$$ 
\endproclaim

\demo{Proof} Recall that if $u \in L^q(\Omega)$, then by Lebesque's
theorem, for almost every $x_0
\in \Omega$ we have 
$$\lim_{r \to 0} \frac1{|\Omega_r(x_0)|}\int\limits_{\Omega_r(x_0)}
|u(x)-u(x_0)|^q\,dx =0.\eqno(12)$$
So choose $ x_0 \in \Omega$ for which (12) holds. For almost every 
$x \in \Omega$, we have
$$\eqalign{
&|a_{(0)}(x_0,r)-u(x_0)|^q \cr
&\leq c_6(q)\Bigl\{|a_{(0)}(x_0,r)-P_k(x,x_0,r)|^q
+ |P_k(x,x_0,r)-u(x)|^q +|u(x)-u(x_0)|^q \Bigr\}\cr
}$$
and hence integrating over $\Omega_r(x_0)$ gives
$$\eqalign{
& |a_{(0)}(x_0,r)-u(x_0)|^q \le
\frac{c_6}{|\Omega_r(x_0)|}\int\limits_{\Omega_r(x_0)}|a_{(0)}(x_0,r)
-P_k(x,x_0,r)|^q\,dx \cr
& +\frac{c_6}{|\Omega_r(x_0)|}\int\limits_{\Omega_r(x_0)}\left|P_k(x,x_0,r)
-u(x)\right|^q
\,dx +\frac{c_6}{|\Omega_r(x_0)|}\int\limits_{\Omega_r(x_0)}|u(x)-u(x_0)|^q
\,dx \cr
&=I_1 +I_2 +I_3. \cr
}$$
By Lebesque's theorem, $I_3 \to 0$ as $r \to 0$. Since $\Omega$ satisfies
property ($I$), as $r \to 0$
$$I_2 \le \frac{c_6}{A r^n}\int\limits_{\Omega_r(x_0)}\left|P_k(x,x_0,r)-u(x)\right|^q
\,dx \le \frac{c_6 \, r^{kq+n}}{A r^n}\omega(r)^q  {[u]'}^q \to 0
\hbox{ as } r \to 0.$$
Finally, for some constant $c_7=c_7(A,n,q,k)$, we have
$$ \eqalign{
I_1 \le& \frac{c_6}{A
r^n}\int\limits_{\Omega_r(x_0)}|a_{(0)}(x_0,r)-P_k(x,x_0,r)|^q\,dx \cr
\le & c_7 \sum_{1\le |p|\le k} |a_p(x_0,r)|^q r^{|p|q} \to 0 
\quad \hbox{as } r \to 0.
\cr}$$  
Thus, $|a_{(0)}(x_0,r)-u(x_0)|^q \to 0$ as $r \to 0$ and so, for
almost every $x_0 \in \Omega$ 
$$\lim_{r \to 0} a_{(0)}(x_0,r)=u(x_0).$$ But then by (7) we have 
$u(x_0)=\lim\limits_{r \to 0} a_{(0)}(x_0,r)=v_{(0)}(x_0) \in
C^{k,\omega_1}(\Omega)$. Since $x_0 \in \Omega$ was arbitrary, $u\equiv
v_{(0)}$ and thus $\forall x,y \in \Omega,\, |p|=k$, Theorems 1.6 and
1.8 give, with $c_6=c_6(k,q,n,A)$
$$|D^pu(x)-D^pu(y)|=|D^p v_{(0)}(x)-D^pv_{(0)}(y)|=|v_p(x)-v_p(y)|\leq
c_6 [u]'\omega_1(|x-y|).$$
That is, 
$$[u]_{k,\omega_1;\Omega}\leq c_6 [u]'_{q,k,\omega;\Omega}\,,$$
which proves our Main Theorem for the case $1\le q< \infty. \quad \qed$
\enddemo 
\noindent {\bf Remark.} For the case $q=\infty$, the proof is the same 
(yet easier), and the space 
$${\Cal M}_\infty^{k,\omega}(\Omega)= \left\{u \in
L^\infty(\Omega):[u]'_{\infty,k,\omega;\Omega} < + \infty\right\}$$
is defined by way of the finite seminorm
$$[u]'_{\infty,k,\omega;\Omega}=\sup\Sb x_0 \in \overline{\Omega}\\ 0<r \leq
d(\Omega)\endSb  {1 \over {r^{k} \omega(r)}}\inf_{p \in {\Cal
P}_k}\|u-p\|_{L^\infty(\Omega_r(x_0))}.$$
When $\Omega$ is convex, it follows by Taylor's formula that 
$C^{k,\omega}(\Omega) \subset {\Cal M}_q^{k,\omega}(\Omega)$, and hence when 
$\Omega$ is convex and satisfies property (I), by our main theorem, we have 
the inclusion $C^{k,\omega}(\Omega) \subset {\Cal M}_q^{k,\omega}(\Omega) 
\subset C^{k,\omega_1}(\Omega)$.
In our applications, we will use only the case $q=\infty$. 
\smallskip
\noindent{\bf Remark.} The inclusion ${\Cal M}_\infty^{k,\omega}\subset
C^{k,\omega_1}$ is sharp in the sense that $\omega_1$ cannot be replaced by a
smaller modulus of continuity. In particular (since $\omega(t)\le
2\omega_1(t)$), ${\Cal M}_\infty^{k,\omega}\not\simeq
C^{k,\omega}$. The following example demonstrates this, as well as provides an 
example of $u \in {\Cal M}_\infty^{k,\omega}$ where 
$u \notin C^{k,\alpha}\forall\alpha>0$. 
\medskip
\noindent {\bf Example.}  Let $k=n=1,\,q=\infty$. Consider the
function
$$u(x)=x \left( \ln\frac1{|x|} \right)^{-1}, \qquad x \in \Omega =B_{1/2}(0).$$ 
Note that 
$$u'(x)=\left( \ln\frac1{|x|} \right)^{-1} +\left( \ln\frac1{|x|} \right)^{-2},$$
and hence $u'(x)$ has modulus of continuity $\sim \left( \ln\frac1{t}
\right)^{-1}$, while $u \in {\Cal M}_\infty^{1,\omega}(B_{1/2}(0))$ for
$\omega(t)= \left( \ln\frac1{t} \right)^{-2}$. But 
$$\omega_1(t)=\int_0^t\frac{\left( \ln\frac1{r}
\right)^{-2}}r\,dr=\left( \ln\frac1{t} \right)^{-1}.$$
That is, $Du=u'$ has modulus of continuity $\sim \omega_1(t)$, hence
our inclusion above is sharp. To verify that $u \in {\Cal M}_\infty^{1,\omega}$, i.e. that
$[u]'_{\infty,1,\omega}<+\infty$, fix $x \in \Omega=
B_{1/2}(0)$ and take $r>0$. For any $y \in B_r(x)$, set $p(y)=T_{1,x}u(y) \in
{\Cal P}_1$. Of course, $u''(x)\le
3\left( \ln\frac1{|x|}\right)^{-2} / |x|$, for all $x \in \Omega$. 
Now if $|x|\ge 2r$, by Taylor's Theorem, for some $z \in (y,x)$, we have
$$\eqalign{
|u(y)-p(y)|=&|\frac{u''(z)}2 (y-x)^2| \le \frac{3\left( \ln\frac1{|z|}
\right)^{-2}|y-x|^2}{2|z|}  \cr
\le& \frac{3\left( \ln\frac1{r} \right)^{-2}r^2}{2r} 
= \frac{3r\left( \ln\frac1{r}\right)^{-2}}2 
=\frac32 r \omega(r).\cr
}$$
On the other hand, if $|x|<2r$, choose $p(y)=y\left( \ln\frac1{r}
\right)^{-1}\in {\Cal P}_1$. Without loss of generality, since $u$ is
an odd function, we may consider $x>0$. By the Mean Value Theorem, we
have, for some $z \in (y,r)$
$$\eqalign{
\sup_{y \in B_r(x)}|u(y)-p(y)|&\le \sup_{y \in B_{3r}(0)}|u(y)-p(y)|
=\sup_{|y|\le 3r} \textstyle{\left|y\left( \ln\frac1{|y|}
\right)^{-1}-y\left( \ln\frac1{|r|}
\right)^{-1}\right| }\cr
&\le \sup_{|y|\le 3r}|y|
\tsize\left| \frac{\left( \ln\frac1{|z|} \right)^{-2}}{|z|}(y-r)\right| \cr
&\le 3r \left( \ln\frac1{r}\right)^{-2}
=3r \omega(r),\cr
}$$
hence $[u]'_{\infty,1,\omega}\le 3$, since $x \in
B_{1/2},\,r>0$ were arbitrary. Thus $u \in {\Cal M}_\infty^{1,\omega}(B_{1/2})$.
 Note however that $u \notin C^{1,\omega}(B_{1/2})$, since
$$ \eqalign{
\sup_{x \not= y \in B_{1/2}(0)} \frac{|u'(x)-u'(y)|}{\omega(|x-y|)} \ge &
\sup_{x \not= 0 \in B_{1/2}(0)} \frac{|u'(x)|}{\omega(|x|)} \cr
=& \sup_{x \not=0 \in B_{1/2}(0)}\tsize\frac{\left( \ln\frac1{|x|}
\right)^{-1} +\left(\ln\frac1{|x|}\right)^{-2}}{\left(
\ln\frac1{|x|}\right)^{-2}} =+\infty\,. \cr
}$$
Thus, $[u]'_{\infty,1,\omega}< +\infty$, while
$[u]_{1,\omega} = + \infty$  and so in general, even if $\omega(t)$
is a Dini modulus of continuity, the seminorms
$[u]'_{q,k,\omega;\Omega}$ and $[u]_{k,\omega;\Omega}$ are not equivalent. 
Moreover, $u \notin
C^{1,\alpha}(B_{1/2}(0))$ for {\it any} $\alpha >0$. Since
$u'(0)=0$, we have
$$ \eqalign{
\sup_{x \not= y \in B_{1/2}(0)} \frac{|u'(x)-u'(y)|}{|x-y|^\alpha} \ge &
\sup_{x \not= 0 \in B_{1/2}(0)} \frac{|u'(x)|}{|x|^\alpha} \cr
=& \sup_{x \not=0 \in B_{1/2}(0)}  \tsize\frac{\left( \ln\frac1{|x|}
\right)^{-1} +\left(\ln\frac1{|x|}\right)^{-2}}{|x|^\alpha} =+\infty\,. \cr
}$$

\head 2. Interior regularity for $\Delta u=f$ \endhead

In this section, we give an application of the inclusion 
${\Cal M}_\infty^{2,\omega}(B)\subset C^{2,\omega_1}(B)$ in the simplest 
setting. We use this inclusion to obtain estimates on the modulus of 
continuity of second derivatives of classical solutions of Poisson's equation 
$\Delta u=f$ in $B$, where $f$ is Dini continuous in $B$, i.e. 
$f \in C^{0,\omega}(B)$. Using potential theory, various authors 
(see [ME], [B], [HW]) have shown that if 
$u \in C^2(\overline B_{2}(x_0))$ satisfies $\Delta u=f$ in
$B_{2}(x_0)$, then for all $0<r<1$, we have 
$$\sup_{|x-y|\le r}|D^2u(x)-D^2u(y)|\le C\Bigl\{\int_0^{r}\frac{\omega(t)}t 
\,dt +r\int_{r}^{c}\frac{\omega(t)}{t^2}\, dt\Bigr\},\eqno(13)
$$
where $C$ depends only on $n,\omega, |u|_{0;B_1}$ and $|f|_{0, \omega;B_1}$ 
and $c$ is independent of $r$. Of course, when $f \in C^\alpha(B_2)$,
i.e. $\omega(t) \sim t^\alpha,\,0<\alpha<1$, the right hand side of (13) is 
$\le Cr^\alpha$. But for general Dini moduli of continuity, neither
of the summands in the right hand side of (13) can be omitted, as
simple examples show. The usual way of obtaining this estimate is by a lengthy
examination of the Newtonian potential of $f$. By using the Dini-Campanato 
inclusion, we can obtain this estimate in a simpler way. Specifically, 
we will show that if $u \in C^2(\overline B_{1}(0))$ satisfies 
$\Delta u=f \in C^{0,\omega}(B_1(0))$, then  $u \in {\Cal M}_\infty^{2,\varphi}
(B_{1/2}(0))\subset C^{2,\varphi_1}(B_{1/2}(0))$, for an appropriate Dini 
modulus of continuity $\varphi$, where $\varphi_1$ will be the right hand 
side of (13). 
It suffices to show $\exists \delta=\delta(n,\omega)>0$, such that if 
$|u|_{0;B_1}\le 1$ and $|f|_{0,\omega} \le \delta$, then  
$u \in {\Cal M}_\infty^{2,\varphi}(B_{1/2}(0)$. The estimate (2.0) will 
follow 
by rescaling. For our solution $u$, consider the function
$$\tilde u(x)=\frac{u(x)}{|u|_{0;B_{1}}
+ \delta^{-1}|f|_{\omega;B_{1}} }:=\frac{u(x)}K, \quad\hbox{ if } K\ge 1$$
(Otherwise, consider $\tilde u=u$.) Note that $\tilde u$ satisfies 
$|\tilde u|_{0;B_1}\le
1$ and $\Delta \tilde u=\frac{f}{K} := \tilde f$ in $B_1$, where $\tilde f$ 
is Dini continuous in $B_1$ and $|\tilde f|_{\omega;B_1}\le
\frac{|f|_{\omega;B_1}}K\le \delta$. That $u \in {\Cal M}_\infty^{2,\varphi}
(B_{1/2}(0)$ follows from the following lemma.

\proclaim{Lemma 2.1} Take any $x_0\in B_{1/2}(0)$. There exists  $0<\mu<1$  
depending only on $n,\omega$ and a sequence of paraboloids 
$$P_k(x)=P_{k, x_0}(x)= a_k + b_k\cdot (x-x_0) +
\frac{(x-x_0)^tC_k(x-x_0)}{2}$$ such that $\forall k \in \Bbb N^+$
$$ \eqalign{
tr(C_k)&=0 \cr 
{\left| u-P_k\right|}_{0;B_{\mu^k}(x_0)} &\le \mu^{2k}\varphi(\mu^k),\cr
}$$
where $P_0\equiv 0$ and $\varphi(t)=\dsize t\int_t^c \frac{\omega(r)}{r^2}\,dr
,\quad t\in (0,c/2]$.
\endproclaim
\demo{Proof} In the upper limit of the integral defining $\varphi$, we usually 
take $c\le 1$, depending on the domain of definition of $\omega$. (e.g. if 
$\omega(t)=t^\alpha \left(\ln\frac1t\right)^\beta,\alpha \in (0,1)$, we can 
take $c=1$.) Note since $\omega$ is nondecreasing, by the definition of 
$\varphi(t)$, we always have 
$$\omega(t) \le \left(\frac{c}{c-t}\right)\varphi(t)\le 2\varphi(t).$$  
We may assume $x_0=0$ and $f(0)=0$. First choose $\mu$ so small, depending on 
$\omega$, so that 
$$N_1 81 \mu c_e\le \frac12, \qquad \mu \leq {7 \over 16},\qquad \omega(\mu)\le \frac12$$ 
and then choose $\delta = \frac{\mu^3}{4N_2}$, where $c_e,N_1,N_2$ are 
constants depending only on $n$. 
Observe that by considering $\varphi(Kt)$ instead of $\varphi(t)$ 
(and considering smaller values of $t$) we may assume $\varphi(1)\ge 1$ and 
hence the claim holds for $k=0$, since $P_0\equiv 0, \,tr(0)=0$ and 
${| u|}_{0;B_1(0)} \leq 1$. Assume it holds for $k=i$. We now show it holds for
 $k=i+1$. So for this fixed $i$, consider the function
$$v(x)=\frac{(u-P_i)(\mu^ix)}{\mu^{2i}\varphi(\mu^i)} \qquad x\in B_1(0),$$
which, by inductive hypothesis, satisfies 
$$\Delta v(x)=\frac{\Delta u(\mu^ix)-tr(C_i)}{\varphi(\mu^i)}
=\frac{f(\mu^ix)}{\varphi(\mu^i)} :=f_i(x) \quad\hbox{ in } B_1(0), 
\quad {| v|}_{0;B_1(0)} \le 1.$$
Let $h \in C^{\infty}(\overline B_{7/8}(0))$ be the solution to the
 Dirichlet problem
$$ \gather
 \Delta h =0 \text{ in } B_{7/8}(0) \\
 h=v \text{ on } \partial B_{7/8}(0) \endgather 
$$
with $$[h]_{4,0;B_{\frac{7}{16}}(0)} \le
\left(\tsize{16 \over 7}\right)^{4}c_e
|v|_{0;\partial B_{7/8}(0)} \le 81 c_e
|v|_{0;B_1 (0)} \le 81c_e,$$ for some constant $c_e=c_e(n)$. By Taylor's 
formula, for 
$$T_{2,0}h(x)=h(0) + Dh(0) x + \frac12 x^t D^2h(0) x \quad \in {\Cal P}_2,$$ 
we have
$$|h-T_{2,0}h|_{0;B_\mu(0)}\le N_1(n) [h]_{4,0;B_{\mu}(0)}\mu^{4} 
\le N_1 [h]_{4,0;B_{\frac{7}{16}}(0)} \mu^{4} \le N_1 81 \mu^{4} c_e\,.
$$
Since $f(0)=0$, the classical a priori estimates yield, for some constant 
$N_2=N_2(n)$ 
$$\eqalign{
|v -h|_{0; B_{7/8}(0)} 
\le & |v -h|_{0; \partial B_{7/8}(0)} +N_2\left(\tsize\frac78\right)^2  
|\Delta v -\Delta h |_{0;B_{7/8}(0)}  \cr
\le &  N_2 | f_i|_{0;B_{7/8}(0)} 
\le  N_2 [f]_{\omega}\frac{\omega(\mu^i)}{\varphi(\mu^{i})} 
\le 2 N_2 [f]_{0,\omega} \cr
}$$
Thus
$$|v-T_{2,0}h|_{0; B_\mu(0)} \le |v-h|_{0; B_\mu(0)} 
+|h-T_{2,0}h|_{0;B_\mu(0)}\le 2 N_2  [f]_{0,\omega} +  N_1 81 \mu^{4} c_e.$$
So for $x \in B_{\mu^{i+1}}(0)$, set 
$P_{i+1}(x)=P_i(x)+ \mu^{2i}\varphi(\mu^i) T_{2,0}h\left (\frac{x}{\mu^i}
\right) \in {\Cal
P}_2$. Rescaling back, plugging in the definition of $v$, 
recalling that $\mu,\delta$ are small and that 
$\mu \varphi(\mu^i) \le \varphi(\mu^{i+1})$ we get, 
$\forall x \in \overline B_{\mu^{i+1}}(0)$
$$\eqalign{
|u(x)-P_{i+1}(x)|&=\tsize\left|u(x)-P_i(x)-\mu^{2i}\varphi(\mu^i)
T_{2,0}h\left(\frac{x}{\mu^i}\right) \right|  \cr
&=\tsize\mu^{2i}\varphi(\mu^i)\left| v\left(\frac{x}{\mu^i}\right) 
-T_{2,0}h\left(\frac{x}{\mu^i} \right )\right| \cr
&\le \mu^{2i}\varphi(\mu^i)|v-T_{2,0}h|_{0; B_\mu(0)}\cr
&\le \mu^{2i}\varphi(\mu^i)\Bigl\{ 2 N_2  [f]_{\omega} +  N_1 81 \mu^{4} c_e
\Bigr\} \cr
&=  \mu^{2(i+1)}\Bigl\{ 2 N_2  [f]_{\omega}\frac{\varphi(\mu^i)}{\mu^2} + 
N_1 81 \varphi(\mu^i)\mu^{2} c_e\Bigr\} \cr
&\le  \mu^{2(i+1)}\Bigl\{ 2 N_2  [f]_{\omega}\frac{\varphi(\mu^{i+1})}{\mu^3} 
+  N_1 81 \varphi(\mu^{i+1})\mu c_e\Bigr\} \cr
&\le  \mu^{2(i+1)} \varphi(\mu^{i+1}) \Bigl\{ \frac{2 N_2 \delta}{\mu^3} 
+ N_1 81 \mu c_e\Bigr\} \cr
&\le  \mu^{2(i+1)} \varphi(\mu^{i+1}),\cr 
}$$
and hence $|u-P_{i+1}|_{0;B_{\mu^{i+1}(0)}} \le \mu^{2(i+1)}\varphi(\mu^{i+1})$.
 Moreover, by definition of $P_{i+1}$, \break $C_{i+1} = C_i 
 +\varphi(\mu^{i})D^2h(0)$ 
from which it follows 
$$tr(C_{i+1})=tr(C_i) + \varphi(\mu^{i}) \Delta h(0)=0, $$ 
which completes the proof of Lemma 2.1. \qed 
\enddemo 
By Lemma 2.1, we know that $\forall\,x_0 \in B_{1/2}(0),\exists \,0< \mu <1$ 
(depending only on $n,\omega$) and a sequence
$\{P_{k}\}=\{P_{k,x_0}\} \subset {\Cal P}_2$ such that 
$${| u-P_k|}_{0;B_{\mu^k}(x_0)} \leq \mu^{2k}
\varphi(\mu^k)\quad\forall k\geq 0.$$
So, $\forall \, 0<r\le 1$, choose $k \ge 0$ so large that
$\mu^{k+1} <r \le \mu^k$. Since $\{P_k\} \subset {\Cal P}_2$, we immediately get
$$\eqalign{
\inf_{p \in {\Cal P}_2} {| u-p|}_{0;B_{r}(x_0)} 
&\leq \inf_{p \in {\Cal P}_2} {| u-p|}_{0;B_{\mu^k}(x_0)} \cr
&\le {|u-P_k|}_{0;B_{\mu^k}(x_0)} \le \mu^{2k}\varphi(\mu^k) 
  =\tsize\frac{\mu^{2(k+1)}}{\mu^2}\varphi\left( \frac{\mu^{k+1}}{\mu}\right )\cr
&\le \frac1{\mu^3} r^2 \varphi(r)
=C_1 r^2 \varphi(r). \cr
}$$
Since $0<r \le 1=d(B_{1/2}(0))$ and $x_0 \in B_{1/2}(0)$, are arbitrary, 
we have with $q=\infty$
$$[u]'_{2,\varphi;B_{1/2}(0)}=\sup \Sb 0 < r \le d(B_{1/2}(0)) \\ 
x_0 \in B_{1/2}(0)\endSb \frac 1{r^2 \varphi(r)}\inf_{p
\in {\Cal P}_2} {| u-p|}_{0;B_{r}(x_0)\cap B_{1/2}(0)  } \le C_1.$$
That is, $u \in {\Cal M}_\infty^{2,\varphi}(B_{1/2}(0))$. Since
$\varphi $ is a Dini modulus of continuity (since $\omega$ is),
by our Dini-Campanato inclusion, we have $ u \in C^{2, \varphi_1}(B_{1/2}(0))$ 
and 
$$[u]_{2, \varphi_1 ;B_{1/2}(0)} \le N [u]'_{2,\varphi;B_{1/2}(0)} \le C_2.$$
But by definition of $\varphi(t)$ and Fubini's theorem, we have 
$$\eqalign{
\varphi_1(t) =&\int_0^t \frac{\varphi(r)}r \,dr = \int_0^t \int_r ^c 
\frac{\omega(\rho)}{\rho^2}\, d\rho \,dr \cr
=& \int_0^t \int_0 ^\rho \frac{\omega(\rho)}{\rho^2}\, dr\, d\rho 
+\int_t^c \int_0 ^t \frac{\omega(\rho)}{\rho^2}\, dr\, d\rho \cr
=& \int_0^t \frac{\omega(\rho)}{\rho} \,d\rho +t\int_t^c \frac{\omega(\rho)}{\rho^2}\,d\rho.\cr
}$$

\head 3.   Interior Regularity for $F(D^2u,x)=f(x)$ \endhead

In this chapter, we use the inclusion  ${\Cal
M}_\infty^{2,\omega}(B)\subset C^{2,\omega_1}(B)$ to estimate the modulus of
continuity of second derivatives $D^2u$ of solutions of fully nonlinear 
elliptic equations
$F(D^2u,x)=f(x)$ in $B=B_1(0)$. Here, we assume that $f$ is Dini
continuous in $B_1$, in the weaker $L^n$ (as opposed to $L^\infty$) sense with 
Dini modulus of continuity $\omega(t)$. That is, we assume that 
$\forall x_0 \in B_1(0)$ 
$$\left\{\frac1{|B_r(x_0)|} \int_{B_r(x_0)} \left| f(x)-f(x_0)\right|^n dx 
\right\}^{1/n} \le C\omega(r),\quad \forall r <1.$$
We further assume that $\omega(t)$ satisfies the following property  
$$\lim_{\mu \to 0+}\sup_{0\le t \le \frac12} \frac{\mu^{\overline
\alpha}\varphi(t)}{\varphi(t\mu)}=0, \quad \hbox{ where }\varphi(t)
:=t^{\overline \alpha} + \omega(t), \eqno(14)$$
where $\overline \alpha=\overline
\alpha (n,\lambda,\Lambda) \in (0,1)$ is the
H\"older exponent given in the Evans-Krylov theorem. This restriction was not 
required in the linear case, since there, we had solvability of the constant 
coefficient Dirichlet problem with any order of smoothness. In the fully 
nonlinear setting however, we have solvability of the constant coefficient 
Dirichlet problem with order of smoothness only $2 +\overline \alpha$.  
\medskip

\noindent {\bf Remark.} As strong a condition as (14) appears, it is satisfied by
$\omega(t) = \nobreak t^\alpha \left( \ln \frac1t \right)^\beta$, $0<\alpha
< \overline\alpha,\beta \in \Bbb R$. This enables us to generalize
the known result for
H\"older continuous $f$, i.e. $f \in
C^{0,\alpha}(B),\, 0< \alpha < \overline\alpha$. Indeed, for 
$\omega(t)= t^\alpha \left( \ln \frac1t \right)^\beta$, $0< \alpha
< \overline\alpha$, integration by parts gives that $\int_0^t
\frac{r^\alpha \left( \ln \frac1r \right)^\beta}r \, dr \le C t^\alpha
\left( \ln \frac1t \right)^\beta$ and hence by Theorem
3.1 below, $D^2u$ has modulus of continuity $\le
C\psi(t)$, where 
$$\psi(t)=t^{\overline\alpha} +\int_0^t
\frac{r^\alpha \left( \ln \frac1r \right)^\beta}r \, dr \le t^{\overline\alpha} +C t^\alpha
\left( \ln \frac1t \right)^\beta \le C_1 t^\alpha
\left( \ln \frac1t \right)^\beta.$$ Taking $\beta =0$, we recover the
well-known result for $f \in C^{0,\alpha}(B), 0< \alpha < \overline\alpha$. 
Note that $\psi(t)$ is a Dini modulus of continuity. This is not always the 
case, as our Example 3.1 shows.


More importantly, (14) holds for $\omega(t)= \left( \ln \frac1t \right)^\beta,
\beta <-1$. The significance of this class of moduli of continuity
satisfying (14) is that it permits us
to consider $f$ whose $L^n$ averages are Dini continuous, yet not in 
$C^{0,\alpha}(B)$ for {\it
any} $\alpha > 0$. (See Example 3.1 below.) Property (14) fails for Dini 
moduli of continuity which are ``nice''
compared to $t^{\overline\alpha}$. Indeed (14) implies
that $\lim\limits_{t \to 0+}\frac{t^{\overline\alpha}}{\omega(t)}=0$, which 
generalizes the $0 <\alpha
<\overline \alpha\,$ condition. Hence
(14) fails for $\omega(t)= t^{\overline\alpha}$, $\omega(t)= t\left( \ln
\frac1t \right)^\beta, \beta \ge 0$ and most notably for $\omega
\equiv 0$. But if $\omega \equiv 0$, then $f$ is constant and by the
Evans-Krylov theorem, $D^2u \in C^{0,\overline \alpha}$. Furthermore, for 
sufficiently small $t>0$, $\,t\left( \ln \frac1t \right)^\beta \le t^\alpha,
\forall \alpha \in (0,1)$. Hence any $f$ whose $L^n$ averages are 
$\sim t\left( \ln \frac1t \right)^\beta,\beta \ge 0$
will automatically have $L^n$ averages belonging to $C^{0,\alpha}(B),
\forall \alpha \in (0,\overline \alpha)$ and hence by Safonov's result 
(see [S1]), $D^2u \in C_{loc}^{0,\alpha}(B)$. We cannot
conclude however, that if $\omega(t)$ fails (14) then $\omega(t)\le C
t^{\overline\alpha}$, since for example, 
$\omega(t)= t^{\overline\alpha} \ln \frac1t$, has limit 1 in (14). Even in 
this case, the regularity of second derivatives is covered by known results, 
since for sufficiently small $t>0$, $t^{\overline\alpha} \ln \frac1t \le 
t^\alpha \,\forall \alpha \in (0,\overline \alpha)$. Thus, property (14) 
enables us to generalize well-known regularity results for H\"older continuous 
$f$ (subject to the restriction $0 <\alpha <\overline \alpha$) and extend these
results to a large class of functions whose $L^n$ averages are Dini, yet 
non-H\"older continuous.
\medskip 

\noindent{\bf Example 3.1.} Consider the uniformly elliptic, concave
equation
$$F(D^2u,x)=f(x):=\left( \ln \frac1{|x|} \right)^{-2} \hbox { in } 
B=B_{1/2}(0).$$
Taking $x_0=0$ (since $f(0)=0$), the inequalities
$$ \eqalign{
C(n) \left( \ln \frac1{r} \right)^{-2} 
\le & \big\{ \frac{n}{r^n} \int_0^r \rho^{n-1} \left( \ln \frac1{\rho} \right)
^{-2n} d\rho \big\}^{1/n} 
=\big\{\int\limits_{B_r(0)}\kern -.6cm \diagup \left( \ln \frac1{|x|} \right)^{-2n} 
 dx  \big\}^{1/n} \cr
\le& \left( \ln \frac1{r} \right)^{-2}  \cr
\cr}$$ 
show that $f$ is not H\"older continuous at $x_0=0$ in the $L^n$ sense for 
{\it any} $\alpha \in (0,1)$. Here, $\int \kern -.4cm \diagup\,$ denotes 
average. Yet clearly, $f$ is Dini continuous in $B$ in the $L^n$ sense, 
since for $x_0 \in B$, by the subadditivity of the function 
$\left( \ln \frac1{t} \right)^{-2}$ for $t>0$ small, we have
$$\eqalign{
\Big\{\int\limits_{B_r(x_0)} \kern -.675cm \diagup \left|  
\left( \ln \frac1{|x|} \right)^{-2} 
- \left( \ln \frac1{|x_0|} \right)^{-2} \right|^n dx \Big\}^{1/n} 
\le& \Big\{\int\limits_{B_r(x_0)} \kern -.675cm \diagup 
\left( \ln \frac1{|x-x_0|} \right)^{-2n}  dx \Big\}^{1/n} \cr
\le& \left( \ln \frac1r \right)^{-2} \cr
}$$
and $\omega(r)=\left( \ln \frac1r \right)^{-2}$ is a Dini modulus of 
continuity which satisfies (14). Hence by our Theorem 3.1 below, locally, 
$D^2u$ has modulus of continuity $\le
C\psi(t)$, where for sufficiently small $t>0$
$$\psi(t)=t^{\overline \alpha} +\int_0^t\frac{\left( \ln \frac1{r}
\right)^{-2}}r\,dr=t^{\overline \alpha} +\left( \ln \frac1{t}
\right)^{-1}\le C_1\left( \ln
\frac1{t} \right)^{-1}.  $$
Observe that $\psi(t)$ is not a Dini modulus of continuity.

Now consider the function 
$$\tilde \beta(x,x_0)=\tilde \beta_F (x,x_0)=\sup_{M \in S} {|F(M,x)-F(M,x_0)| \over {\|M \| +1}},$$
which measures the oscillation of $F$ in $x$ near the point $x=x_0 \in
B$. For our Theorem 3.1, we must impose some sort of continuity restriction on
$\tilde \beta(\cdot, x_0)$, since even in the linear
case $Lu=tr\left[A(x) D^2u
\right]=a_{ij}(x)D_{ij}u=f(x)$ (for H\"older continuity) we require that $f$ 
and $a_{ij}$
belong to $C^{0,\alpha}$.
Hence we require that both $f$ and all $\tilde
\beta(\cdot,x_0)$ belong to $C^{0,\omega}(B)$ in the $L^n$ sense. 
The following is a generalization of the argument used by Caffarelli in 
[C1],[CC] to prove pointwise $C^{2,\alpha}$ estimates for viscosity solutions 
of $F(D^2u,x)=f(x)$.

\proclaim{Theorem 3.1} Let $F$ be concave, uniformly elliptic (with
ellipticity constants $\lambda$ and $\Lambda$), $F$ and $f$ are
continuous in $x$. Suppose that $f$, as well as all the oscillations of
$F$ in $x$, belong to $C^{0,\omega}(B_{1})$ in the $L^n$ sense, where 
$\omega(t)$ is a Dini modulus of continuity satisfying property (14). 
If $u \in C^2(B_{1})$ is a solution of $F(D^2u,x)=f(x)$ in $B_{1}(0)$, 
then $u \in C^{2,\psi}(B_{1/2}(0))$, where for $0\le t \le 1/2$
$$\psi(t)=t^{\overline \alpha}+\int_0^t\frac{\omega(r)}r \,dr,$$ 
where $\overline \alpha=\overline
\alpha (n,\lambda,\Lambda) \in (0,1)$ is the
H\"older exponent given in the Evans-Krylov theorem.
\endproclaim

\demo{Proof.} Since $\omega(t)$ is a Dini modulus of continuity, assume for 
definiteness that $\int_0^1\frac{\omega(r)}r \,dr <+\infty$. Following routine 
normalizations (see [CC] p.75), we may assume ${| u |}_{0;B_1}\le 1$. 
It suffices to prove $\exists\delta >0$ (small enough) depending only on 
$n, \lambda,\Lambda,\omega$ such that if $u\in C^2(B_1)$ is a solution of 
$F(D^2u,x)=f(x)$ in $B_1 =B_1(0)$ and if $\forall x_0 \in B_{1/2}(0)$
$$\big\{\int\limits_{B_r(x_0)}\kern -.675cm \diagup \tilde \beta(x,x_0)^n \,dx 
\big\}^{1/n} \le \delta \omega(r),\quad  
\big\{\int\limits_{B_r(x_0)}\kern -.675cm \diagup |f(x)-f(x_0)|^n \,dx 
\big\}^{1/n} \le\delta \omega(r)\quad\forall r\le 1,$$
then $u \in C^{2,\psi}(\overline {B_{1/2}(0)})$. It suffices to prove the following lemma.
\enddemo

\proclaim{Lemma 3.2} Take any $x_0 \in B_{1/2}(0)$. There exists  $0<\mu<1$  
depending only on $n,\lambda,\Lambda,\omega$ and a sequence of polynomials 
$$P_k(x)=a_k+ b_k\cdot (x-x_0) + {1\over 2}(x-x_0)^tC_k(x-x_0)$$ 
such that $ F(C_k,x_0)=0$ for all $k\geq 0$,  
${| u-P_k|}_{0;B_{\mu^k}(x_0)} \leq \mu^{2k} \varphi(\mu^k)$ for all 
$k\geq 0$ and $$|a_k-a_{k-1}| +\mu^{k-1}|b_k-b_{k-1}| 
+\mu^{2(k-1)}\| C_k-C_{k-1}\| \leq 13 c_e \mu^{2(k-1)}\varphi(\mu^{k-1}),$$ 
where $P_0\equiv P_{-1}\equiv 0, \,c_e$ is a universal constant and 
$\varphi(t)=t^{\overline\alpha} +\omega(t).$
\endproclaim 

\demo{Proof} As before, we assume that $x_0=0$ and  that $F(0,0)=0=f(0)$. 
First choose $\mu$ small enough (depending only on $n, \omega,\lambda,\Lambda$)
 such that (14) holds and
$\omega(\mu)\leq {1/2}$, $\mu \leq {7/16}$.
Then choose $\delta$ such that
$$2N_1 \omega_n^{1/n}\delta (52c_e\varphi_1(1)\,+1) (9c_e +2)
\le c_e \mu^{2+\overline \alpha},$$
where $c_e=c_e(n,\lambda,\Lambda)$ is the constant in the Evans-Krylov theorem,
 $N_1=N_1(n,\lambda,\Lambda)$ is from the Alexandrov estimates and $\omega_n$ 
is the volume of the unit ball. Note that $\delta$ depends only on 
$n,\lambda,\Lambda,\omega$. The claim holds for $k=0$ since $P_0\equiv P_{-1}
\equiv 0, \,F(0,0)=0$ and $ {| u|}_{0;B_1(0)}\leq 1$.
Assume it holds for $k=i$. We now show it holds for $k=i+1$. So for this 
fixed $i$, consider the function
$$ v(x)=\,{(u-P_i)(\mu^i x) \over {\mu^{2i}\varphi(\mu^i)}} \quad x\in B_1(0),$$
which satisfies
$F\bigl(\varphi(\mu^i)\,D^2v(x)+C_i,\mu^ix\bigr) = f(\mu^ix)$
and hence $F_i(D^2v, x)=f_i (x)$ in $B_1(0)$, where
$$F_i(M,x) ={ {F(\varphi(\mu^i)M+C_i, \mu^ix) - F(C_i,\mu^i x}) \over 
{\varphi(\mu^i)}}, \quad f_i(x) ={ {f(\mu^ix)- F(C_i, \mu^ix)} \over 
{\varphi(\mu^i)}}.$$
Now $F_i(M,x)$ is concave in $M$ and has ellipticity constants 
$\lambda, \Lambda$ (since $F$ does), and $F_i(0,x)=0$. 
By the Evans-Krylov theorem, $\exists\, h \in C_{loc}^{2, \overline 
\alpha}(\overline B_{7/8}(0))$ solving
$$ \gather
F_i(D^2h,0)=0 \text{ in } B_{7/8}(0) \\
h=v \text{ on } \partial B_{7/8}(0) \endgather
$$
and 
$${\| h\|}_{C^{2,\overline \alpha} (B_{7/16}(0))} ^*
\leq c_e {| v |}_{0;\partial B_{7/8}(0)} \leq c_e {| v |}_{0;B_1(0)}
\leq c_e\,$$
where $c_e=c_e(n,\lambda,\Lambda)$. By Taylor's formula, for 
$$T_{2,0}h(x)=h(0) \,+ Dh(0) x + {1 \over 2}\,x^tD^2h(0)x \in {\Cal P}_2,
$$ we have
$$\eqalign{
{| h-T_{2,0}h |}_{0;B_{\mu}(0)} \leq&  
[h]_{2,\overline\alpha;B_{\mu}(0)}\mu^{2+\overline\alpha}
\leq  [h]_{2,\overline\alpha;{B_{\frac7{16}(0)}}}\mu^{2+\overline\alpha}  \cr
\leq& \tsize\left({16 \over 7}\right)^{2+\overline\alpha}c_e 
{\mu}^{2+\overline\alpha}\leq c_e 27\mu^{2+\overline\alpha}. \cr
}$$  
By the classical Alexandrov estimates, we have, for some constant 
$N_1=N_1(n,\lambda,\Lambda)$
$$\eqalign{
| v-h |_{0;B_{7/8}(0)} &\le | v-h |_{0;\partial B_{7/8}(0)} +N_1 \| F_i(D^2h,
\cdot)-F_i(D^2v,\cdot )\|_{L^n(B_{7/8})} \cr
& = N_1 \| F_i(D^2h,\cdot)-f_i\|_{L^n(B_{7/8})} \cr
& \le N_1 \biggl\{ \| F_i(D^2h,\cdot)-F_i(D^2h,0)\|_{L^n(B_{7/8})} 
+\|f_i\|_{L^n(B_{7/8})} \biggr\} \cr
& \le N_1 \biggl\{ \|\tilde{\beta}_{F_i}(\cdot,0)\|_{L^n(B_1)}   (9c_e +1) 
+ \|f_i\|_{L^n(B_1)} \biggr\}              \cr
}$$ 
We need to estimate both $\|\tilde{\beta}_{F_i}(\cdot,0)\|_{L^n(B_1)}$ and 
$\|f_i\|_{L^n(B_1)}$. For $x \in B_1(0)$,
$$\eqalign{
&\tilde{\beta}_{F_i}(x,0) =\sup_{M \in \Cal S} {{|F_i(M,x)-F_i(M,0)|} 
\over {\| M\| +1}}\cr
 &=\sup_{M \in \Cal S}\Bigg|{ {{\left[F(\varphi(\mu^i)M+C_i ,
 \mu^ix)-F(\varphi(\mu^i)M+C_i,0)\right]} -{\left[F(C_i, 
 \mu^ix)-F(C_i,0)\right]}} \over {\varphi(\mu^i)(\| M \| +1)}}\Biggr| \cr
&\leq \sup_{M \in \Cal S}\Biggl( { {\| \varphi(\mu^i)M+C_i \| +1 +\| C_i \| +1}
 \over {\| M \| + 1}}\Biggr) 
 { {\widetilde {\beta} (\mu^ix,0)} \over {\varphi(\mu^i)}} \cr
&\leq \sup_{M \in \Cal S} \Biggl( {{\varphi(\mu^i)\| M \| +2(\| C_i \| +1)} 
\over { \| M\| +1}} \Biggr) 
{ {\widetilde {\beta} (\mu^ix,0)} \over {\varphi(\mu^i)}} \cr
}$$
Since $\omega$ (hence $\varphi$) is a Dini modulus of continuity, the integral 
test yields
$$\eqalign{
\| C_i \| \leq& \sum_{k=1}^i \| C_k -C_{k-1}\| \le 13c_e \sum_{k=1}^{\infty} 
\varphi(\mu^{k-1}) \cr
\le& 13c_e\left( \varphi(1) + \ln\left(\tsize{\frac1{\mu}}\right)^{-1} 
\int_0^1 \frac{\varphi(r)}r\,dr \right) 
\le 52 c_e \varphi_1(1). \cr
}$$
Hence for $x \in B_1(0)$
$$\eqalign{
\widetilde {\beta_{F_i}} (x,0) \le& \sup_{M \in \Cal S} 
\Biggl( {\varphi(\mu^i)\| M \| +2\bigl(52c_e\varphi_1(1)+1 \bigr) 
\over { \| M\| +1}
} \Biggr) { {\widetilde {\beta} (\mu^ix,0)} \over {\varphi(\mu^i)}} \cr
\le& 2\left( 52 c_e \varphi_1(1)+1 \right)  
{ {\widetilde {\beta} (\mu^ix,0)} \over {\varphi(\mu^i)}}\,, \cr
}$$
and thus since $\omega \le \varphi$ and the $L^n$ average of 
$\tilde\beta(\cdot,0)$ is small, we get
$$\eqalign{
\|\tilde{\beta_{F_i}}(\cdot,0)\|_{L^n(B_1)}\le&2\left( 52 c_e \varphi_1(1)+1 
\right) \frac{\| {\widetilde{\beta}}(\mu^i\cdot,0)
\|_{L^n(B_1)}}{\varphi(\mu^i)} \cr
\le&  2\left( 52 c_e \varphi_1(1)+1 \right) \frac{ \omega_n^{1/n} \delta 
\omega(\mu^i)} {\varphi(\mu^i)}
\le  2\omega_n^{1/n} \delta \left( 52 c_e \varphi_1(1)+1 \right).\cr
}$$
Similarly, for $x \in B_1(0)$
$$\eqalign{
|f_i(x)| = \frac{|f(\mu^ix)-F(C_i, \mu^ix)|}{\varphi(\mu^i)} 
&\le \frac{|f(\mu^i x)| +|F(C_i,0)-F(C_i,\mu^ix)|}{\varphi(\mu^i)}\cr
&\le \frac{ |f(\mu^i x)|+ \widetilde{\beta}(\mu^ix,0) (\| C_i\| +1)}
{\varphi(\mu^i)} \cr
&\le  \frac{ |f(\mu^i x)| +  \widetilde{\beta}(\mu^ix,0)\left( 52 c_e 
\varphi_1(1) +1 \right)  }{{\varphi(\mu^i)}}, \cr 
}$$
which implies, since the $L^n$ average of $f$ is small
$$\eqalign{
\|f_i\|_{L^n(B_1)} &\le \frac{ \|f(\mu^i \cdot)\|_{L^n(B_1)} + \| \widetilde{\beta}(\mu^i
\cdot,0)\|_{L^n(B_1)} \left( 52 c_e \varphi_1(1) +1 \right)  }{{\varphi(\mu^i)}} \cr
&\le \frac{\omega_n^{1/n} \delta \omega(\mu^i) + \omega_n^{1/n} \delta \omega(\mu^i) 
 \cdot ( 52 c_e \varphi_1(1) +1) }{\varphi(\mu^i)} \cr
&\le  2\omega_n^{1/n} \delta \left( 52 c_e \varphi_1(1)+1 \right). \cr
}$$
Returning to our a priori estimates and recalling that $\delta$ is small, 
we get 
$$\eqalign{
| v-h |_{0;B_{7/8}} & \le N_1 \biggl\{ \|\tilde{\beta}_{F_i}(\cdot,0)\|_
{L^n(B_1)}   (9c_e +1) + \|f_i\|_{L^n(B_1)} \biggr\} \cr  
&\le N_1 \Bigl\{ 2\delta \omega_n^{1/n}\left( 52 c_e \varphi_1(1)
+ 1\right) \left(9c_e +1 \right)
+2\delta\omega_n^{1/n} \left( 52 c_e \varphi_1(1)+ 1\right)\Bigr\} \cr
&\le N_1 2\delta \omega_n^{1/n} \left( 52 c_e \varphi_1(1)+ 1\right)\left(9c_e +2 \right) \cr
&\le c_e \mu^{2+\overline\alpha}, \cr
}$$
and hence, since $\mu \le \frac7{16}$, we have 
$${| v-T_{2,0}h |}_{0;B_{\mu}(0)} \leq {| v-h |}_{0;B_{\mu}(0)} 
+{| h-T_{2,0}h |}_{0;B_{\mu}(0)} \le 28c_e\mu^{2+\overline\alpha}.$$
Now, for $x \in
B_{\mu^{i+1}}(0)$, set $P_{i+1}(x)=P_i(x)
+\mu^{2i}\varphi(\mu^i)T_{2,0}h\left({x \over {\mu^i}}\right) \in {\Cal
P}_2$. Rescaling back, plugging in the definition of $v$ and recalling that 
$\omega(t)$ satisfies (14), we get
$$\eqalign{
|u(x)-P_{i+1}(x)|&=\Big|u(x)-P_i(x)-\mu^{2i} \varphi(\mu^i) T_{2,0} h
\tsize\left({x \over {\mu^i}}\right)\Big| \cr
&=\mu^{2i} \varphi(\mu^i)\Big|\tsize v\left({ x \over {\mu^i}}\right)
-T_{2,0}h \left({x \over {\mu^i}} \right)\Big| \cr
&\le \mu^{2i} \varphi(\mu^i) 28c_e \mu^{2+\overline\alpha}  \cr
& \le \mu^{2(i+1)} \varphi(\mu^{i+1}),\cr
}$$
i.e. $|u-P_{i+1}|_{0;B_{\mu^{i+1}}(0)} \le \mu^{2(i+1)} \varphi(\mu^{i+1})$, 
completing the induction step. Note that $P_{i+1}$'s  coefficients satisfy
$$C_{i+1}=C_i +\varphi(\mu^i) D^2h(0),\quad b_{i+1}=b_i 
+\mu^i \varphi(\mu^i)Dh(0),\quad a_{i+1} =a_i + \mu^{2i}\varphi(\mu^i)h(0).$$ 
Hence $F(C_{i+1},0)=F(\varphi(\mu^i) D^2h(0)+C_i,0)
=\varphi(\mu^i)F_i(D^2h(0),0) +F(C_{i},0)=0$.  Since ${\| h\|}_
{C^{2,\overline\alpha}({B_{\frac7{16}}(0)})}^*\leq c_e$, we have 
$$\eqalign{
&|a_{i+1} -a_i| + \mu^i |b_{i+1}-b_i| + \mu^{2i} \| C_{i+1}-C_i\| \cr
&\leq \mu^{2i}\varphi(\mu^i)\Bigl( |h(0)| +|Dh(0)| +\| D^2h(0)\| \Bigr) \cr
&\leq \mu^{2i}\varphi(\mu^i)\Bigl( c_e +{16 \over 7}c_e +\left( {16 \over 7}
\right )^2\ c_e \Bigr) \cr
&\leq 13c_e  \mu^{2i}\varphi(\mu^i).\cr
}$$
This completes the proof of Lemma 3.2. \qed
\enddemo
The above argument holds at any fixed $x_0 \in B_{1/2}(0)$ since for concave 
$F$, the Evans-Krylov theorem guarantees the solvability of the Dirichlet 
problem for $F(D^2h,x_0)=0$, with universal constant $c_e$. The same argument 
which follows Lemma 2.1 now gives us that 
$u \in {\Cal M}_\infty^{2,\varphi}(B_{1/2}(0)) \subset 
C^{2,\varphi_1}(B_{1/2}(0))$. But by definition of $\varphi(t)$, we have 
$$\varphi_1(t) =\int_0^t \frac{\varphi(r)}r \,dr = \int_0^t r^{\overline 
\alpha-1}\,dr + \int_0^t \frac{\omega(r)}{r}\,dr\sim \psi(t),$$
which completes the proof of Theorem 3.1. 


\bigskip

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\endRefs
\enddocument