1)}.$ Then, for $2

1,$ where $c$ is a nonzero complex constant. Simple calculations show that equality holds for $f$ in Conjecture 1. Equality holds for $f$ in Conjecture 2 when $1< p \leq 2\;$ and for $\overline {f}$ when $2\leq p <\infty.$ In section 6, we'll see that equality holds in Conjectures 1 and 2 for a large class of functions in $\dot W^{1,2}$ and $\dot W^{1,p},$ respectively. By contrast, it seems plausible that when $p\ne 2$ equality never holds in Conjecture 3. To construct sequences which saturate the upper bound in Conjecture 3, take $p\in (1, \infty)$ and $\alpha \in (0,1/p).$ Define $f_{\alpha}(z) = z|z|^{-2\alpha},\; \text{if}\;|z|\leq 1,\; f_{\alpha}(z) = 1/{\zb},\;\text{if}\; |z|\geq 1.$ One computes that $\icc |\bd f_{\alpha}|^p / \icc |\be f_{\alpha}|^p \rightarrow (p -1)^p$ as $\alpha \rightarrow 1/p.$ Thus, $\icc |\bd (\overline{f_{\alpha}})|^p / \icc |\be\,(\overline{f_{\alpha}})|^p \rightarrow (p -1)^{-p}$ as $\alpha \rightarrow 1/p.$ From these two relations, and the interchangeability of $\bd f$, $\be f$ in the conjectures, it follows that the constant on the right hand side of Conjecture 3 must be at least $\max^p (p-1, {1\over {p-1}}) = (p^*-1)^p.$ Here are three reasons we find these conjectures of interest. (a) Truth of Conjecture 3 would imply in the limiting case $p\rightarrow\infty$ a notable recent theorem of Astala [As1] about area distortion of quasiconformal mappings in the plane. (b) Let $S$ be the singular integral operator in the plane defined by $$Sf(z) = -{1\over {\pi}} \icc {f(\ze)\over {(z-\ze)^2}}\,|d\ze|^2.\tag1.3$$ Truth of Conjecture 3 would show that the norm of $S$ on $L^p(\cc, \cc)$ is precisely $p^* - 1.$ (c) Falsity of Conjecture 1 would prove, for $2\times 2$ matrix valued functions, a conjecture of Morrey in the calculus of variations which asserts that rank one functions are not necessarily quasiconvex. \medskip In Sections 2-5 we'll elaborate on statements (a), (b), and (c). In Sections 6-8 we'll present some evidence in favor of the conjectures. We are grateful to Professors Astala, Ba\~nuelos, and Iwaniec for helpful communications, especially to Professor Iwaniec for sharing some of his unpublished notes with us. Thanks go also to N. Arcozzi, D.Burkholder, R. Laugesen and the referee for corrections and comments on the first version of the manuscript. The first author thanks the organizers of the Uppsala conference for their marvelous hospitality and efficiency. Above all, he thanks Matts and Agneta for many years of inspiration and friendship. \bigskip \noindent 2. \bf Area distortion by quasiconformal mappings. \rm The integral in (1.3) is a Cauchy principal value. The operator $S$ is sometimes called the Beurling-Ahlfors transform. The general theory of such operators, as developed by Calder\'on, Zygmund, and others, is presented, for example, in [S]. Among its consequences are the facts that $S$ is bounded on $L^p$ for $1

2$ which depends only on $K.$ Via H\"older's inequality, this enhanced integrability leads to an inequality for the distortion of area by qc maps. One way to state the area distortion property is as follows: If $F(0) = 0$ and $F(1) = 1,$ then for all measurable sets $E\subset (|z|<1),$ $$ |F(E)| \leq C |E|^{\kappa},\tag2.3$$ where $|\cdot|$ denotes Lebesgue measure, and $C$ and $\kappa$ depend only on $K.$ Gehring and Reich [GR] conjectured in 1966 that the best possible, i.e. smallest, $\kappa$ for which (2.3) is valid should be $\kappa = 1/K.$ Prototypical conjectured extremals were the radial stretch maps $$F_K(z) = z|z|^{{1\over K}-1}.$$ $F_K$ is $K-qc$ for each $ K\geq1,$ and satisfies $ |F_K(B)| = {\pi}^{1-{1\over K}} |B|^{1/K}$ for balls $B$ centered at the origin. The Gehring-Reich conjecture withstood many assaults before it was finally proved in the 1990's by Astala [As1], by means of very innovative considerations involving holomorphic dynamics and thermodynamical formalism. Eremenko and Hamilton [EH] gave a shorter proof of the conjecture using a distillation of Astala's ideas. More background and related results can be found in the survey [As2]. In [N] and [AsM], the distortion results are applied to problems about ``homogenization" of composite materials. To continue our story requires a backup. The weak 1-1 and $L^2$ boundedness of $S$ imply existence of absolute constants $c$ and $\alpha$ such that for all $E\subset (|z| < 1),$ $$\int_{|z|<1}|S(1_E)| \leq c |E|\,\log({{\alpha}\over {|E|}}). $$ Gehring and Reich showed that their area distortion conjecture is more or less equivalent to proving that the smallest $c$ for which some $\alpha$ exists is $c = 1.$ Let $||S||_p$ denote the norm of $S$ acting as an operator from $L^p(\cc, \cc)$ into itself. Iwaniec [I1] found that ``c=1" is implied by $$\liminf_{p\rightarrow \infty}{1\over p}||S||_p = 1.$$ This implication, together with the examples $f_{\alpha}$ in Section 1, led Iwaniec to Conjecture 3, which, as noted in (b) at the end of section 1, can be restated as $$||S||_p = (p^*-1),\quad 1

1\}.$ Then
$$\align L(\bd f(z), \be f(z)) & = r^{-1}g(r) + g'(r) - 1, \quad r\in E, \\
& =r^{-1}g(r)g'(r), \quad r\notin E.\tag6.6\endalign$$
Thus,
$$\align rL(\bd f(z), \be f(z)) & = {d\over {dr}}(rg - \hf r^2), \quad r\in E,
\\
& = \hf {d\over {dr}} g^2, \quad r\notin E.\tag6.7\endalign$$
Now $E$ is either empty, or is a single interval $(0,R],$ with $0 2.$
Suppose that $f\in \dot W^{1,p}(\cc)$ is harmonic in $|z|<1$ and in $1 < |z| \leq
\infty.$ Then there
exist holomorphic functions $g$ and $h$ in $|z|<1$ such that
$$\align f(z) &= g(z) + \overline{h}(z),\quad |z|<1,\\
& = f(1/\overline{z}), \quad |z|>1.\endalign$$
Let $p>2$. Then computation gives
$$\align {1\over {{\alpha}_p}}\int_{\cc} &{\Phi}_p(\bd f, \be f)
= \int_{|z|<1}((p-1)|g'(z)| - |h'(z)|)(|g'(z)| + |h'(z)|)^{p-1}\,dx\,dy\\
& + \int_{|z|>1} ((p-1)|h'(1/\overline{z})| - |g'(1/\overline{z})|)
(|h'(1/\overline{z})| + |g'(1/\overline{z})|)^{p-1} |z|^{-2p}\,dx\,dy\\
& = 2\pi \int_0^1((p-1) - r^{2p-4})I_1(r)\, r\,dr \,+\,
2\pi \int_0^1 ((p-1)r^{2p-4} - 1) I_2(r)\,r\,dr, \endalign$$
where $I_1(r),\;I_2(r)$ are the respective mean values on the circle $|z| = r$
of the functions $|g'| (|g'| + |h'|)^{p-1}$ and $|h'|(|g'| + |h'|)^{p-1}.$ The
logarithms
of these functions are subharmonic, hence so are the functions themselves.
Thus, $I_1$ and $I_2$ are nondecreasing functions of $r$ on $[0,1].$
From $I_2 \nearrow,$ one easily shows that the integral containing $I_2$ is
nonnegative. The integral containing $I_1$ is clearly nonnegative, because its
integrand is. Hence, $\int_{\cc} {\Phi}_p(\bd f, \be f) \geq 0.$ \bigskip
\bigskip
\def\T{\Bbb T}
\noindent 8. \bf Numerical Evidence. \rm
In this section, we present numerical evidence in favor of Conjecture~1.
Let $\T$ be the space $[0,1]$ with $0$ and $1$ identified. Then
$W^{1,2}(\T^2,\cc)$ will denote the Sobolev space of complex valued functions
$f:[0,1]^2 \to \cc$ such that $f(0, y) \equiv f(1, y)$, $f(x, 0) \equiv f(x,
1)$, and both $f$ and its distributional derivatives are in $L_2$. We will
work with the following conjecture, which is equivalent to Conjecture~1.
\proclaim {Conjecture 4} Let $f\in W^{1,2}(\T^2, \cc).$ Then
$$\int_{\T^2} L(\bd f, \be f)
\geq 0. $$ \endproclaim
The approach is to consider piecewise linear functions described as
follows. Let $N$ be a natural number. Let $p_n$ be the fractional
part of $n/N$ (so that $p_{N+n} = p_n$).
Split $\T^2$ into triangles $\Delta^+_{m,n}$ with corners
$(p_m,p_n)$, $(p_{m+1},p_n)$, $(p_m,p_{n+1})$, and triangles
$\Delta^-_{m,n}$ with corners
$(p_m,p_n)$, $(p_{m-1},p_n)$, $(p_m,p_{n-1})$.
We will say that
$u:\T^2 \to \cc$ is an element of $\Cal P_N$ if $u$ is continuous, and
linear on
each of the triangles $\Delta^+_{m,n}$ and $\Delta^-_{m,n}$. In this
way, once one knows that $u$ is an element of $\Cal P_N$, then $u$
is totally determined by its values at $(p_m,p_n)_{0\le m,n \le N-1}$.
Thus $\Cal P_N$ is a $2 N^2$ real dimensional space. Let
$\iota : \rr^{2N^2} \to \Cal P_N$ denote an isomorphism. Our goal
is to check whether the function $F_N:\rr^{2 N^2} \to \rr$ always takes
positive
values, where
$$ F_N(x) = \int_{\T^2} L(\bd (\iota x), \be (\iota x)) .$$
In fact, by an approximation argument, Conjecture~4 is equivalent to showing
that $F_N(x) \ge 0$ for all $x \in \rr^{2N^2}$ and all $N \ge 1$.
We obtained much numerical evidence to support this conjecture.
The algorithm was to choose a vector $x \in \rr^{2N^2}$ at random,
then minimize $F_N$, with $x$ as starting point, using the
conjugate gradient method described in Chapter~10.6 in [PTVF]. This was
done for various values of $N$, ranging from $6$ to $100$. In every
case, it was found, up to machine precision, that $F_N$ always
takes non-negative values. The results were verified independently
using Maple.
To implement this algorithm, it was necessary to compute
the gradient $\nabla F_N$. Because of the special nature of
this function, the computations needed to do this were not much more arduous
than the computations required for $F_N$. The formulae required to
find $\nabla F_N$ were determined using Maple.
Other interesting facts emerged. For a given $x \in \rr^{2N^2}$, we
may consider the function $h:\rr \to \rr$ given by
$$ h(t) = F_N(t x) .$$
It was found that this function is always increasing for $t \ge 0$, and
always decreasing for $t \le 0$. However, it was also found
that the function $h$ is {\it not\/} necessarily convex.
This last fact is interesting, because if in Conjecture~4 the function
$L$ were to be replaced by a convex function, then $h$ would be convex.
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