\input amstex \loadbold \documentstyle{amsppt} \vsize=7.5 in \hsize=6.0 in \parskip\smallskipamount \overfullrule=0pt \magnification\magstep1 \pageno=1 \document \baselineskip=18pt \define\dd{\Bbb D} \define\eh{e^{i\theta}} \define\th{\theta} \define\cc{\Bbb C} \define\si{\sigma} \define\bd{\partial} \define\be{\overline{\bd}} \define\iny{\infty} \define\nui{{\nu}_1} \define\zb{\overline{z}} \define\Rf{\text{Re}} \define\hf{{1\over 2}} \define\ze{\zeta} \define\zeb{\overline{\zeta}} \define\rf{\text{Re}\,} \define\lio{Lip_0(\cc)} \def\icc{\int_{\cc}} \def\rr{\Bbb R} \define\rpl{{\Bbb R}^+} \define\loc{\text{loc}} \define\1op{{1\over \pi}} \topmatter \title Some conjectures about integral means of $\partial f$ and $\overline{\partial} f.$ \endtitle \author Albert Baernstein II and Stephen J. Montgomery-Smith \endauthor \address Washington University, St. Louis, Missouri 63130, \newline University of Missouri, Columbia, Missouri 65211-0001 \endaddress \email al\@math.wustl.edu, Stephen\@math.missouri.edu \endemail \thanks The first author was supported in part by NSF grants DMS-9206319 and DMS-9501293. The second author was supported in part by NSF grant DMS-9424396 \endthanks \endtopmatter \noindent 1. \bf The problems. \rm In this note we shall discuss some conjectural integral inequalities which are related to quasiconformal mappings, singular integrals, martingales and the calculus of variations. For a function $f:\cc \rightarrow \cc,$ denote the formal complex derivatives by $$\bd f = {{\bd f} \over {\bd z}} = \hf ({{\bd f} \over {\bd x}} - i{{\bd f} \over {\bd y}}), \quad \be f = {{\bd f} \over {\be z}} = \hf ({{\bd f} \over {\bd x}} + i{{\bd f} \over {\bd y}}).$$ \noindent Define a function $L:\cc \times \cc \rightarrow \rr$ by \align L(z, w) & = |z|^2 - |w|^2,\quad \text{if} \quad |z| + |w| \leq 1, \\ & = 2|z| - 1, \quad \text{if} \quad |z| + |w| > 1.\endalign \proclaim {Conjecture 1} (V. \v Sver\'ak) Let $f\in \dot W^{1,2}(\cc, \cc).$ Then $$\icc L(\bd f, \be f) \geq 0. \tag1.1$$ \endproclaim Here we denote by $\dot W^{1,2}(\cc, \cc)$ denotes the homogeneous" Sobolev space of complex valued locally integrable functions in the plane whose distributional first derivatives are in $L^2$ on the plane. Integrals without specified variables are understood to be with respect to Lebesgue measure. Since $\bd (\overline{f}) = \overline{\be f},$ Conjecture 1 is true if and only if (1.1) always holds when $L(\bd f,\be f)$ in the integral is replaced by $L(\be f, \bd f).$ The function $L,$ with a plus 1 added to the right hand side, was introduced by Burkholder [Bu4], [Bu5, p.20]. In his setting, the variables $z$ and $w$ are taken from an arbitrary Hilbert space. It appears independently in work of \v Sver\'ak [Sv1], who considered the question, as yet unresolved, of whether functions belonging to a certain class which contains a function naturally associated to $L$ are quasiconvex". As will be explained in \S 5, quasiconvexity of this function implies (1.1). For $p\in (1, \infty),$ set $$p^* = \max\; (p, p'), \quad \text{where}\quad {1\over p} + {1\over {p'}} = 1.$$ Again following Burkholder [Bu5, p.16], [Bu3, p.77], [Bu4, p.8], define functions ${\Phi}_p :\cc \times \cc \rightarrow \Bbb R\,$ by $${\Phi}_p(z,w) = {\alpha}_p ((p^* - 1)|z| - |w|)\,(|z| + |w|)^{p-1},\quad {\alpha_p} = p(1 - {1\over {p^*}})^{p-1}.$$ For $1 1)}.$ Then, for $21,$ 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 $12$ 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$$Now E is either empty, or is a single interval (0,R], with 0 1\}. Then, from (6.1),$$\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.\tag6.11\endalign$$For r\in [0,\infty), define F(r) = r^{-1}g + g' - 1. If r\in E, then$$ F(r) \leq L(\bd f(z), \be f(z)),\tag6.12$$by (6.11). If r\notin E, then the third equation in (6.1) implies that g(r)\leq r and |g'(r)|\leq 1. Hence, for r\notin E,$$ F(r) - L(\bd f(z), \be f(z)) = r^{-1}g + g' - 1 - r^{-1}gg' =(1-r^{-1}g)(g'-1) \leq 0.$$Thus, (6.12) holds for all r\in [0,\infty). If the set \{r\in [0,\infty): g(r)\geq r\}, is nonempty, let R denote its supremum. If the set is empty, define R=0. Then 0\leq R <\infty, since g=o(1) at \infty, and g(R) = R. Since (6.12) holds for r\in [0, \infty), we have$${1\over {2\pi}} \int_{|z|R} L(\bd f, \be f) = \int_R^{\infty} gg'\,dr + \int_{E\cap (R,\infty)} (rG - gg')\,dr.\tag6.14$$From (6.13) and (6.14), it follows that$${1\over {2\pi}} \int_{\cc} L(\bd f, \be f) \geq \int_{E\cap (R,\infty)} (rG - gg')\,dr.\tag6.15$$Since g(r) < r for r>R, it follows from (6.1) that E\cap (R,\infty) = \{r\in (R,\infty): |g'(r)| > 1\}. If E\cap (R,\infty) is empty, then (6.9) follows from (6.15). Assume E\cap (R,\infty) is nonempty. Then it is a finite or countable union of open intervals (r_1, r_2)\subset (R, \infty), on each of which either g' is everywhere >1 or g' is everywhere <-1. The hypothesis f\in \dot W^{1,2} insures that all endpoints r_2 are finite. To prove (6.9), it suffices, in view of (6.15), to prove that, for each such (r_1, r_2),$$\int_{r_1}^{r_2} (rG-gg')\,dr \geq 0.\tag6.16$$Suppose that g'>1 on (r_1, r_2). Then, on (r_1, r_2),$$rG-gg' = g + rg' -r -gg' = - \hf {d\over {dr}}(r-g)^2.$$Hence,$$\int_{r_1}^{r_2} (rG-gg')\,dr = \hf [(r_1-g(r_1) )^2 - (r_2-g(r_2))^2].\tag6.17$$But$$g'>1 \implies g(r_2) - g(r_1) > r_2 - r_1 \implies r_1 - g(r_1) > r_2 - g(r_2) > 0.$$Thus, the integral in (6.17) is >0. Suppose that g'< -1 on (r_1, r_2). Then, on (r_1, r_2),$$rG-gg' = - g - rg' -r -gg' = - \hf {d\over {dr}}(r+g)^2.$$Hence,$$\int_{r_1}^{r_2} (rG-gg')\,dr = \hf [(r_1 + g(r_1) )^2 - (r_2 + g(r_2))^2].\tag6.18$$But$$g'<-1 \implies g(r_2) - g(r_1) < - r_2 + r_1 \implies r_1 + g(r_1) > r_2 + g(r_2) > 0.$$Thus, the integral in (6.18) is >0. The proof of (6.9) is complete.\enddemo \bigskip \noindent 7. \bf Some other partial results. \rm There are a few other classes of functions, in addition to the stretch functions, for which we can confirm Conjectures 1 and or 2. \proclaim {Theorem 3} For a,b\in \cc,\; k = 1,2,3,..., Conjecture 1 is true for$$\align f(z) & = az^k + b{\overline{z}}^k, \quad |z| \leq 1, \\ & = a{\overline{z}\,}^{-k} + bz^{-k}, \quad |z| \geq 1. \endalign$$\endproclaim \proclaim {Theorem 4} For 12. Suppose that f\in \dot W^{1,p}(\cc) is harmonic in |z|<1 and in 1 < |z| \leq \infty. Then there exist holomorphic functions g and h in |z|<1 such that$$\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. 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