%Latest version 11/4/03 e-mailed by Haim %Barbara corrections 10/8/03 %Barbara 9/16/03-9/19/03 %Petru 09/14/03 %Latest Version 9/23/03 corrections by fax \input amstex \documentstyle{amsppt} \magnification 1200 \pageheight{8.5truein} \def\Mint{\diagup\hskip-.50truecm\int} \def\mint{\diagup\hskip-.39truecm\int} \def\la{\langle} \def\ra{\rangle} \NoRunningHeads \def\d{\displaystyle} \def\rank{\text{rank\,}} \def\Spec{\text{Spec\,}} \def\ov{\overline} \def\loc{\text{loc\,}} \def\varep{\varepsilon} \def\vp{\varphi} \def\wh{\widehat} \def\wt{\widetilde} \def\supp{\text {supp\,}} \def\dist{\text {dist\,}} \def\mes{\text {mes\,}} \def\diam{\text {diam\,}} \def\Re{\text {\,Re\,}} \def\Im{\text {\,Im\,}} \def\1{1 \!\!\!1} \def\bn{{\bar n}} \def\k{\lower 3pt\hbox{$\bar$}k} \def\tl{\bar{\topsmash \ell}} \def\kk{\overset{\raise3pt\hbox{$\scriptscriptstyle\approx$}}\to {\topsmash k}}\def\bm{{\bar m}} \font\biggermath=cmex10 scaled\magstep4 \def\frownover#1{\lower4pt\hbox{$\topshave{\operatornamewithlimits{#1} \limits^{\hbox{\biggermath\char'147}}}$}} \def\tphi{\widetilde{\topsmash\phi}^{-1}} \NoBlackBoxes \parskip=\medskipamount %\null\vskip 1 true in \topmatter \title Lifting, Degree and Distributional Jacobian Revisited\endtitle \author {JEAN BOURGAIN$^{(1)}$, HA\"IM BREZIS$^{(2),(3)}$ AND PETRU MIRONESCU$^{(4)}$} \endauthor \bigskip \address ${}^{\text {(1)}}$ INSTITUTE FOR ADVANCED STUDY\endgraf PRINCETON, NJ 08540\endgraf \endaddress \email bourgain\@math.ias.edu \endemail \null \address ${}^{\text {(2)}}$ LABORATOIRE J. -L. LIONS\endgraf UNIVERSIT\'E P. ET M. CURIE, B.C. 187\endgraf 4 PL. JUSSIEU\endgraf 75252 PARIS CEDEX 05\endgraf \endaddress \email brezis\@ccr.jussieu.fr \endemail \address ${}^{\text{(3)}}$ RUTGERS UNIVERSITY\endgraf DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS\endgraf 110 FRELINGHUYSEN RD, PISCATAWAY, NJ 08854\endgraf \endaddress \email brezis\@math.rutgers.edu\endemail \null \address ${}^{\text {(4)}}$ D\'EPARTEMENT DE MATH\'EMATIQUES\endgraf UNIVERSIT\'E PARIS-SUD\endgraf 91405 ORSAY\endgraf \endaddress \email Petru.Mironescu\@math.u-psud.fr\endemail \bigskip \endtopmatter \document \subhead 0. Introduction\endsubhead Let $g:I = (0, 1) \to S^1$. If $g\in VMO$, we may write $g=e^{i\varphi}$ for some $\vp\in$VMO ; this $\varphi$ is unique modulo $2\pi$ (see [BN] and the earlier work [CM]). There is no control of $|\varphi|_{BMO}$ in terms of $|g|_{BMO}$, since we always have $|g|_{BMO}\leq 2$ and $|\varphi|_{BMO}$ can be arbitrarily large ; recall, however, that, when $|g|_{BMO}$ is sufficiently small, there is a linear estimate $|\varphi|_{BMO} \leq C|g|_{BMO}$ (see [CM], Theorem 4 in [BN], and Remark 1 below). We are going to establish that a norm slightly stronger than $|g|_{BMO}$ {\bf does} control $|\varphi|_{BMO}$. Consider, for $1 < p < \infty, \; 0 < s < 1,$ the fractional Sobolev space $W^{s,p}(I)$, equipped with its standard seminorm $$ |g|_{s,p} = \left( \int\limits_I\int\limits_I \frac{|g(x) - g(y)|^p}{|x-y|^{1+sp}} dxdy\right)^{1/p}. $$ Set $$ W^{s,p} (I; S^1) = \{ g\in W^{s, p}(I;\Bbb R^2); |g| = 1\ \hbox{\rm a.e.}\}. $$ Recall (see [BBM1]) that, if $g\in W^{1/p,p} (I;S^1)$, then $g=e^{i\varphi}$ for some $\varphi\in W^{1/p,p} (I;\Bbb R)$ ; this $\varphi$ is unique modulo $2\pi$ . Again, there is no estimate of $|\varphi|_{1/p,p}$ in terms of $|g|_{1/p,p}$. The canonical example (see [BBM1]) is the following : let $$ \varphi_n(x) = \cases 0, & \text{ if }\quad 0 1/2 + 1/n\endcases . $$ Then $|\varphi_n|_{1/p, p} \to \infty$, while $|e^{i\varphi_n}|_{1/p, p}\leq C$. In view of the injection $$ W^{1/p, p} (I) \hookrightarrow VMO (I), \; 1 < p< \infty, $$ (see, e.g., [BN] and [T]), it is natural to ask whether a control of $|g|_{1/p,p}$ yields a control of $|\varphi|_{BMO}$. This is indeed true : \proclaim{Theorem 1} Let $1< p < \infty$. Let $\varphi \in W^{1/p,p}(I;\Bbb R)$ and $g = e^{i\varphi}$. Then $$ |\varphi|_{BMO} \leq C_p (|g|^p_{1/p, p} + |g|_{1/p, p}).\tag 0.1 $$ \endproclaim \remark{Remark 1} The $p-th$ power growth in (0.1) is optimal when $|g|_{1/p, p}$ is large. This is easily seen by choosing $\varphi_n (x) = n x$. When $|g|_{1/p,p}$ is small, the linear growth in (0.1) is a special case of a result of [CM], namely $$ |\varphi|_{BMO} \leq C|g|_{BMO} \; \text{ if } \; | g|_{BMO} \leq \delta,\tag 0.2 $$ where $\delta$ is a sufficiently small constant. \endremark \remark{Remark 2} When $p=2$, estimate (0.1) can be derived from Theorem 3 in [BBM4] (announced in [BBM2] ; see also [BB]) which asserts that, if $g\in H^{1/2} (I; S^1)$, then we may write $g=e^{i(\varphi_1 + \varphi_2)}$, with $$ |\varphi_1|_{1/2, 2} \leq C|g|_{1/2, 2}\tag 0.3 $$ and $$ |\varphi_2|_{W^{1,1}} \leq C|g|^2_{1/2, 2}.\tag 0.4 $$ Since $$ |\varphi_1 + \varphi_2|_{BMO} \leq C(|\varphi_1|_{1/2, 2} + |\varphi_2|_{W^{1,1}}), $$ estimate (0.1) for $p=2$ follows from (0.3), (0.4). \bigskip Note that, if Theorem 1 holds for some $p$, it also holds for every $q\in (1, p)$ ; this follows from (0.1) and (0.2). Hence Theorem 1 for $1< p \leq 2$ is a consequence of (0.3) - (0.4). The main novelty concerns the case $p> 2$ ; our argument relies on a completely different approach. In fact, we do not know whether (0.3) - (0.4) still hold when 2 is replaced by $p$ : \bigskip \noindent{\bf Open Problem 1.} Let $\varphi\in C^\infty(\bar I; \Bbb R)$, $g = e^{i\varphi}$ and $p>2$. Does there exist a decomposition $\varphi = \varphi_1 + \varphi_2$, with $$ |\varphi_1|_{1/p, p} \leq C|g|_{1/p, p}\tag 0.3$'$ $$ and $$ |\varphi_2|_{W^{1,1}} \leq C|g|_{1/p, p}^p\ ? \tag 0.4$'$ $$ Same question when $I$ is replaced by $(0, 1)^N$. \bigskip An immediate consequence of Theorem 1 is the following \endremark \proclaim{Corollary 1} Set $Q=(0,1)^N$. Let $N N$ and $g\in W^{N/p, p} (S^N; S^N)$. Then $$ |\deg g |\leq C_{p, N} \; |g|^p_{N/p, p}.\tag 0.9 $$ \endproclaim Although the conclusions of Theorems 1 and 2 are different in nature, the proofs we present below bear some similarities. \remark{Remark 3} For $g\in W^{1,N}(S^N; S^N)$, the estimate $$ |\deg g| \leq C_N \int_{S^N} |\nabla g |^N\tag 0.9$'$ $$ is well-known and follows from Kronecker's formula $$ \deg g = \Mint_{S^N} \det (\nabla g)=\Mint_{S^N} \det (\nabla g , g)\ \tag 0.10 $$ (in the first integral, $g$ is regarded as a map from $S^N$ into itself and ``det'' denotes the determinant of an $N\times N$ matrix ; in the second integral, $g$ is considered as a $\Bbb R^{N+1}$-valued map, and ``det'' denotes the determinant of an $(N+1)\times (N+1)$ matrix). \endremark In fact, we will use (0.10) in the proof of Theorem 2. It is presumably possible to rederive (0.9$'$) as a limiting case of (0.9) via a careful analysis of $C_{p, N}$ as $p\searrow N$, in the spirit of [BBM3]. Estimate (0.9), which asserts that for every $p>N$, $$ |\deg g|\leq C_{p,N}\int_{S^N}\int_{S^N} \frac{|g(x) - g(y)|^p}{|x-y|^{2N}}dxdy, $$ suggests the following stronger estimate: \noindent{\bf Open Problem 2.} Is it true that, for every $g\in C^0(S^N; S^N)$, $$ |\deg g| \leq C_N\iint_{\{(x, y) \in S^N \times S^N; |g(x)-g(y)|> 1/10\}} |x-y|^{-2N} dxdy \,\quad { } ? $$ The answer to Open Problem 2 is positive when $N=1$; the proof is given in [BBM5], where we also present an improvement of Theorem 1 in the same spirit. \medskip We next discuss the distributional Jacobian of maps $g\in W^{N/p,p}(S^{N+1}; S^N)$. Recall that, if $g$ is a smooth map from $S^{N+1}$ into $\Bbb R^{N+1}$, its distributional Jacobian is defined through its action on smooth functions $\zeta\in C^\infty(S^{N+1};\Bbb R)$ by the formula $$ \la\text{\rm Det} (\nabla g), \zeta \ra = -\frac{1}{N+1} \sum^{N+1}_{j=1} \int_{S^{N+1}} \zeta_{x_j}\det (g_{x_1}, \ldots, g_{x_{j-1}},g, g_{x_{j+1}},\ldots, g_{x_{N+1}})\ ;\tag 0.11 $$ here, the derivatives are computed pointwise in an orthonormal frame such that \newline $(x_1, \ldots, x_{N+1}, n)$ is direct, where $n$ is the outward normal to $S^{N+1}$ (this integrand is frame invariant). Note that formula (0.11) still makes sense when $$ g\in W^{1,N} (S^{N+1};\Bbb R^{N+1})\cap L^\infty $$ and $\zeta \in W^{1,\infty} (S^{N+1}; \Bbb R)$. Observe also that if $g\in C^1(S^{N+1}; S^N)$, then its Jacobian determinant vanishes pointwise. By density, it follows that Det$(\nabla g)=0$ for every $g\in W^{1, N+1}(S^{N+1} ; S^N)$. On the other hand, it is standard to construct maps in $W^{1, N}(S^{N+1} ; S^N)$ (and even in $W^{1, q}$, $\forall\ q\lambda\}$. Note that, for a.e. $\lambda$, $g_{|\Gamma_\lambda} \in W^{N/p,p}(\Gamma_\lambda; S^N)\subset$ VMO $(\Gamma_\lambda; S^N),$ so that $\deg (g;\Gamma_\lambda)$ makes sense (by [BN]). \remark{ Remark 4} a) Since $W^{1,N}(S^{N+1} ; S^N)\cap W^{N/p,p}$ is dense in $W^{N/p,p}(S^{N+1} ; S^N)$, $Np$, and the two definitions of $T(g)$ (relative to $p$ and to $q$) coincide. \noindent d) We have reached here the ``largest'' Sobolev class to which one can extend the distributional Jacobian ; when $sp p$, and the two definitions of Det$(\nabla g)$ (relative to $p$ %and to $q$) coincide. We have reached here the ``largest'' Sobolev %class in which one can define Det$(\nabla g)$ ; when $sp N$ and $g\in W^{N/p, p} %(S^{N+1}; S^N)$. As in [BCL] and [BBM2], we define a distribution %$T(g)$ which describes the location and degree of the topological %singularities of $g$. Its action on a smooth function $\zeta\in %C^\infty (S^{N+1};\Bbb R)$ is %$$ %\la T(g), \zeta\ra = \int \deg (g, \Gamma_\lambda) d\lambda\tag 0.11 %$$ %where $\Gamma_\lambda=\{x; \zeta (x) = \lambda\}$. It is not clear %from this definition neither that $\la T(g), \zeta\ra$ makes sense, %nor that it is linear in $\zeta$. % %\proclaim{Theorem 3} There are points $P_i, N_i \in S^{N+1}$ such %that %$$ %\sum^\infty_{i=1} |P_i - N_i|\leq C(|g|^p_{N/p,p} + 1)\tag 0.12 %$$ %and %$$ %\la T(g), \zeta \ra = |S^N|\sum^\infty_{i=1} (\zeta(P_i) - %\zeta(N_i)), \forall\ \zeta \in C^\infty (S^{N+1}; \Bbb R).\tag 0.13 %$$ %\endproclaim % %As a consequence of Theorem 3, we may define the length of the minimal connection of $g$, %$$ %L(g)= \frac{1}{|S^N|} \text{\rm Max}\{\la T(g), \zeta\ra; |\nabla %\zeta|\leq 1\text{ on } S^{N+1}\}\leq C(|g|^p_{N/p,p} + 1).\tag 0.14 %$$ % % %The proof of Theorem 3 relies heavily on Theorem 2 combined with a %density result concerning maps with a finite number of singularities, established in Appendix A. \subhead 1. Proofs of Theorems 1 and 2 \endsubhead Let $g\in VMO(S^N; S^N)$ and let $u$ be its harmonic extension to $B^{N+1}$ (with values into $B^{N+1}$). Let $v(x, \varep) = u((1-\varep) x)$, $x\in S^N$, $0< \varep \leq 1$. We have $$|v(x, \varep)|\to 1\text{ uniformly in $x$ as } \varep \to 0,\tag 1.1$$ %$$\exists \gamma > 0 \text{ such that } |g|_{BMO(B_\varep(x))} \leq \gamma %\Rightarrow |v(x, \varep)|\geq 1/2,\tag 1.2$$ $$|\nabla v(x, \varep)|\leq C/\varep, \; \forall\ x \in S^N, \text{ where } C \text{ is an absolute constant}\ ; \tag 1.2$$ (for the proof of (1.1), see [BN]). Set, for every $x\in S^N$, $$ d (x) = \cases 1/2, \text{ if }\quad |v(x, \varep) |>1/2, \text{ for every } \varep \in (0, 1/2]\\ \text{Min}\{\varep \in (0,1/2]\ ;\ |v(x, \varep)|\leq 1/2 \}, \text{ otherwise }\endcases . $$ \noindent In other words, $d(x)$=min$(\ell(x),1/2)$, where $\ell(x)$ is the length of the largest radial interval coming from $x\in S^N$, on which $|u|\geq 1/2.$ Clearly, $$ G = \{ y \in B^{N+1}\ ;\ |u(y)|\leq 1/2\} \subset \bigcup_{x\in S^N} [0, (1-d(x))x]. \tag 1.3 $$ We start with the following ingredient which is of interest in itself. \proclaim{Theorem 4} For $g\in C^1 (S^N; S^N)$, we have $$|\deg g | \leq C\ I(g), $$ where $$ I(g)=\int_{S^N}\frac{1}{(d(x))^N}.$$ \endproclaim \noindent The proof of Theorem 4 relies on \proclaim{Lemma 1} We have $$ \int_G |\nabla u |^{N+1} \leq C \; I(g).\tag 1.4 $$ \endproclaim \demo{Proof of the lemma} By (1.2) and (1.3), we have $$ \aligned \int_G |\nabla u |^{N+1} dy &\leq C\int_{S^N}\bigg( \int^{1-d(x)}_{0} \frac{r^N}{(1-r)^{N+1}} \; dr \bigg) d{x} \\ &\leq C\int_{S^N} \bigg( \int^{1-d(x)}_0\frac{1}{(1- r)^{N+1}} \; dr\bigg) d{x} = C'I(g). \endaligned $$ \enddemo \demo{Proof of Theorem 4} Set, for $y\in B^{N+1}$, $$ \tilde u (y) = \cases u(y)/|u(y)|, & \text{ if } \; |u(y)|> 1/2\\ 2u (y), & \text{ if } \; |u(y)|\leq 1/2.\endcases $$ Note that $\tilde u = g$ on $S^N$ and thus, by Kronecker's formula (0.10), we have $$ \deg g = \Mint_{S^N} \det (\nabla g) = \Mint_{B^{N+1}}\det (\nabla \tilde u). $$ [To prove the last equality, consider the vector field $$ D=(D_1, \ldots, D_{N+1}) $$ where $$ D_j = \det (\tilde u_{x_1}, \ldots, \tilde u_{x_{j-1}}, \tilde u,\tilde u_{x_{j+1}}, \ldots, \tilde u_{x_{N+1}}). $$ Clearly, we have $$ \text{ div } D= (N+1) \det (\nabla \tilde u) $$ and thus $$ \Mint_{B^{N+1}} \det (\nabla \tilde u) = \frac{(N+1)^{-1}}{|B_{N+1}|}\int_{S^N} D\cdot \nu, $$ where $\nu$ is the outward normal to $S^N$. On the other hand, it is easy to see that $D\cdot \nu= \det(\nabla g)$, where the $N\times N$ Jacobian determinant $\det(\nabla g)$ is computed with respect to any orthonormal frame in the tangent space to $S^N$ at $x$ and in the tangent space to $S^N$ at $g(x)$.] Since $|\tilde u(y)|=1$ on $ B^{N+1}\setminus G$ we have $\det(\nabla \tilde u) = 0$ on $B^{N+1}\setminus G$ and thus $$ \deg g = \frac{1}{|B^{N+1}|}\int_G \det (\nabla \tilde u) = \frac{2^{N+1}}{|B^{N+1}|} \int_G \det (\nabla u ). $$ Hence $$ |\deg g | \leq C\int_G |\nabla u |^{N+1} \leq C' I(g) \text{ by Lemma 1}. $$ \noindent [There is an alternative proof of the first inequality above using differential forms. As is well-known $$ \deg g = \deg (u, B^{N+1},0). $$ \noindent The latter can be given as the integral of the pull back, under the map $u$, of any smooth $(N+1)$-form $\mu$, with compact support in the open ball $B^{N+1}$, and whose integral is 1. Take $\mu= h(z)dz$, where $h$ is any smooth function with support in $\{z\in\Bbb R^{N+1}; |z|<1/2\}$, and whose integral is 1. Then we find $$ \deg g = \deg (u, B^{N+1},0)=\int_{B^{N+1}}h(u(y)) \det(\nabla u(y))dy, $$ which yields the desired estimate.] \bigskip In the proof of Theorem 2 we will also use the following \enddemo \proclaim{Lemma 2} Let $p> N$, $g \in W^{N/p, p}(S^N; S^N)$. Then $$ \int_{S^N}\frac{1}{(d(x))^N} \leq C(|g|^p_{N/p, p} + 1).\tag 1.5 $$ \endproclaim \demo{Proof} It suffices to consider only the $x$'s such that $d(x)< 1/2$. For any such $x$, we have $$ 1/2 \leq |u ((1-d(x))x) - g(x)| \leq d(x)^{N/p} |v|_{C^{0,N/p}(\{x\} \times(0,1/2))}\leq Cd(x)^{N/p}|v|_{(N+1)/p, p (\{x\}\times (0, 1/2))}, $$ by the embedding $W^{s,p} (0,1) \subset C^{0,\alpha}(0,1)$ where $sp > 1$ and $\alpha = s-1/p$. Thus $$ \frac{1}{d(x)^N} \leq C|v|^p_{(N+1)/p, p (\{x\} \times (0, 1/2))}.\tag 1.6 $$ Let, for $f$ defined on $B^{N+1}$ and $x\in S^{N}$, $f^x(r)=f(rx)$, $1/2\le r\le 1$. Recall the Besov type inequality (see, e.g., [A], p. 208-214) $$ \int_{S^{N}}|f^x|^p_{s,p(1/2 , 1)} dx \leq C|f|^p_{s,p(B^{N+1})}, \quad\forall\ f \in W^{s,p}(B^{N+1}).\tag 1.7 $$ Inequality (1.5) follows by combining (1.6) and (1.7) with the standard estimate \noindent $|v|_{(N+1)/p,p(S^N\times (0 , 1/2))}\le C|u|_{(N+1)/p, p} \leq C|g|_{N/p,p}$. \enddemo \demo{Proof of Theorem 2} We want to establish that, for every $g\in W^{N/p,p}(S^N; S^N)$, $$ |\deg g| \leq C|g|_{N/p,p}^p.\tag 1.8 $$ By density of $C^1(S^N;S^N)$ in $W^{N/p,p}(S^N;S^N)$ and continuity of the degree under $VMO$ convergence, it suffices to prove (1.8) for $g\in C^1(S^N; S^N)$. When $|g|_{N/p,p}$ is sufficiently small, we have $\deg g =0$, once more by continuity of the degree under VMO convergence, and thus (1.8) holds. Otherwise, (1.8) follows from Theorem 4 and Lemma 2. \enddemo \demo{Proof of Theorem 1} We will prove that $$ |\varphi|_{BMO(I)} \leq C(|g|^p_{1/p,p} + |g|_{1/p, p}).\tag 1.9 $$ As above, we may assume that $g$ is smooth. When $|g|_{1/p, p}$ is sufficiently small, (1.9) follows from the estimate $$ |\varphi|_{BMO(I)}\leq C|g|_{BMO(I)}, \text{ if } |g|_{BMO(I)} \leq \delta $$ ($\delta$ small constant) of Coifman and Meyer [CM]. In view of this and scale invariance, it suffices to establish the following weaker form of (1.9) $$ \Mint_I\Mint_I |\varphi(x) - \varphi(y)|\leq C(|g|^p_{1/p, p} + 1).\tag 1.10 $$ Extending $g$ by symmetry, we may always assume $g$ and $\varphi$ periodic, and thus defined on a circle(with $g$ of degree zero). We will prove that $$ \int_{S^1}\int_{S^1} |\varphi(x_1) - \varphi(x_2)|dx_1 dx_2\leq C(|g|^p_{1/p, p} + 1),\tag 1.11 $$ where $\varphi \in W^{1/p, p}(S^1; \Bbb R)$ and $g = e^{i\varphi}$. As in the proof of Lemma 1, and by Lemma 2, we have $$ \int_{\{y=rx ; r\leq 1 - d(x)\}} |\nabla u |^{2}dy\leq C(|g|^p_{1/p, p} + 1).\tag 1.12 $$ By the coarea formula, (1.3), and (1.12), we have $$\aligned \int^{1/2}_{1/3} \left(\int_{\{y\in B^2; |u(y)|= t\}}|\nabla u |\right) dt & = \int_{\{y\in B^2; 1/3 < |u(y)|< 1/2\}} |\nabla u |\; |\nabla |u|| \\ \\ & \leq \int_G |\nabla u |^2\leq C(|g|^p_{1/p, p} + 1).\endaligned $$ Thus we may find some $t\in (1/3, 1/2)$ such that $$ \int_\Gamma |\nabla u | \leq C(|g|^p_{1/p, p} + 1),\tag 1.13 $$ where $\Gamma =\{ y\ ;\ |u(y) | = t\}$. Let $\gamma_1, \gamma_2,\ldots $, be the connected components of $\Gamma$. By (1.13), we have $$ \sum_j |\deg(u, \gamma_j)|\leq \frac{1}{2\pi t} \sum_j\int_{\gamma_j} |\nabla u | \leq C(|g|^p_{1/p,p} + 1).\tag 1.14 $$ On the other hand, if $j\neq k$, then the domains enclosed by $\gamma_j$ and $\gamma_k$ have disjoint interiors, by the maximum principle. Let now $x, y \in S^1$ and consider the domains $$ U=\{ z\ ;\ |u(z)|>t\}, V\text{ as on the figure below and } \tilde W = U\cap V. $$ Let $W$ be the connected component of $\tilde W$ whose boundary contains $x$ and $y$. 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to 12.446 22.767 \putrule from 12.446 22.767 to 12.446 22.786 \putrule from 12.446 22.786 to 12.446 22.794 \putrule from 12.446 22.794 to 12.446 22.796 \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\putrule from 11.589 23.050 to 11.589 23.114 \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\putrule from 6.509 20.003 to 6.509 20.005 \plot 6.509 20.005 6.505 20.013 / \plot 6.505 20.013 6.498 20.032 / \plot 6.498 20.032 6.488 20.064 / \plot 6.488 20.064 6.475 20.108 / \plot 6.475 20.108 6.460 20.157 / \plot 6.460 20.157 6.443 20.206 / \plot 6.443 20.206 6.430 20.252 / \plot 6.430 20.252 6.418 20.295 / \plot 6.418 20.295 6.405 20.333 / \plot 6.405 20.333 6.397 20.367 / \plot 6.397 20.367 6.388 20.398 / \plot 6.388 20.398 6.382 20.426 / \plot 6.382 20.426 6.378 20.451 / \plot 6.378 20.451 6.371 20.483 / \plot 6.371 20.483 6.367 20.513 / \plot 6.367 20.513 6.363 20.542 / \plot 6.363 20.542 6.361 20.572 / \putrule from 6.361 20.572 to 6.361 20.604 \putrule from 6.361 20.604 to 6.361 20.633 \plot 6.361 20.633 6.363 20.663 / \plot 6.363 20.663 6.369 20.693 / \plot 6.369 20.693 6.375 20.720 / \plot 6.375 20.720 6.382 20.748 / \plot 6.382 20.748 6.392 20.771 / \plot 6.392 20.771 6.403 20.796 / \plot 6.403 20.796 6.416 20.822 / \plot 6.416 20.822 6.433 20.845 / \plot 6.433 20.845 6.449 20.870 / \plot 6.449 20.870 6.471 20.896 / \plot 6.471 20.896 6.494 20.923 / \plot 6.494 20.923 6.519 20.949 / \plot 6.519 20.949 6.547 20.972 / \plot 6.547 20.972 6.576 20.995 / \plot 6.576 20.995 6.604 21.014 / \plot 6.604 21.014 6.634 21.031 / \plot 6.634 21.031 6.663 21.048 / \plot 6.663 21.048 6.695 21.061 / \plot 6.695 21.061 6.720 21.071 / \plot 6.720 21.071 6.748 21.080 / \plot 6.748 21.080 6.778 21.086 / \plot 6.778 21.086 6.809 21.093 / \plot 6.809 21.093 6.841 21.095 / \plot 6.841 21.095 6.877 21.099 / \putrule from 6.877 21.099 to 6.913 21.099 \putrule from 6.913 21.099 to 6.949 21.099 \plot 6.949 21.099 6.987 21.095 / \plot 6.987 21.095 7.025 21.093 / \plot 7.025 21.093 7.061 21.086 / \plot 7.061 21.086 7.097 21.080 / \plot 7.097 21.080 7.133 21.071 / \plot 7.133 21.071 7.171 21.061 / \plot 7.171 21.061 7.199 21.052 / \plot 7.199 21.052 7.228 21.042 / \plot 7.228 21.042 7.258 21.031 / \plot 7.258 21.031 7.290 21.021 / \plot 7.290 21.021 7.324 21.006 / \plot 7.324 21.006 7.358 20.991 / \plot 7.358 20.991 7.394 20.976 / \plot 7.394 20.976 7.432 20.959 / \plot 7.432 20.959 7.468 20.940 / \plot 7.468 20.940 7.506 20.921 / \plot 7.506 20.921 7.546 20.900 / \plot 7.546 20.900 7.584 20.879 / \plot 7.584 20.879 7.622 20.858 / \plot 7.622 20.858 7.658 20.836 / \plot 7.658 20.836 7.696 20.813 / \plot 7.696 20.813 7.732 20.790 / \plot 7.732 20.790 7.768 20.767 / \plot 7.768 20.767 7.806 20.743 / \plot 7.806 20.743 7.838 20.722 / \plot 7.838 20.722 7.872 20.699 / \plot 7.872 20.699 7.906 20.673 / \plot 7.906 20.673 7.942 20.648 / \plot 7.942 20.648 7.980 20.623 / \plot 7.980 20.623 8.016 20.595 / \plot 8.016 20.595 8.056 20.568 / \plot 8.056 20.568 8.094 20.540 / \plot 8.094 20.540 8.134 20.510 / \plot 8.134 20.510 8.177 20.483 / \plot 8.177 20.483 8.217 20.453 / \plot 8.217 20.453 8.257 20.426 / \plot 8.257 20.426 8.295 20.398 / \plot 8.295 20.398 8.335 20.373 / \plot 8.335 20.373 8.371 20.348 / \plot 8.371 20.348 8.410 20.322 / \plot 8.410 20.322 8.446 20.301 / \plot 8.446 20.301 8.479 20.280 / \plot 8.479 20.280 8.513 20.259 / \plot 8.513 20.259 8.547 20.240 / \plot 8.547 20.240 8.583 20.221 / \plot 8.583 20.221 8.619 20.201 / \plot 8.619 20.201 8.655 20.185 / \plot 8.655 20.185 8.693 20.168 / \plot 8.693 20.168 8.733 20.151 / \plot 8.733 20.151 8.774 20.136 / \plot 8.774 20.136 8.814 20.121 / \plot 8.814 20.121 8.858 20.106 / \plot 8.858 20.106 8.901 20.094 / \plot 8.901 20.094 8.945 20.081 / \plot 8.945 20.081 8.989 20.068 / \plot 8.989 20.068 9.036 20.058 / \plot 9.036 20.058 9.081 20.047 / \plot 9.081 20.047 9.127 20.036 / \plot 9.127 20.036 9.174 20.028 / \plot 9.174 20.028 9.218 20.019 / \plot 9.218 20.019 9.265 20.011 / \plot 9.265 20.011 9.313 20.003 / \plot 9.313 20.003 9.358 19.996 / \plot 9.358 19.996 9.404 19.988 / \plot 9.404 19.988 9.453 19.979 / \plot 9.453 19.979 9.502 19.973 / \plot 9.502 19.973 9.555 19.964 / \plot 9.555 19.964 9.608 19.956 / \plot 9.608 19.956 9.665 19.947 / \plot 9.665 19.947 9.720 19.939 / \plot 9.720 19.939 9.777 19.928 / \plot 9.777 19.928 9.836 19.920 / \plot 9.836 19.920 9.891 19.909 / \plot 9.891 19.909 9.948 19.899 / \plot 9.948 19.899 10.003 19.890 / \plot 10.003 19.890 10.056 19.880 / \plot 10.056 19.880 10.105 19.869 / \plot 10.105 19.869 10.154 19.859 / \plot 10.154 19.859 10.198 19.848 / \plot 10.198 19.848 10.240 19.840 / \plot 10.240 19.840 10.281 19.829 / \plot 10.281 19.829 10.319 19.816 / \plot 10.319 19.816 10.370 19.801 / \plot 10.370 19.801 10.416 19.784 / \plot 10.416 19.784 10.458 19.765 / \plot 10.458 19.765 10.501 19.746 / \plot 10.501 19.746 10.539 19.725 / \plot 10.539 19.725 10.575 19.704 / \plot 10.575 19.704 10.607 19.681 / \plot 10.607 19.681 10.636 19.657 / \plot 10.636 19.657 10.662 19.636 / \plot 10.662 19.636 10.683 19.613 / \plot 10.683 19.613 10.702 19.590 / \plot 10.702 19.590 10.717 19.569 / \plot 10.717 19.569 10.731 19.547 / \plot 10.731 19.547 10.742 19.526 / \plot 10.742 19.526 10.753 19.501 / \plot 10.753 19.501 10.763 19.478 / \plot 10.763 19.478 10.772 19.450 / \plot 10.772 19.450 10.778 19.425 / \plot 10.778 19.425 10.784 19.395 / \plot 10.784 19.395 10.789 19.367 / \plot 10.789 19.367 10.791 19.340 / \plot 10.791 19.340 10.793 19.310 / \plot 10.793 19.310 10.795 19.285 / \putrule from 10.795 19.285 to 10.795 19.257 \putrule from 10.795 19.257 to 10.795 19.234 \putrule from 10.795 19.234 to 10.795 19.209 \putrule from 10.795 19.209 to 10.795 19.183 \putrule from 10.795 19.183 to 10.795 19.160 \putrule from 10.795 19.160 to 10.795 19.133 \plot 10.795 19.133 10.793 19.107 / \plot 10.793 19.107 10.791 19.078 / \plot 10.791 19.078 10.789 19.050 / \plot 10.789 19.050 10.784 19.022 / \plot 10.784 19.022 10.778 18.993 / \plot 10.778 18.993 10.772 18.967 / \plot 10.772 18.967 10.763 18.940 / \plot 10.763 18.940 10.753 18.917 / \plot 10.753 18.917 10.742 18.891 / \plot 10.742 18.891 10.729 18.866 / \plot 10.729 18.866 10.712 18.843 / \plot 10.712 18.843 10.696 18.817 / \plot 10.696 18.817 10.676 18.790 / \plot 10.676 18.790 10.653 18.762 / \plot 10.653 18.762 10.630 18.737 / \plot 10.630 18.737 10.604 18.709 / \plot 10.604 18.709 10.577 18.684 / \plot 10.577 18.684 10.552 18.661 / \plot 10.552 18.661 10.526 18.639 / \plot 10.526 18.639 10.503 18.618 / \plot 10.503 18.618 10.478 18.599 / \plot 10.478 18.599 10.456 18.584 / \plot 10.456 18.584 10.435 18.570 / \plot 10.435 18.570 10.412 18.555 / \plot 10.412 18.555 10.389 18.542 / \plot 10.389 18.542 10.361 18.527 / \plot 10.361 18.527 10.334 18.514 / \plot 10.334 18.514 10.304 18.502 / \plot 10.304 18.502 10.272 18.489 / \plot 10.272 18.489 10.240 18.479 / \plot 10.240 18.479 10.204 18.470 / \plot 10.204 18.470 10.171 18.462 / \plot 10.171 18.462 10.132 18.453 / \plot 10.132 18.453 10.094 18.447 / \plot 10.094 18.447 10.054 18.440 / \plot 10.054 18.440 10.025 18.438 / \plot 10.025 18.438 9.993 18.434 / \plot 9.993 18.434 9.959 18.432 / \plot 9.959 18.432 9.923 18.430 / \plot 9.923 18.430 9.887 18.428 / \plot 9.887 18.428 9.847 18.426 / \putrule from 9.847 18.426 to 9.804 18.426 \putrule from 9.804 18.426 to 9.760 18.426 \putrule from 9.760 18.426 to 9.716 18.426 \putrule from 9.716 18.426 to 9.669 18.426 \plot 9.669 18.426 9.622 18.428 / \plot 9.622 18.428 9.574 18.430 / \plot 9.574 18.430 9.527 18.432 / \plot 9.527 18.432 9.478 18.436 / \plot 9.478 18.436 9.432 18.440 / \plot 9.432 18.440 9.385 18.445 / \plot 9.385 18.445 9.341 18.449 / \plot 9.341 18.449 9.296 18.455 / \plot 9.296 18.455 9.252 18.462 / \plot 9.252 18.462 9.207 18.468 / \plot 9.207 18.468 9.163 18.476 / \plot 9.163 18.476 9.119 18.485 / \plot 9.119 18.485 9.074 18.493 / \plot 9.074 18.493 9.028 18.504 / \plot 9.028 18.504 8.981 18.514 / \plot 8.981 18.514 8.932 18.527 / \plot 8.932 18.527 8.884 18.540 / \plot 8.884 18.540 8.833 18.555 / \plot 8.833 18.555 8.782 18.570 / \plot 8.782 18.570 8.731 18.584 / \plot 8.731 18.584 8.680 18.601 / \plot 8.680 18.601 8.630 18.618 / \plot 8.630 18.618 8.579 18.635 / \plot 8.579 18.635 8.530 18.652 / \plot 8.530 18.652 8.481 18.669 / \plot 8.481 18.669 8.435 18.688 / \plot 8.435 18.688 8.388 18.705 / \plot 8.388 18.705 8.344 18.724 / \plot 8.344 18.724 8.299 18.741 / \plot 8.299 18.741 8.255 18.758 / \plot 8.255 18.758 8.211 18.777 / \plot 8.211 18.777 8.166 18.796 / \plot 8.166 18.796 8.122 18.815 / \plot 8.122 18.815 8.077 18.836 / \plot 8.077 18.836 8.031 18.857 / \plot 8.031 18.857 7.982 18.881 / \plot 7.982 18.881 7.933 18.904 / \plot 7.933 18.904 7.885 18.927 / \plot 7.885 18.927 7.836 18.953 / \plot 7.836 18.953 7.785 18.978 / \plot 7.785 18.978 7.736 19.003 / \plot 7.736 19.003 7.690 19.029 / \plot 7.690 19.029 7.643 19.056 / \plot 7.643 19.056 7.597 19.082 / \plot 7.597 19.082 7.552 19.107 / \plot 7.552 19.107 7.510 19.135 / \plot 7.510 19.135 7.470 19.160 / \plot 7.470 19.160 7.429 19.185 / \plot 7.429 19.185 7.391 19.211 / \plot 7.391 19.211 7.355 19.234 / \plot 7.355 19.234 7.315 19.264 / \plot 7.315 19.264 7.275 19.293 / \plot 7.275 19.293 7.237 19.323 / \plot 7.237 19.323 7.197 19.355 / \plot 7.197 19.355 7.156 19.387 / \plot 7.156 19.387 7.118 19.420 / \plot 7.118 19.420 7.078 19.454 / \plot 7.078 19.454 7.040 19.488 / \plot 7.040 19.488 7.004 19.522 / \plot 7.004 19.522 6.968 19.556 / \plot 6.968 19.556 6.932 19.590 / \plot 6.932 19.590 6.900 19.624 / \plot 6.900 19.624 6.869 19.653 / \plot 6.869 19.653 6.841 19.685 / \plot 6.841 19.685 6.816 19.713 / \plot 6.816 19.713 6.790 19.740 / \plot 6.790 19.740 6.767 19.765 / \plot 6.767 19.765 6.748 19.791 / \plot 6.748 19.791 6.723 19.820 / \plot 6.723 19.820 6.699 19.850 / \plot 6.699 19.850 6.678 19.880 / \plot 6.678 19.880 6.657 19.911 / \plot 6.657 19.911 6.636 19.943 / \plot 6.636 19.943 6.612 19.977 / \plot 6.612 19.977 6.591 20.013 / \plot 6.591 20.013 6.570 20.051 / \plot 6.570 20.051 6.549 20.087 / \plot 6.549 20.087 6.532 20.117 / \plot 6.532 20.117 6.519 20.140 / \plot 6.519 20.140 6.513 20.155 / \plot 6.513 20.155 6.509 20.161 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 13.517 20.906 12.914 20.113 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 13.568 21.679 12.425 20.282 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 13.449 23.000 12.402 21.539 / \plot 12.402 21.539 12.465 21.507 / \special{ps: 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setrgbcolor}\plot 12.524 19.241 11.985 18.669 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 13.113 23.368 12.414 22.352 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 13.144 21.177 13.176 21.145 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 11.938 19.907 11.271 19.209 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 12.287 19.812 11.430 18.923 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 11.779 22.162 10.795 20.765 / \special{ps: grestore}% % Fig POLYLINE object % 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setrgbcolor}\plot 7.017 21.050 6.382 20.384 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 10.795 19.431 10.001 18.415 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 9.013 20.102 7.933 18.927 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 10.516 19.774 9.468 18.504 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 9.970 19.844 8.922 18.574 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 9.493 19.939 8.414 18.764 / \special{ps: grestore}% % Fig POLYLINE object % 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grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 9.978 9.629 10.518 8.994 / % % arrow head % \plot 10.305 9.146 10.518 8.994 10.402 9.228 / % \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 2.957 7.641 3.099 8.181 / % % arrow head % \plot 3.096 7.919 3.099 8.181 2.973 7.951 / % \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 6.322 8.416 5.656 7.717 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 6.646 8.352 5.774 7.432 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 6.875 8.113 6.018 7.224 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 6.911 7.751 6.371 7.180 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 3.365 9.779 3.461 10.604 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 15.272 13.652 15.304 9.366 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 18.343 12.598 19.437 13.693 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\putrule from 13.938 21.876 to 14.287 21.876 \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 11.669 12.573 11.320 12.270 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 15.837 17.799 11.074 13.322 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 14.501 17.742 11.168 14.472 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 13.796 9.762 15.255 11.271 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 14.620 9.540 15.240 10.192 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 17.416 12.954 19.272 14.747 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 15.240 13.684 19.226 12.287 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 3.156 8.448 2.838 7.241 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 9.946 9.644 10.613 8.913 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 8.429 11.635 8.939 11.017 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 18.447 16.447 15.558 13.652 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 15.176 12.287 13.018 10.096 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 18.923 15.684 16.383 13.335 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 15.208 13.494 12.256 10.668 / \special{ps: grestore}% % Fig POLYLINE object % \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \special{ps: gsave 0 0 0 setrgbcolor}\plot 16.859 17.526 11.462 12.383 / \special{ps: grestore}\linethickness=0pt \putrectangle corners at 0.540 25.855 and 19.653 5.093 \endpicture} $$ Let $\gamma$ be the arc from $x$ to $y$ as above. Let $h$: $\displaystyle U\to S^1$, $\displaystyle h(z) = \frac{u(z)}{|u(z)|}$. Since $u\in W^{2/p, p}$, we have $h\in W^{2/p, p}$. Next we note that, since $g\in W^{1/p, p} \cap L^\infty$, it suffices to establish (1.10) for ${ p\geq 2}$. Assuming $p\geq 2$, we have $|h|_{2/p,p} \leq C|u|_{2/p, p} \leq C|g|_{1/p, p}$, so that it suffices to prove that $$ \int_{S^1}\int_{S^1} |\varphi(x) - \varphi(y) |dxdy\leq C(|v|_{2/p, p (S^N\times (0,1/2))}^p + |g|^p_{1/p, p} + 1) .\tag 1.15 $$ Let $\psi$ be the lifting of $h$ on $\gamma$ such that $\psi(x)=\varphi(x).$ Then $$ \varphi(y) - \varphi(x) = \psi(y) - \psi(x) \pm 2 \pi \sum\deg (u, \gamma_j), $$ where the above summation is done over the $j$'s such that $\gamma_j\subset W$. By (1.14), we have $$ |\varphi(y) - \varphi(x)|\leq |\psi(y) - \psi(x)|+ C(|g|^p_{1/p, p} + 1).\tag 1.16 $$ We next note that, if $\tilde \gamma$ is an arc on $\gamma\cap \Gamma$ with endpoints $a, b$, then $$|\psi(b) - \psi(a)|\leq 1/t\int_{\tilde \gamma}|\nabla u |.\tag 1.17$$ We write $$ \gamma = I_1 \cup \tilde \gamma_1 \cup I_2 \cup\ldots \cup I_n, $$ where $I_1, \ldots, I_n$ are line segments, $\tilde \gamma_1, \ldots, \tilde \gamma_{n-1}$ are on $\gamma \cap \Gamma$, $I_1$ has endpoints $a_1= x$ and $b_1, \tilde \gamma_1$ has endpoints $b_1$ and $a_2$, etc. By (1.13), (1.16) and (1.17), we find that $$ |\psi(y) - \psi(x)|\leq C(|g|^p_{1/p, p} + 1) + \sum^n_1 |\psi(b_j)- \psi(a_j)|.\tag 1.18 $$ We estimate the terms $|\psi(b_1)-\psi(a_1)|$ and $|\psi(b_n)-\psi(a_n)|$ in (1.18) with the help of \enddemo \proclaim{Lemma 3} Let $\psi\in C^{0,\alpha}((0, l) ; \Bbb R)$ with $0<\alpha\le 1$ and set $h=e^{i\psi}$. Then $$ |\psi(l) - \psi(0)|\leq 4(l|h|^{1/\alpha}_{C^{0,\alpha}} + l^{\alpha} |h|_{C^{0,\alpha}}).\tag 1.19 $$ \endproclaim \demo{Proof of Lemma 3} After scaling, we may always take $l=1$. Suppose first that $|h|_{C^{0,\alpha}}\le 1$. Then, clearly, $$ |\psi (1)-\psi (0)|\le 2|h(1)-h(0)|\le 2 |h|_{C^{0,\alpha}} $$ and the desired conclusion follows. \noindent When $|h|_{C^{0,\alpha}}>1$, let $n$ be the integer part of $|h|^{1/\alpha}_{C^{0,\alpha}}+1$. For $j=0,\ldots , n$, set $a_j=j/n$. Since $$ |h(a_{j+1})-h(a_j)|\le |h|_{C^{0,\alpha}}(1/n)^\alpha\le 1, $$ we deduce as above that $$ |\psi (a_{j+1})-\psi (a_j)|\le 2 |h(a_{j+1})-h(a_j)|\le 2|h|_{C^{0,\alpha}}(1/n)^\alpha,\quad j=0,\ldots , n-1. $$ Summing these inequalities for $j=0,\ldots , n-1$, we find $$ |\psi (1)-\psi (0)|\le 2|h|_{C^{0,\alpha}} n^{1-\alpha}\le 4 |h|^{1/\alpha}_{C^{0,\alpha}}, $$ since $\displaystyle n\le |h|^{1/\alpha}_{C^{0,\alpha}}+1\le 2 |h|^{1/\alpha}_{C^{0,\alpha}}\ ; $ this is again the desired conclusion. \enddemo %\proclaim{Lemma 3} Let $1< p < \infty$, $h\in W^{2/p, p}((0,l); S^1)$, %$h=e^{i\psi}$. Then %$$ %|\psi(l) - \psi(0)|\leq C(l^{1/2}|h|^{p/2}_{2/p,p} + l^{1/p} |h|_{2/p, %p}).\tag 1.19 %$$ %\endproclaim % %\demo{Proof of Lemma 3} If $l^{1/p}|h|_{2/p, p}$ is sufficiently % small, then %$$ %|h(x) - h(y)|\leq Cl^{1/p}|h|_{2/p,p} < 1,\ \forall\ x, y\in (0, l), %$$ %so that $|\psi(l) - \psi(0)|\leq Cl^{1/p}|h|_{2/p, p},$ and the %conclusion follows. When $l^{1/p} |h|_{2/p, p}$ is not small, %let $\delta > 0$ be such that $l^{1/p}|h|_{2/p, p} \leq \delta %\Rightarrow |\psi(l) - \psi(0)|\leq 1.$ Consider a partition of %$(0,l)$ as follows: % %\bigskip % %insert Figure 2 % %\bigskip % %Pick $l_1$ such that $l_1^{1/p} |h|_{2/p,p(I_1)} = \delta$, next $l_2$ %such that $l_2^{1/p}|h|_{2/p, p(I_2)} = \delta,$ the last interval of this %kind being $I_n$ ; therefore, $l^{1/p}_{n+1} |h|_{2/p, p (I_{n+1})} %\leq\delta$. % %Then %$$ %|h|^p_{2/p, p(0, l)} \geq \sum^n_1 |h|_{2/p, p(I_j)}^p = % \delta^p\sum^n_1\frac{1}{l_j}. %$$ %Since %$$ %\sum^n_1 \frac{1}{l_j} \geq n^2\bigg/\bigg(\sum^n_1 l_j\bigg)\geq %\frac{n^2}{l}, %$$ %we find %$$ %|\psi(l) - \psi(0) |\leq n+1 \leq C(l^{1/2} |h|_{2/p,p}^p + 1), %$$ %and the lemma follows. %\enddemo \demo{Proof of Theorem 1 completed} Using Lemma 3, the 1-d embedding $W^{2/p,p}\hookrightarrow C^{0,1/p}$ and the inequality $$ |\nabla u(y)|\le C\quad\text{ if }|y|\le 1/2,\tag 1.20 $$ we find that $$ |\psi (b_1)-\psi(a_1)|+|\psi (b_n)-\psi(a_n)|\le C(|v|^p_{2/p , p(\{ x\}\times (0,1/2))}+ |v|^p_{2/p , p(\{ y\}\times (0,1/2))}+1).\tag 1.21 $$ The ingredient for estimating the terms $|\psi(b_j)-\psi(a_j)|$, $j=2,\ldots , n-1$, is the inequality $$ |\psi(b_j)-\psi(a_j)|=\bigg| \int\limits_{[a_j,b_j]}\overline h\frac{\partial h}{\partial\tau}\bigg|\le C\int\limits_{[a_j,b_j]}|\nabla u|. \tag 1.22 $$ Estimate (1.22), used in conjunction with (1.20), yields $$ \sum^{n-1}_2 |\psi(b_j)- \psi(a_j)|\le C\bigg(\int\limits_{\{ rx\ ;\ 1/2\le r\le 1-d(x)\}}| \nabla u|+\int\limits_{\{ ry\ ;\ 1/2\le r\le 1-d(y)\}}|\nabla u|+1\bigg). \tag 1.23 $$ By (1.18), (1.21) and (1.23), we find that $$ \aligned |\varphi(x) - \varphi(y)| \leq C\bigg( & \int\limits_{\{ rx\ ;\ 1/2\le r\le 1-d(x)\}}|\nabla u|+\int\limits_{\{ ry\ ;\ 1/2\le r\le 1-d(y)\}}|\nabla u|\\ & +|g|^p_{1/p, p} + |v|^{p}_{2/p, p(\{x\}\times (0, 1/2))} + |v|^{p}_{2/p, p(\{y\}\times (0, 1/2))}+1\bigg). \endaligned \tag 1.24 $$ \enddemo The conclusion follows, with the help of (1.7) and (1.12), by integrating (1.24). \demo{Proof of Corollary 1} Recall that we want to obtain the estimate $$ |\varphi|_{BMO} \leq C(|g|^p_{N/p,p} + |g|_{N/p,p}).\tag 1.25 $$ When $|g|_{N/p,p}$ is small, the conclusion follows from Theorem 4 in [BN]. Otherwise, assume, e.g., $N= 2$. It suffices (after scaling) to prove that $$ J= \iint_{(0,1)^2 \times (0,1)^2} |\varphi(x) - \varphi(y)| \leq C (|g|_{2/p, p}^p+1). \tag 1.26 $$ This follows from $$ |\varphi(x) - \varphi(y)|\leq |\varphi(x_1, x_2)-\varphi(x_1, y_2) |+|\varphi(x_1, y_2) - \varphi(y_1, y_2)|,\tag 1.27 $$ which, combined with Theorem 3, yields $$ J\leq C \left(1+\int|g|^p_ {1/p,p (\{s\} \times [0, 1])} ds + \int|g|_{1/p,p([0, 1]\times\{t\})}^p dt \right) \leq C(|g|^p_{2/p,p} + 1).\tag 1.28 $$ \enddemo \subhead 2. Proof of Theorem 3 \endsubhead We want to prove that the distribution Det$(\nabla g)$, initially defined in (0.11) for $g\in W^{1,N}(S^{N+1}; S^N)$, makes sense for $g\in W^{N/p,p}(S^{N+1}; S^N)$, $NN+1$. In this case, we are going to choose a {\bf special} extension $F$ of $g$ such that : \noindent (i) $F\in C^\infty(B^{N+2} ; \Bbb R^{N+1})$ ; \noindent (ii) $F\in W^{(N+1)/p, p}(B^{N+2})$ ; \noindent (iii) $H$ (defined by (2.2)) belongs to $L^1$. \medskip For every $g\in W^{N/p , p}(S^{N+1} ; S^N)$, let $u$ be the harmonic extension of $g$ to $B^{N+2}$ (with values into $B^{N+1}$). \medskip \noindent [{\bf Warning :} Here, $g$ need not be VMO, in contrast with the situation we encountered in the proofs of Theorems 1 and 2. In general, $|u(y)|$ does {\bf not} tend to $1$ as $|y|\to 1$ and the set $\{ y\in \overline{B^{N+2}}\ ;\ |u(y)|\le 1/2\}$ is {\bf not} a compact subset of the open ball $B^{N+2}$. This will become particularly transparent later on at the points of $S^{N+1}$ where $g$ has topological singularities.] \medskip Fix any map $\Phi\in C^\infty (\Bbb R^{N+1} ; \Bbb R^{N+1})$ such that $\d\Phi (X)=X/|X|$ if $|X|\ge 1/2$. The {\bf special} $F$ we will use is defined by $$ F(y)=\Phi (u(y)),\quad\quad\forall\ y\in B^{N+2}.\tag 2.4 $$ Note that $F\in C^\infty (B^{N+2} ; B^{N+1})$ and that $F(y)\in S^N$ when $|u(y)|\ge 1/2$. Consider the vector-field $H$ defined by (2.2) for this $F$ and observe that $H=0$ in the open set $\{ y\in B^{N+2}\ ;\ |u(y)| > 1/2\}$. For every $\xi\in W^{1,\infty}(B^{N+2} ; \Bbb R)$, define $$ Y(\xi)=X(F,\xi)\tag 2.5 $$ as in (2.1)-(2.2). This requires a justification, since it is not clear that $H\in L^1$. A key ingredient in the proof of Theorem 3 is \proclaim{Lemma 4} For each $g\in W^{N/p,p} (S^{N+1}; S^N)$, we have $H\in L^1(B^{N+2} ; \Bbb R^{N+2})$, so that the quantity $Y(\xi)$ is well-defined . Moreover : \noindent a) $Y(\xi_1)=Y(\xi_2)$ when $\xi_1=\xi_2$ on $S^{N+1}$. \noindent Set $\la T(g) , \zeta\ra =Y(\xi)$, where $\xi$ is any Lipschitz extension of a given $\zeta\in W^{1,\infty}(S^{N+1} ; \Bbb R)$. Then : \noindent b) $$ |\la T(g) , \zeta\ra|\leq C|g|_{N/p,p}^p\|\nabla\zeta \|_{L^\infty},\quad\forall\ \zeta\in W^{1,\infty}(S^{N+1} ; \Bbb R)\ ;\tag 2.6 $$ c) the map $g\mapsto T(g)$ is strongly continuous from $W^{N/p,p}$ into $(W^{1,\infty} (S^{N+1}))^*$. \endproclaim \demo{Proof of Lemma 4} We start by proving that $H\in L^1$. Assume first that $NN+1$. In the open set $\{y\in B^{N+2}\, ; |u(y)|> 1/2\}$, $F$ is $S^N$-valued, and thus $H=0$ pointwise. Therefore, $$ \int\limits_{B^{N+2}}|H| = \int_{\{y ; |u(y)|\leq 1/2\}} |H|. $$ Clearly, $|\nabla F|\le C|\nabla u|$ and therefore $|H|\le C|\nabla u|^{N+1}$. By the proof of Lemma 1, we have $$ \int_{\{y ; |u(y)|\leq 1/2\}} |H| \leq C \int_{\{y ; |u(y)|\leq 1/2\}} |\nabla u |^{N+1} \leq C\int_{S^{N+1}}\frac{1}{(d(x))^N}, $$ where $d(x)$ is defined as in Section 1. \enddemo By the proof of Lemma 2, we further obtain that $$ \int_{S^{N+1}}\frac{1}{(d(x))^N} \leq C (|g|^p_{N/p, p} + 1), $$ and thus $$ \int_{\{y ; |u(y)|\leq 1/2\}} |H| \leq C(|g|^p_{N/p,p} + 1). $$ Hence $H\in L^1$ and consequently $Y(\xi)$ is well-defined. \medskip We now turn to the proof of a). Let $\xi_1,\xi_2\in W^{1,\infty}(B^{N+2} ; \Bbb R)$ be such that $\xi_1=\xi_2$ on $S^{N+1}$ and set $\eta=\xi_1-\xi_2\in W^{1,\infty}_0(B^{N+2})$. Consider a sequence $(\eta_j)\subset C^\infty_c(B^{N+2})$ such that $\nabla\eta_j\to\nabla\eta$ a.e. and $\d \|\nabla\eta_j\|_{L^\infty}\le C$. Since div $H=0$, we clearly have $\d\int\limits_{B^{N+2}}H\cdot\nabla\eta_j=0$, $\forall\ j$, and thus $\d\int\limits_{B^{N+2}}H\cdot\nabla\eta=0$. \medskip We next establish b). It suffices to estimate $\la T(g),\zeta\ra$ when $$ \int_{S^{N+1}} \zeta = 0.\tag 2.7 $$ In view of (2.7), we may find an extension $\xi$ of $\zeta$ to $B^{N+2}$ such that $$ \|\nabla \xi\|_{L^\infty} \leq C\|\nabla \zeta\|_{L^\infty} \tag 2.8 $$ and $$ \text{Supp } \xi \subset \{ y\in \overline{B^{N+2}}\ ; |y|\geq 1/2\}.\tag 2.9 $$ For such a~ $\xi$, we have $$ |\la T(g) , \zeta\ra |\le \int\limits_{B^{N+2}}|H||\nabla\xi|\le C\|\nabla\zeta\|_{L^\infty}\int\limits_{\{y ; |y| \geq 1/2 \text{ and } |u(y)\leq 1/2\}} |\nabla u |^{N+1}.\tag 2.10 $$ Going back to the proofs of Lemmas 1 and 2, we see that $$ \int\limits_{\{y ; |y| \geq 1/2 \text{ and } |u(y)\leq 1/2\}} |\nabla u |^{N+1} \leq C|g|^p_{N/p,p},\tag 2.11 $$ so that b) is a consequence of (2.10) and (2.11). \medskip Finally, we prove c). As we already observed, it suffices to consider the case $p>N+1$. Let $g_n$, $g\in W^{N/ p,p}(S^{N+1}; S^N)$ be such that $g_n \to g$ in $W^{N/p,p}$ and let $H_n, H$ be the corresponding vector-fields. We claim that $$ \int_{B^{N+2}} |H_n - H| \to 0.\tag 2.12 $$ By the uniqueness of the limit, it suffices to establish (2.12) for a subsequence. With $u_n$, $u$ the corresponding harmonic extensions, we have $u_n \to u$ in $C^\infty(B^{N+2})$ and in $W^{(N+1)/ p,p}$. For $x\in S^{N+1}$ and $t\in I=(0,1/2)$, set $$ v_n(x,t)=u_n((1-t)x)\quad\text{ and }\quad v(x,t)=u((1-t)x). $$ In view of (1.7), we know that $$ v_n\to v\quad\quad\text{ in }L^p(S^{N+1} ; W^{s,p}(I)), $$ where $s=(N+1)/p$. Passing to a subsequence (still denoted $v_n$) we obtain a function $K\in L^1(S^{N+1})$ such that $$ |v_n(x, \cdot )|^p_{s,p(I)}\le K(x),\quad\forall\ n, \text{ and a.e. }x\in S^{N+1}.\tag 2.13 $$ As in the proof of Lemma 2 we find, using (2.13), $$ \frac 1{d_n(x)^N}\le CK(x),\quad\forall\ n, \text{ and a.e. }x\in S^{N+1},\tag 2.14 $$ (where $d_n$, corresponding to $g_n$, is defined as in Section 1). Next we have, (using (1.2) and (1.3)), $$|H_n(rx)| \leq\cases 0, &\text{ if }\quad 1-d_n(x) < r< 1\\ C/(1-r)^{N+1}, &\text{ if }\quad 0\leq r< 1\endcases.\tag 2.15 $$ Combining (2.14) and (2.15), we obtain $$ |H_n(y)|\leq M(y), \quad\quad \forall\ y\in B^{N+2},\tag 2.16 $$ for some $M\in L^1$. Since, clearly, $H_n \to H$ in $C^\infty (B^{N+2})$, (2.12) follows from (2.16). \medskip \noindent {\bf Step 2.} Proof of (0.13) \medskip As we already observed, we may still define $T( g)$ if $g\in W^{N/(N+1), N+1}(S^{N+1} ; \Bbb R^{N+1})$ (note that here $g$ need not be $S^N$-valued). Indeed, for such a $g$, we have $u\in W^{1, N+1}(B^{N+2} ; \Bbb R^{N+1})$ and thus $H\in L^1$. Similarly, the definition (0.11) of Det$(\nabla g)$ still makes sense for $g\in W^{1, N}(S^{N+1} ; \Bbb R^{N+1})\cap L^\infty$. An easy adaptation of the proof of Lemma 1 in [BBM4] yields, in $(W^{1,\infty})^*$, the equality $$ \text{\rm Det}(\nabla g) = T(g) ,\quad\forall\ g\in W^{1,N}(S^{N+1} ; \Bbb R^{N+1})\cap W^{N/(N+1), N+1}\cap L^\infty.\tag 2.17 $$ This completes the proof of (0.13) when $N\ge 2$. Indeed, if $N\ge 2$ we have $W^{1,N}(S^{N+1} ; S^N)\subset W^{N/p,p}$, $\forall\ p>N$, so that (0.13) is a special case of (2.17). We now turn to the proof of (0.13) when $N=1$, i.e., $$ \text{Det}(\nabla g)=T(g),\quad\quad\forall\ p>1,\ \forall\ g\in W^{1,1}(S^2 ; S^1)\cap W^{1/p , p}.\tag 2.18 $$ It is useful to introduce the class $$ \aligned \Cal R= &\{g \in W^{1,q}(S^{N+1}; S^N) \text{ for every } 1\leq q < N+1\ ;\\ &\ g\in C^\infty(S^{N+1}\setminus A) \text{ for some finite set } A\}.\endaligned $$ Note that every $g\in\Cal R$ belongs to $W^{1,N}$ and also to $W^{N/(N+1), N+1}$. Thus (2.17) holds for every $g\in\Cal R$. Equality (2.18) follows from \noindent a) Lemma 5 below ; \noindent b) (2.17) applied to $g\in\Cal R$ ; \noindent c) the continuity of $g\mapsto T(g)$ from $W^{1/p,p}(S^2 ; S^1)$ into $(W^{1,\infty})^*$ ; \noindent d) the continuity of $g\mapsto\,$Det$(\nabla g)$ from $W^{1,1}(S^2 ; S^1)$ into $(W^{1,\infty})^*$ (which is obvious from (0.11)). \proclaim{Lemma 5} Let $p>1$. For every $g\in W^{1,1}(S^2 ; S^1)\cap W^{1/p,p}$, there is a sequence $(g_n)\subset\Cal R$ such that $g_n\to g$ in $W^{1,1}$ and in $W^{1/p,p}$. \endproclaim The proof of Lemma 5 is given in the Appendix. \medskip \noindent {\bf Step 3.} Proof of (0.14)-(0.16) \medskip The proof of (0.14)-(0.15) is a straightforward adaptation --left to the reader-- of the proof of Theorem 1 in [BBM4]. It relies on four facts : \noindent a) $\Cal R$ is dense in $W^{N/p , p}(S^{N+1} ; S^N)$ (see Appendix) ; \noindent b) $g\mapsto T(g)$ is continuous from $W^{N/p , p}(S^{N+1} ; S^N)$ into $(W^{1,\infty})^*$ ; \noindent c) the equality $$ \text{Det}(\nabla g)=T(g)=\omega_{N+1}\sum_{\text{\rm finite}}d_a\delta_a,\quad\forall\ g\in\Cal R,\tag 2.19 $$ where $\omega_{N+1}$ is the volume of the unit ball in $\Bbb R^{N+1}$ and $d_a$ denotes the degree of $g$ restricted to a small sphere around $a$ in $S^{N+1}$ (with appropriate orientation). Equality (2.19) is proved as in [BBM4], Lemma 2 ; \noindent d) if $g,h\in\Cal R$ and we write $$\text{Det}(\nabla g) - \text{ Det}(\nabla h) = \omega_{N+1} \sum_{a\in A} d_a \delta_a,\tag 2.20 $$ then (see [BCL]) $$ \|\text{Det}(\nabla g) - \text{Det}(\nabla h)\|_{(W^{1,\infty})^*}=\omega_{N+1}L, \tag 2.21 $$ where $$ L= \underset{\sigma \in S_k}\to{\text{Min}} \sum^k_{i=1} d(P_i, N_{\sigma(i)})\ ;\tag 2.22 $$ here $P_i, N_i$ are the points $a\in A$ repeated according to their multiplicity and $d$ is the geodesic distance on $S^{N+1}$. \medskip The proof of (0.16) relies on the following variant of Theorem 4 in [BMP] : \proclaim{Lemma 6} Let $g,h\in \Cal R$. Then, for $\zeta\in C^\infty(S^{N+1};\Bbb R)$, we have $$ \int|\deg (g;\Gamma_\lambda) - \deg(h;\Gamma_\lambda)|d\lambda \leq \frac{1}{\omega_{N+1}} \|\nabla\zeta\|_{L^\infty}\|\text{\rm Det}(\nabla g) - \text{\rm Det}(\nabla h)\|_{(W^{1,\infty})^*}. $$ \endproclaim \demo{Proof} Let $g,\, h\in \Cal R$. Assume that $$ T(g)=\omega_{N+1}\sum_{i=1}^I(\delta_{P_i}-\delta_{N_i}),\quad\quad T(h)=\omega_{N+1}\sum_{j=1}^J(\delta_{\tilde P_j}-\delta_{\tilde N_j}). $$ If $\lambda$ is a regular value of $\zeta$ such that $\zeta (P_i)\neq\lambda$, $\zeta (N_i)\neq\lambda$, $\zeta (\tilde P_j)\neq\lambda$, $\zeta (\tilde N_j)\neq\lambda$, for every $i$ and $j$, then $$ \text{deg}(g ; \Gamma_\lambda)=\text{card}\{ 1\le i\le I\ ;\ \zeta(P_i)>\lambda\}-\text{card}\{ 1\le i\le I\ ;\ \zeta(N_i)>\lambda\}, $$ so that, clearly, $$ \text{deg}(g ; \Gamma_\lambda)=\frac 12\sum_{i=1}^I(\text{sgn}(\zeta(P_i)-\lambda)-\text{sgn}(\zeta(N_i)-\lambda)). \tag 2.23 $$ It follows from (2.23) that $$ \text{deg}(g ; \Gamma_\lambda)-\text{deg}(h ; \Gamma_\lambda)=\frac 12\sum_{k=1}^{I+J}(\text{sgn}(\zeta(P^*_k)-\lambda)-\text{sgn}(\zeta(N^*_k)-\lambda)), \tag 2.24 $$ where the set $\{ P_i\}\cup \{ \tilde N_j\}$, resp. $\{ N_i\}\cup\{ \tilde P_j\}$, is now labeled as $\{ P^*_k\}$, resp. $\{ N^*_k\}$. Assume, e.g., that the length of the minimal connection in (2.22) is given by $L=\d\sum_{k=1}^{I+J}d (P^*_k, N^*_k)$ and let $\gamma_k$ be a geodesic from $P_k^*$ to $N_k^*$, $\forall\ k$. Since, clearly, $$ \frac 12|(\text{sgn}(\zeta(P^*_k)-\lambda)-\text{sgn}(\zeta(N^*_k)-\lambda)|\le \text{card}\{ x\in\gamma_k\ ;\ \zeta (x)=\lambda\}, $$ we find, using the area formula and (2.22), that $$\aligned \int|\deg (g;\Gamma_\lambda) - \deg(h;\Gamma_\lambda)|d\lambda &\leq \sum_k \int \text{card }\{ x\in\gamma_k\ ;\,\zeta(x) = \lambda\} d\lambda = \sum_k\int_{\gamma_k} \bigg|\frac{\partial \zeta}{\partial \tau}\bigg|\\ & \le L\|\nabla\zeta\|_{L^\infty}=\frac{1}{\omega_{N+1}} \|\nabla\zeta\|_{L^\infty}\|\text{\rm Det}(\nabla g) - \text{\rm Det}(\nabla h)\|_{(W^{1,\infty})^*} \endaligned .\tag 2.25 $$ \enddemo \demo{Proof of (0.16)} As in [BBM4], we have $$ \la\text{Det}(\nabla g),\zeta\ra=\int\limits_{\Bbb R}\text{deg}(g; \Gamma_\lambda )d\lambda ,\quad\forall\ \zeta\in C^\infty(S^{N+1} ; \Bbb R),\, \forall\ g\in{\Cal R}.\tag 2.26 $$ Let $g\in W^{N/p,p}(S^{N+1} ; S^N)$ and let $(g_n)\subset \Cal R$ be such that $g_n\to g$ in $W^{N/p,p}$ and $$ \sum_n \|\text{\rm Det}(\nabla g_{n+1}) - \text{\rm Det}(\nabla g_n)\|_{(W^{1, \infty})^*}<\infty. $$ By Lemma 6 we have, for a fixed $\zeta\in C^\infty(S^{N+1};\Bbb R),$ $$ \sum_n\int\limits_\Bbb R|\deg(g_{n+1}; \Gamma_\lambda) - \deg(g_n; \Gamma_\lambda) | d\lambda < \infty.\tag 2.27 $$ On the other hand, passing to a subsequence, we have, for a.e. $\lambda$, ${g_n}_{|\Gamma_\lambda} \to g_{|\Gamma_\lambda}$ in $W^{N/p,p}$ and thus in VMO. Therefore, $$ \deg(g_n; \Gamma_\lambda) \to \deg(g;\Gamma_\lambda)\text{ for a.e. } \lambda . \tag 2.28 $$ >From (2.27) and (2.28) we obtain $$ \deg (g_n ;\Gamma_\lambda)\to \deg (g ;\Gamma_\lambda)\quad\quad\text{in }L^1(\Bbb R).\tag 2.29 $$ Property (0.16) follows by combining (2.26), (2.29) and the continuity of $T$. \enddemo \medskip We conclude this section by showing, in the spirit of [BCL], [BBM4] and [BMP], that, given points $(P_i)$ and $(N_i)$ in $S^{N+1}$, the minimal ``energy'' (in the $W^{N/p,p}$ sense) required to produce topological singularities at the $P_i$'s and $N_i$'s, is of the same order as the length of a minimal connection connecting the $P_i$'s to the $N_i$'s. Let $\Cal P= (P_i)$, $\Cal N= (N_i) \subset S^{N+1}$ be such that $\sum_i|P_i - N_i|<\infty$. We define the length of a minimal connection to be $$ L(\Cal P, \Cal N) =\text{Inf }\left\{\sum d(\tilde P_j,\tilde N_j)\ ; \sum(\delta_{P_i} - \delta_{N_i}) = \sum (\delta_{\tilde P_j} - \delta_{\tilde N_j})\right\}. $$ As observed in [BBM4], if $$ T = \omega_{N+1} \sum (\delta_{P_i} - \delta_{N_i}), $$ then $$ \|T\|_{(W^{1, \infty})^*} = \omega_{N+1} L(\Cal P, \Cal N).\tag 2.30 $$ \proclaim{Theorem 5} Given $\Cal P$ and $\Cal N$, we have, for $N1. $$ The proof of Theorem 5 is complete. \enddemo \medskip \noindent{\bf Appendix. Density of the class $\Cal R$.} The appendix is devoted to density results for classes of $S^N$-valued maps. Recall that, if $0