2.9.23 Projection densities and incidence3E5438AF5E6E

Proof. We write the inner integral in polar coordinates on $H$ and show that the normalized surface measures of the unit spheres of the hyperplanes $H$ average to the normalized surface measure of $S^{2n-1}$. For $H\in \mathcal H$, let $S_ H$ be the unit sphere of $H$, and let $\tau _ H$ be the image under $T_ H$ of the normalized surface measure $\mathrm dS/s_{2n-3}$ of the unit sphere $S^{2n-3}$ of $\mathbb C^{n-1}=\mathbb R^{2n-2}$. We regard $\tau _ H$ as a Borel probability measure on $S^{2n-1}$ carried by $S_ H$; it does not depend on the frame, since surface measure is invariant under orthogonal transformations (subsection 3.1.1). Polar coordinates (3.1.2) on all of $\mathbb C^{n-1}$, transported to $H$ by $T_ H$, give

Measurability. Let $h$ be a nonnegative Borel function on $(0,\infty )\times S^{2n-1}$. By the definition of $\tau _ H$,

The map $(H,r,\omega )\mapsto (r,T_ H\omega )$ is Borel by the frame construction of subsection 2.9.17, since its two components are Borel. Hence $(H,r,\omega )\mapsto h(r,T_ H\omega )$ is a nonnegative Borel function on the product of the separable metric spaces $\mathcal H\times (0,\infty )$ and $S^{2n-3}$, and by lemma 2.2.8 and Tonelli's theorem the right side of (2.9.25) is a Borel function of $(H,r)$. In particular, taking $h(r,\theta )=g(\theta )$, we see that $H\mapsto \int g\, \mathrm d\tau _ H$ is Borel for every nonnegative Borel function $g$ on $S^{2n-1}$; for instance, $H\mapsto \tau _ H(B)$ is Borel for every Borel set $B\subset S^{2n-1}$.

Averaging. For a Borel set $B\subset S^{2n-1}$ put $\tau (B)=\int \tau _ H(B)\, \mathrm d\nu (H)$. By monotone convergence, $\tau $ is countably additive, so it is a Borel probability measure on $S^{2n-1}$. For a unitary matrix $U$, the map $UT_ H$ is a complex linear isometry from $\mathbb C^{n-1}$ onto $UH$, so $U$ maps $\tau _ H$ to $\tau _{UH}$, that is, $\tau _ H(U^{-1}B)=\tau _{UH}(B)$. Since $\nu $ is invariant, it follows that $\tau (U^{-1}B)=\int \tau _{UH}(B)\, \mathrm d\nu (H)=\tau (B)$, so $\tau $ is unitarily invariant. The normalized surface measure $\mathrm dS/s_{2n-1}$ of $S^{2n-1}$ is also unitarily invariant, because unitary matrices are orthogonal transformations of $\mathbb R^{2n}$. By the uniqueness in proposition 2.9.18, $\tau =\mathrm dS/s_{2n-1}$. By the definition of $\tau $ for indicator functions, by linearity for simple functions, and by monotone convergence in general, $\int _{\mathcal H}\int g\, \mathrm d\tau _ H\, \mathrm d\nu (H)=\int g\, \mathrm d\tau $ for every nonnegative Borel function $g$ on $S^{2n-1}$.

Conclusion. Integrate the polar-coordinate identity over $H$ with respect to $\nu $. The function $(r,\theta )\mapsto f(r\theta )$ is Borel on $(0,\infty )\times S^{2n-1}$, so by the measurability step $(H,r)\mapsto r^{2n-3}\int f(r\theta )\, \mathrm d\tau _ H(\theta )$ is a nonnegative Borel function on $\mathcal H\times (0,\infty )$, which is product-measurable by lemma 2.2.8. Tonelli's theorem therefore allows us to integrate first over $H$, and the averaging identity with $g(\theta )=f(r\theta )$ shows that the left side of the lemma equals

In polar coordinates $\xi =r\theta $ on all of $\mathbb C^ n=\mathbb R^{2n}$, (3.1.2) gives $|\xi |^{-2}\, \mathrm d\xi =r^{2n-3}\, \mathrm d r\, \mathrm dS(\theta )$, so this is the right side. All integrals may be infinite. $\square$

Proof. On $H$ we use the Fourier transform defined by the Euclidean inner product $\operatorname{Re}\langle \cdot ,\cdot \rangle $ and the measure $m_ H$; in an orthonormal frame it is the transform of subsection 2.9.1 on $\mathbb R^{2n-2}$. For $y\in \mathbb C^ n$ and $\xi \in H$ we have $\operatorname{Re}\langle P_ Hy,\xi \rangle =\operatorname{Re}\langle y,\xi \rangle $, because $y-P_ Hy$ is orthogonal to $H$. Hence the Fourier transform of $(P_ H)_*\mu $ at $\xi \in H$ is $\int \mathrm e^{-2\pi \mathrm i\operatorname{Re}\langle P_ Hy,\xi \rangle }\, \mathrm d\mu (y)=\widehat\mu (\xi )$; that is, it is the restriction of $\widehat\mu $ to $H$. Apply lemma 2.9.24 to the continuous function $|\widehat\mu |^2$, and then theorem 2.9.6 with $d=2n$ and $s=2n-2$, for which $|\xi |^{s-d}=|\xi |^{-2}$. We obtain

The inner integral is a Borel function of $H$, by the frame construction in subsection 2.9.17; hence it is finite for $\nu $-almost every $H$. For each such $H$, we apply proposition 2.9.5 on $\mathbb C^{n-1}=\mathbb R^{2n-2}$ to the image of $\mu $ under $T_ H^{-1}P_ H$. Since $T_ H$ preserves $\operatorname{Re}\langle \cdot ,\cdot \rangle $, the Fourier transform of this image is $w\mapsto \widehat\mu (T_ Hw)$, whose squared $L^2$ norm is $\int _ H|\widehat\mu |^2\, \mathrm d m_ H<\infty $. Hence the image has a density in $L^1\cap L^2$, and composing it with $T_ H^{-1}$ we obtain a density $g_ H$ of $(P_ H)_*\mu $ with respect to $m_ H$, with $\| g_ H\| _2^2=\int _ H|\widehat\mu |^2\, \mathrm d m_ H$. In particular, $H\mapsto \| g_ H\| _2^2$ is Borel, being equal to this Fourier integral, so no measurable choice of the densities themselves is needed. Integrating over $H$ proves the identity.

To compute $\beta _ n$, recall from (2.9.8) that $s_{k-1}=2\pi ^{k/2}/\Gamma (k/2)$ for every $k\geq 1$. Hence $a_ n=\Gamma (n)/(\pi \Gamma (n-1))$, while $b_{2n,2n-2}=\pi ^{n-2}/\Gamma (n-1)$, and therefore $\beta _ n=\Gamma (n)/\pi ^{n-1}$. $\square$

Proof. We parametrize the affine complex lines that can meet $K$ by a compact set carrying a finite measure $\lambda $, and bound from below, independently of $j$, the $\lambda $-measure of a set of lines meeting $E_ j$. Choose $M>0$ with $K\subset B(0,M)$. Every affine complex line can be written as $c+H^\perp $ with $H\in \mathcal H$ and $c\in H$, and $c=P_ Hy$ for every point $y$ of the line. Indeed, if the line is $a+\mathbb Cb$, let $H$ be the hyperplane orthogonal to $b$, so that $H^\perp =\mathbb Cb$, and put $c=P_ Ha$; then $P_ H(a+\zeta b)=c$ for every $\zeta \in \mathbb C$. If the line meets $K$ at a point $y$, then $|c|=|P_ Hy|\leq |y|<M$. Hence the lines that can meet $K$ are parametrized by the set

which is compact, being closed in $\mathcal H\times \overline{B(0,M)}$ because $c\in H$ means $P_ Hc=c$. We equip $\mathcal L_ M$ with the finite measure $\lambda $ defined by

for nonnegative Borel functions $h$. In the frames of subsection 2.9.17, $\lambda $ is the image of the product of $\nu $ and Lebesgue measure on the closed ball of radius $M$ in $\mathbb C^{n-1}$. Thus it is a Borel measure, and its total mass is the volume of a ball of radius $M$ in $\mathbb R^{2n-2}$.

By inner regularity (proposition 2.2.2), choose compact sets $F_ j\subset E_ j$ with $\mu _ j(F_ j)\geq 1/2$. The set

parametrizes the lines meeting $F_ j$. It is compact, as the image of $\mathcal H\times F_ j$ under the continuous map $(H,y)\mapsto (H,P_ Hy)$. Its section over $H$ is $P_ HF_ j$, so $\lambda (J_ j)=\int m_ H(P_ HF_ j)\, \mathrm d\nu (H)$. Let $g_{j,H}$ be the densities given by theorem 2.9.26 for $\mu _ j$. Since $F_ j\subset P_ H^{-1}(P_ HF_ j)$, the Cauchy–Schwarz inequality gives, for $\nu $-almost every $H$,

Integrate this inequality over $H$ with respect to the probability measure $\nu $, and apply the Cauchy–Schwarz inequality once more. Squaring the result and using theorem 2.9.26, we obtain

In particular $C>0$, and $\lambda (J_ j)\geq \delta =1/(4\beta _ n C)$ for every $j$. The sets $\bigcup _{j\geq m}J_ j$ decrease in $m$ and have measure at least $\delta $. Since $\lambda $ is finite, continuity from above gives

so this set is nonempty. If $(H,c)$ belongs to it, then $c\in P_ HF_ j$ for infinitely many $j$. For each such $j$ there is $y\in F_ j$ with $c=P_ Hy$, so $y\in c+H^\perp $. Hence the line $c+H^\perp $ meets $F_ j$, and therefore $E_ j$, for infinitely many $j$. $\square$

Proof. If $n=1$, the only affine complex line is $\mathbb C$ itself. Thus $u$ is subharmonic on every component of $D$, hence upper semicontinuous and locally bounded above.

Let $n\geq 2$, and suppose that $u$ is not bounded above on any neighborhood of some point $p\in D$. Then there are points $x_ j\to p$ with $A_ j=u(x_ j)\to \infty $. Choose $R>0$ with $\overline{B(p,2R)}\subset D$; discarding finitely many terms, we may assume that $|x_ j-p|<R$ for all $j$. Then $\overline{B(x_ j,R)}\subset B(p,2R)\subset D$, and theorem 2.9.22 gives Borel probability measures $\mu _ j$ carried by the sets

with $I_{2n-2}(\mu _ j)\leq C_ nR^{-(2n-2)}$ for every $j$. All the sets $E_ j$ lie in the bounded set $B(p,2R)$. By theorem 2.9.27, some affine complex line $L$ meets $E_ j$ for all $j$ in an infinite set $J$. Choose $y_ j\in L\cap E_ j$ for $j\in J$. Passing to a subsequence, we may assume that $y_ j\to y\in L\cap \overline{B(p,2R)}\subset D$. A small closed disc in $L$ centered at $y$ lies in $D$, so for large $j$ the points $y_ j$ lie in the same connected component of $D\cap L$ as $y$. On this component $u$ is subharmonic, hence upper semicontinuous, and therefore $\varlimsup _ ju(y_ j)\leq u(y)<\infty $. This contradicts $u(y_ j)\geq A_ j\to \infty $. $\square$