3.2.11 Bounded boundary dataFA92650B78EB

Proof. We first verify the normalization

Translation and the substitution $z=x'+tw$ reduce the integral to $\frac{2}{s_{d-1}}\int _{\mathbb R^{d-1}}(1+|w|^2)^{-d/2}\, dw$. The map

parametrizes the upper hemisphere. With $m=d-1$, its metric matrix is $DF^{\mathsf T}DF=(1+|w|^2)^{-1}I-(1+|w|^2)^{-2}ww^{\mathsf T}$. Its eigenvalues are $(1+|w|^2)^{-1}$ in the $m-1$ perpendicular directions and $(1+|w|^2)^{-2}$ in the direction of $w$. The square root of its determinant is $(1+|w|^2)^{-d/2}$. At $w=0$ this formula holds by continuity. The surface area formula and reflection in the equatorial plane show that the last integral is $s_{d-1}/2$, proving (3.2.13). This also treats $d=2$, when $m=1$ and there are no perpendicular directions.

The weighted measure $|f(z)|\, dz$ satisfies (3.2.8), by the same integral at $(0,1)$. Applying the differentiation estimates in the proof of Theorem 3.2.6 to the positive and negative parts of $f$ proves harmonicity. The positivity and normalization of $P_ H$ give the stated bound.

Fix $\varepsilon >0$. At a continuity point $z_0$, choose $\delta >0$ so that $|f(z)-f(z_0)|<\varepsilon $ for $|z-z_0|<\delta $. The contribution of this set to $|\int P_ H(x,z)(f(z)-f(z_0))\, dz|$ is at most $\varepsilon $. If $|x'-z_0|<\delta /2$ and $|z-z_0|\geq \delta $, then $|z-x'|\geq \delta /2$. Writing $s_{d-2}$ for the area of the unit sphere in $\mathbb R^{d-1}$ (in particular $s_0=2$), polar integration gives

Thus

near $(z_0,0)$. Letting $(x',t)\to (z_0,0)$ and then $\varepsilon \downarrow 0$ proves the boundary limit. Finally $|u(0,t)/t-c|\leq \| f\| _\infty /t\to 0$. $\square$