The proof is similar to the Euclidean versions, i.e. the one-dimensional version or the multi-dimensional version .

1. Apply complex lemma on spherical caps to get an estimate where some supremum is involved. Link to heading

Let \(K\) denote a spherical cap with radius \(r\).

It follows from (0x669e0e75) that a point \(p\in B_r\) exists such that

\[ \lvert u(p)\rvert \ge \lVert u\rVert_{L^2(K)} \lvert K\rvert^{-1/2}. \]

Set \(g=\lVert f\rVert_{L^2(B_r)}^{-1}\Vol(B_r)^{1/2} f\). Then, \(\lvert g(p)\rvert\ge 1\) and we may apply (0x67adb01e) (note, that \(g\) suffices the needed analyticity by assumption) and thus

\[\lVert f\rVert_{L^2(B_r)} \le \Vol(B_r)^{1/2} (c_1r^d)^{2\frac{\log M}{\log 2}} \biggl( \frac{1}{\Vol(S\cap B_r)}\biggr)^{2\frac{\log M}{\log 2}+\frac{1}{2}} \lVert f\rVert_{L^2(S\cap B_r)}, \]

with

\[M=\sup_{\text{\(\gamma\) geodesic starting in \(p\)}}\max_{\lvert z\rvert\le 4\lvert \gamma\cap B_r\rvert} \Vol(B_r)^{1/2} \frac{\lvert f\circ \gamma(z)\rvert}{\lVert f\rVert_{L^2(B_r)}}.\]

If small \(r\) is small enough we may estimate \(r^d\le C\Vol(B_r)\) where \(C\) is some constant (depending on the curvature?) and by adjusting the constant \(c_1\) we obtain

\[\lVert f\rVert_{L^2(B_r)} \le \biggl( \frac{c_1 \Vol(B_r)}{\Vol(S\cap B_r)}\biggr)^{2\frac{\log M}{\log 2}+\frac{1}{2}} \lVert f\rVert_{L^2(S\cap B_r)}. \]

{#eq:eq1}

2. Estimate the supremum on good balls Link to heading

There is finite sequence of good geodesic balls \(B_1,\ldots ,B_N\) with radius \(r>0\), such that for every \(k\ge 1\) and \(i \in \{1,\ldots ,N\}\) and

\[\lVert f\rVert_{L^2(\bigcup_{i=1}^N B_i)}^2 \ge \frac{1}{2} \lVert f\rVert_{L^2(M)}^2.\]

{#eq:eq2} Using (0x66e2d74c) we estimate \(M\) on each good ball \(B_i\) with

\[ M \le \sqrt{2} \exp(5Cr\sqrt{E}). \]

We modify [@eq:eq1] with the above estimate and obtain on each good ball \(B\)

\[\lVert f\rVert_{L^2(B_i)} \le \biggl(\frac{c_1 \Vol(B_i)}{\Vol(S\cap B_i)}\biggr)^{c_2(r\sqrt{E} + 1/2)} \lVert f\rVert_{L^2(S\cap B_i)}, \]

{#eq:eq3} where \(c_1\) and \(c_2\) are constants depending on \(d\).

3. Combine the estimates of step 1 and 2 to obtain the result. Link to heading

Combining the results in step 1 and step 2, we obtain

\begin{align} \lVert f\rVert_{L^2(M)} &\le \sqrt{2} \lVert f\rVert_{L^2(\bigcup_{i=1}^{n} B_i)}\\ &\le \sqrt{2} \sum_{i=1}^{n} \lVert f\rVert_{L^2(B_i)}\\ &\le \sqrt{2} \sum_{i=1}^{n} \biggl(\frac{c_1 \Vol(B_i)}{\Vol(S\cap B_i)}\biggr)^{c_2(r\sqrt{E} + 1/2)} \lVert f\rVert_{L^2(S\cap B_i)}\\ &\le \sqrt{2} \biggl(\frac{c_1}{\gamma}\biggr)^{c_2(r\sqrt{E} + 1/2)} \sum_{i=1}^{n} \lVert f\rVert_{L^2(S\cap B_i)}\\ &\le \sqrt{2}\kappa\biggl(\frac{c_1}{\gamma}\biggr)^{c_2(r\sqrt{E} + 1/2)} \lVert f\rVert_{L^2(S)}. \end{align}