Suppose \((M,g)\) is a real-analytic Riemannian manifold with non negative sectional curvature and \(B_r\subseteq M\) is a geodesic ball with radius \(r>0\) on \(M\) with \(B_{6r} \subseteq M\). Let \(x\in B_r\) and \(f\colon B_r\to \mathbb{C}\) a function, such that \(f\circ \gamma\) is analytic on \(D(0,10r)\) if \(\gamma\) is a geodesic starting in \(x\).
Suppose \(S\subseteq B_r\) is a measurable subset with nonzero measure and \(p\ge 1\). If \(\lvert f(x)\rvert\ge 1\) and \(M=\sup_{\gamma}\max_{\lvert z\rvert\le 4\lvert \gamma\cap B_r\rvert} \lvert f\circ \gamma(z)\rvert\), then there is a constant \(C>0\) depending only on \(d\) such that
\[ 1 \le \biggl(C \frac{r^d}{\Vol(S\cap B_r)}\biggr)^{2\frac{\log M}{\log 2}+\frac{1}{p}} \lVert f\rVert_{L^p(S)}. \]Using (0x6698db36) we find a geodesic \(\gamma\) starting in \(x\) which satisfies
\[ \Vol(S\cap B_r) \le C r^{d-1} \lvert S \cap \gamma\rvert \]{#eq:eq1}
We apply (0x669a1456) on \(f\circ \gamma\). This implies
\[ 1 \le \sup_{t\in [0,R]}\lvert f\circ \gamma(t)\rvert\le \biggl(\frac{12 \lvert \gamma\rvert}{\lvert S \cap \gamma\rvert}\biggr)^{2 \frac{\log M_\gamma}{\log 2}} \sup_{t\in A} \lvert f\circ \gamma(t)\rvert, \]{#eq:eq2} with \(R=\max \{t>0 \mid \gamma(t)\in B_r\}=\lvert \gamma\cap B_r\rvert\), \(M_\gamma=\max_{\lvert z\rvert \le 4r} \lvert f\circ \gamma\rvert\) and \(A=\{t\in [0,R]\mid \gamma(t) \in S\cap B_r\}\). Note, \(f\circ \gamma\) is analytic on \(D(0,5R)\) by assumption.
Using [@eq:eq1] and the estimate \(\lvert \gamma\rvert \le 2r\) yields
\[ \frac{\lvert \gamma\rvert}{\lvert S \cap \gamma\rvert} \le \frac{Cr^d}{\Vol(S\cap B)}. \]{#eq:eq3}
We apply [@eq:eq3] in [@eq:eq2] and estimate \(\sup_{t\in A} \lvert f\circ \gamma(t)\rvert \le \sup_{y\in S\cap B} \lvert f(y)\rvert\), then
\[ 1\le \biggl(\frac{Cr^d}{\Vol(S\cap B)}\biggr)^{2\frac{\log M_\gamma}{\log 2}} \sup_{x\in S\cap B} \lvert f(x)\rvert. \]{#eq:eq4} Since \(r^d\ge \Vol(S\cap B)\) (is it true?) we allowed to replace \(M_{\gamma}\) in [@eq:eq4] with \(M\).
Since \(S\) was arbitrary, we may apply (0x6698f127) and this yields
\[ 1\le (Cr^d)^{2\frac{\log M_\gamma}{\log 2}} \biggl(\frac{1}{\Vol(S\cap B)}\biggr)^{2\frac{\log M_\gamma}{\log 2} + \frac{1}{p}} \lVert f\rVert_{L^p(S\cap B)}. \]See also Link to heading
Questions Link to heading
- Is it sufficient and necessary to consider functions that are analytic on \(B_{5r}(x)\)?
- To be able to use (0x6698f127) \((M, \Vol(\cdot )\) needs to be a measure space. Is this the case?
CanDos Link to heading
- Perhaps, the condition of non negative curvature can be relaxed. In this case the constant \(C\) will depend on the curvature of \(M\).
- Address to the relation of the injectivity radius.