We consider restrictions along closed geodesics and geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann surface. We obtain a non-trivial bound on the L^2-norm of such restrictions as the eigenvalue tends to infinity. We use methods from the theory of automorphic functions and in particular the uniqueness of invariant functionals on irreducible unitary representations of PGL(2,R).
For $2\leq p<4$, we study the $L^p$ norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact $2$-dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp} found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For $p=4$, we will also obtain a logarithmic analogous estimate, by using a…
If $(M,g)$ is a compact Riemannian surface then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negative curvature and $γ(t)$ is a geodesic parameterized by arc length, the measures $e_j(γ(t))\, dt$ on $\R$ tend to zero in the sense of distributions as the eigenvalue $\la_j\to \infty$, and so integrals of eigenfunctions over periodic geodesics tend to zero as $\la_j\to \infty$. The assumption of negative curvature is necessary for the latter result.
We show that for a quantum completely integrable system in two dimensions,the $L^{2}$-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds ofthe form $ \int_γ |ϕ_{j}^{\hbar}|^2 ds = {\mathcal O}(|\log \hbar|)$ for generic curves $γ$ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov are always attained for certain exceptional subsequences of eigenfunctions.
We consider the hyperbolic Casimir operator C defined on the tangent sphere bundle SY of a compact hyperbolic Riemann surface Y. We prove a nontrivial bound on the L^2-norm of the restriction of eigenfunctions of C to certain natural hypersurfaces in SY. The result that we obtain goes beyond known (sharp) local bounds of L. Hormander.
Let X be an arithmetic hyperbolic surface, ψa Hecke-Maass form, and l a geodesic segment on X. We obtain a power saving over the local bound of Burq-Gérard-Tzvetkov for the L^2 norm of ψrestricted to l, by extending the technique of arithmetic amplification developed by Iwaniec and Sarnak. We also improve the local bounds for various Fourier coefficients of ψalong l.
We give uniform upper and lower bounds for the L^2 norm of the restriction of eigenfunctions of the Laplacian on the three-dimensional standard flat torus to surfaces with non-vanishing curvature. We also present several related results concerning the nodal sets of eigenfunctions.
Let $(M,g)$ be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $μ$ in the sense that $ M \setminus \text{supp}(π_* μ) \neq \emptyset$ where $π:T^*M \to M$ is the canonical projection. Using Carleman estimates we prove that for any real-smooth closed hypersurface $H \subset (M\setminus \text{supp} (π_* μ))$ sufficiently close to $ \text{supp}(π_* μ),$ and for all $δ>0,$ $$ \int_{H} |u_h|^2 dσ\geq C_δ\, e^{- [\, d(H, \text{supp}(π_* μ)) + \,δ] /h} $$ as $h \to 0^+$. We also show that the result holds…
We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. We also show how our approach apply to higher dimensional manifolds.
We estimate the $L^2$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace-Beltrami operator on the standard flat torus $\mathbb{T}^d$, $d\ge2$. We reduce getting correct bounds to counting lattice points in the intersection of some $ν$-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain-Rudnick on $L^2$-restriction estimates for rational hyperplanes. On $\mathbb{T}^2$, we prove the uniform $L^2$ restriction bounds …
This article is about two types of restrictions of eigenfunctions $ϕ_j$ on a compact Riemannian manifold $(M,g)$: First, we restrict to a submanifold $H \subset M$, and expand the restriction $γ_H ϕ_j$ in eigenfunctions $e_k$ of $H$. We then Fourier restrict $γ_H ϕ_j$ to a short interval of eigenvalues of $H$. Laplace eigenvalues of $M$ are denoted $λ_j^2$ and those of $H$ are denoted $μ_k^2$. The Fourier coefficients are negligible unless the $H$- eigenvalues lie in the interval $μ_k \in [-λ_j, λ_j]$. The short windows have the form $|μ_k - c λ_j| < ε$. The goal is to obtain asymptotics and e…
We show that one can obtain improved $L^4$ geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge's strategy in proving improved critical $L^p$ estimates. We first combine the improved $L^2$ restriction estimate of Blair and Sogge and the classical improved $L^\infty$ estimate of Bérard to obtain an improved weak-type $L^4$ restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger. This estimate improves the $L^4$ restriction esti…
We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we are able to improve the corresponding $L^2$-restriction bounds of Burq, Gérard and Tzvetkov and Hu. Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved $L^4$-estimates for restrictions to geodesics, which, by Hölder's inequality and interpolation, implies improved $L^p$-bounds for all exponents $p\ge 2$. We do…
Let $(M,g)$ be a compact Riemannian surface with nonpositive sectional curvature and let $γ$ be a closed geodesic in $M$. And let $e_λ$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator $Δ_g$ with $-Δ_g e_λ= λ^2 e_λ$. Sogge, Xi, and Zhang showed using the Gauss-Bonnet theorem that $$ \int_γe_λ\, ds = O((\logλ)^{-1/2}),$$ an improvement over the general $O(1)$ bound. We show this integral enjoys the same decay for a wide variety of curves, where $M$ has nonpositive sectional curvature. These are the curves $γ$ whose geodesic curvature avoids, pointwise, the geodesic curvatur…
Let $(Ω,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$ and $u_λ:= ϕ_λ |_{\partial Ω}$ the associated Dirichlet data (ie. boundary restriction of $ϕ_λ$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary corner points) and any $δ\in [0,1),$ $$ \| ϕ_h \|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for…
We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_λ$ over geodesics go to zero at the rate of $O((\logλ)^{-1/2})$ if $λ$ are their frequencies. As discussed in \cite{CSPer}, no such result is possible in the constant curvature case if the curvature is $\ge0$. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius $r\le 1$ are pinched from above by $-δr^N$ …
Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets $Γ$ of $M$. The sets $Γ$ that we consider are Borel measurable, Lebesgue-null but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on $Γ$. For large Lebesgue exponents $p$, these estimates provide a natural generalization of $L^p$ bounds for eigenfunctions restricted to submanifolds, previously obtained in \cite{Ho68, Ho71, Sog88, B…
In this article we develop new techniques for studying concentration of Laplace eigenfunctions $ϕ_λ$ as their frequency, $λ$, grows. The method consists of controlling $ϕ_λ(x)$ by decomposing $ϕ_λ$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $λ^{-\frac{1}{2}}$. We control $ϕ_λ(x)$ by the $L^2$-mass of $ϕ_λ$ on each geodesic tube and derive a purely dynamical statement through which $ϕ_λ(x)$ can be studied. In particular, we obtain estimates on $ϕ_λ(x)$…
Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{ϕ_λ\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-Δ_gϕ_λ=λ^2 ϕ_λ$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(M,H)$, even when $H=\{x\}$, for which $$ \Big|\int_Hϕ_λdσ_H\Big|=O\Big(\frac{λ^{\frac{k-1}{2}}}{\sqrt{\log λ}}\Big)\qquad \text{or}\qquad |ϕ_λ(x)|=O\Big(\frac{λ^{\frac{n-1}{2}}}{\sqrt{\log λ}}\Big), $$ as $λ\to \infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions i…
This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions $ϕ_j$ of a compact Riemannian manifold to a submanifold $H \subset M$. We fix a number $c \in (0,1)$ and study the asymptotics of the thin sums, $$ N^{c} _{ε, H }(λ): = \sum_{j, λ_j \leq λ} \sum_{k: |μ_k - c λ_j | < ε} \left| \int_{H} ϕ_j \overline{ψ_k}dV_H \right|^2 $$ where $\{λ_j\}$ are the eigenvalues of $\sqrt{-Δ}_M,$ and $\{(μ_k, ψ_k)\}$ are the eigenvalues, resp. eigenfunctions, of $\sqrt{-Δ}_H$. The inner sums represent the `jumps' of $ N^{c} _{ε, H }(λ)$ and reflect the geometry o…
We prove an analogue of Sogge's local $L^p$ estimates for $L^p$ norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov, Hu, and Chen-Sogge. The improvements are logarithmic on negatively curved manifolds (without boundary) and by $o(1)$ for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get $o(1)$ improvements on $L^\infty$ estimates of Cauchy data away from a shrinking neighborhood of…
Let $(M,g)$ be a compact, smooth, Riemannian manifold and $\{ ϕ_h \}$ an $L^2$-normalized sequence of Laplace eigenfunctions with defect measure $μ$. Let $H$ be a smooth hypersurface. Our main result says that when $μ$ is $\textit{not}$ concentrated conormally to $H$, the eigenfunction restrictions to $H$ and the restrictions of their normal derivatives to $H$ have integrals converging to 0 as $h \to 0^+$.
We show that one can obtain logarithmic improvements of $L^2$ geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a $(\logλ)^{-\frac12}$ gain for the $L^2$-restriction bounds, which improves the corresponding bounds of Burq, Gérard and Tzvetkov, Hu, Chen and Sogge. We achieve this by adapting the approaches developed by Chen and Sogge, Blair and Sogge, Xi and the author. We derive an explicit formula for the wave kernel on 3D hyperbolic space, which improves the kernel estimates from the Hadamard parametrix…
Let $(M,g)$ be an $n$-dimensional compact boudaryless Riemannian manifold with nonpositive sectional curvature, then our conclusion is that we can give improved estimates for the $L^p$ norms of the restrictions of eigenfunctions to smooth submanifolds of dimension $k$, for $p>\dfrac{2n}{n-1}$ when $k=n-1$ and $p>2$ when $k\leq n-2$, compared to the general results of Burq, Gérard and Tzvetkov \cite{burq}. Earlier, Bérard \cite{Berard} gave the same improvement for the case when $p=\infty$, for compact Riemannian manifolds without conjugate points for $n=2$, or with nonpositive sectional curvat…
We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $ϕ_h$, $(h^2 Δ- 1)ϕ_h = 0$, to $H$ in the region exterior to the coball bundle of $H$, on $h^δ$-scales ($0\leq δ< 2/3$). We use this estimate to obtain an $O(1)$ $L^2$-restriction bound for the Neumann data along $H.$ The estimate also applies to eigenfunctions of semiclassical Schrödinger operators.
Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{ψ_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle γ_H e_j, ψ_k \rangle_{L^2(H)} = \int_H e_j \overline ψ_k \, dV_H, \] of restrictions $γ_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{(μ_k, λ_j)\}_{j,k - 0}^{\infty}$ of the (square roots …
Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let $Δ_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_λ$ be $L^2$-normalized eigenfunctions of $Δ_g$ with eigenvalue $λ$, i.e. \[ -Δ_g e_λ= λ^2 e_λ. \] We prove \[ \left| \int_{\mathbb R} b(t) e_λ(γ(t)) \, dt \right| = o(1) \quad \text{ as } λ\to \infty \] where $b$ is a smooth, compactly supported function on $\mathbb R$ and $γ$ is a curve parametrized by arc-length whose geodesic curvature $κ(γ(t))$ avoids two critical curvatures $\mathbf k(γ'^\perp(t))$ a…
We prove a new quantum variance estimate for toral eigenfunctions. As an application, we show that, given any orthonormal basis of toral eigenfunctions and any smooth embedded hypersurface with nonvanishing principal curvatures, there exists a density one subsequence of eigenfunctions that equidistribute along the hypersurface. This is an analogue of the Quantum Ergodic Restriction theorems in the case of the flat torus, which in particular verifies the Bourgain-Rudnick's conjecture on $L^2$-restriction estimates for a density one subsequence of eigenfunctions in any dimension. Using our quant…
We consider upper bounds on the growth of $L^p$ norms of restrictions of eigenfunctions and quasimodes to geodesic segments in a nonpositively curved manifold in the high frequency limit. This sharpens results of Chen and Sogge as well as Xi and Zhang, which showed that the crux of the problem is to establish bounds on the mixed partials of the distance function on the covering manifold restricted to geodesic segments. The innovation in this work is the development of a formula for the third variation of arc length on the covering manifold, which allows for a coordinate free expressions of the…
We study the $L^2$ norm of the Eisenstein series $E(z,1/2+iT)$ restricted to a segment of a geodesic connecting infinity and an arbitrary real. We conjecture that on slightly thickened geodesics of this form, the Eisenstein series satifies restricted QUE. We prove a lower bound that matches this predicted asymptotic. We also prove an upper bound that nearly matches the lower bound assuming the Riemann Hypothesis (unconditionally, the sharp upper bound holds for almost all $T$). Finally, we show the restricted QUE conjecture for geodesics with rational endpoints.
We prove a quantum ergodic restriction theorem for the Cauchy data of a sequence of quantum ergodic eigenfunctions on a hypersurface $H$ of a Riemannian manifold $(M, g)$. The technique of proof is to use a Rellich type identity to relate quantum ergodicity of Cauchy data on $H$ to quantum ergodicity of eigenfunctions on the global manifold $M$. This has the interesting consequence that if the eigenfunctions are quantum unique ergodic on the global manifold $M$, then the Cauchy data is automatically quantum unique ergodic on $H$ with respect to operators whose symbols vanish to order one on th…
Burq-Gérard-Tzvetkov and Hu established $L^p$ estimates ($2\le p\le \infty$) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint $L^2$ estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be removed in general, while the problem is still open. So this paper is devoted to the study of sharp endpoint restriction estimates for eigenfunctions in this case. Chen and Sogge removed the log loss for the geodesics on 3-dimensional manifolds. In this paper, we generalize their resu…
We obtain restrictions on the persistence barcodes of Laplace-Beltrami eigenfunctions and their linear combinations on compact surfaces with Riemannian metrics. Some applications to uniform approximation by linear combinations of Laplace eigenfunctions are also discussed.
Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the general results of Burq, Gérard and Tzvetkov \cite{burq}. By earlier results of Bourgain \cite{bourgainef} and the first author \cite{Sokakeya}, they are equivalent to improvements of the general $L^p$-estimates in \cite{soggeest} for $n=2$ and $2<p<6$. The proof uses the fact that the exponential map from any point in $x_0\in M$ is a universal covering map f…
We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $λ$. The condition that ensures this is that their $L^2$ norms over $O(λ^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $λ$ is large (which is not the case for the highest weight spherical harmonics on $S^2$ for instance). The proof exploits Gauss' lemma and the fact that the bilinear oscillatory integrals in Hörmander's proo…
We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines techniques from semiclassical analysis with harmonic theory on reductive groups, and makes use of new asymptotic bounds for spherical functions that are of independent interest.
Let $M$ be a compact manifold with or without boundary and $H\subset M$ be a smooth, interior hypersurface. We study the restriction of Laplace eigenfunctions solving $(-h^2Δ_g-1)u=0$ to $H$. In particular, we study the degeneration of $u|_H$ as one microlocally approaches the glancing set by finding the optimal power $s_0$ so that $(1+h^2Δ_H)_+^{s_0}u|_H$ remains uniformly bounded in $L^2(H)$ as $h\to 0$. Moreover, we show that this bound is saturated at every $h$-dependent scale near glancing using examples on the disk and sphere. We give an application of our estimates to quantum ergodic re…
In the recent work arXiv:1311.3999, the authors proved that real analytic manifolds $(M, g)$ with maximal eigenfunction growth must have a self-focal point p whose first return map has an invariant L1 measure on $S^*_p M$. In this addendum we add a purely dynamical argument on circle maps to improve the conclusion to: all geodesics from p are smoothly closed.
We prove that the Cauchy data of Dirichlet or Neumann $Δ$- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e. a point at which a positive measure of geodesics leaving the point return to the point. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds.
We study the number of intersections of the nodal lines of an eigenfunction of the Laplacian on the standard torus with a fixed reference curve, that is, the number of zeros of the eigenfunction restricted to the curve. An upper bound is the wave number k. When the curve has nowhere zero curvature, we conjecture that, up to a constant multiple, this should also be the correct a lower bound. We give a lower bound which differs from this by an arithmetic quantity, given in terms of the maximal number of lattice points in arcs of size square root of the wave number k on a circle of radius k. Acco…
Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $ϕ_j |_H$ to $H$ of $Δ$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic on $H$. We prove two kinds of results: First (i) for any smooth hypersurface $H$, the Cauchy data $(ϕ_j|H, \partial ϕ_j|H)$ is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly (ii) we give conditions on $H$ so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equalit…
Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.
We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.
Husimi distributions of Laplace eigenfunctions are special types of `microlocal lifts' of eigenfunctions to phase space. Their weak * limits are the well-known quantum limits or microlocal defect measures of an orthonormal basis $\{ ϕ_j\}$ of eigenfunctions on a Riemannian manifold $(M,g)$ . Husimi distributions are normalized mod squares of analytic continuations of eigenfunctions to the complexification of $M$, which may be identified with an open subset of the cotangent bundle $T^*M$. Husimi distributions are probability measures whose density at $ζ$ is the probability density of a quantum …
Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $φ^\ell_m$ be the orthonormal basis of joint eigenfunctions of $Δ_g$ and $\partial_θ$, the generator of the rotation action. The main result is an explicit formula for the weak-* limit of the normalized empirical measures, $Σ_{m = -\ell}^\ell ||φ^\ell_m||^2_{L^2(H)} δ_{\frac{m}{\ell}}(c)$ on $[-1,1]$. The explicit formula shows that, asymptotically, the $L^2$ norms of restricted eigenfunctions are minimal for the zonal eigenfunction $m = 0$, maximal for Gaussian beams $m = \pm 1$…
Let $φ_λ$ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$, i.e., $Δ_g φ_λ + λφ_λ=0$. We show that $φ_λ$ satisfies a local Bernstein inequality, namely for any geodesic ball $B_g(x,r)$ in $M$ there holds: $\sup_{B_g(x,r)}|\nablaφ_λ|\leq C_δ\max\left\{\frac{\sqrtλ\log^{2+δ}λ}{r},λ\log^{2+δ}λ\right\}\sup_{B_g(x,r)}|φ_λ|$. We also prove analogous inequalities for solutions of elliptic PDEs in terms of the frequency function.
