We use the Gauss-Bonnet theorem and triangle comparison theorems of Rauch Toponogov to show that on compact Riemannian surfaces negative curvature period integrals eigenfunctions e λ over geodesics go zero at rate O((log λ) -1/2 ) if are their frequencies.As discussed in [4], no such result is possible constant case ≥ 0. Notwithstanding, we also these bounds for valid provided all geodesic balls radius r ≤ 1 pinched from above by -δr N some fixed δ > This allows, instance, be nonpositive...

We expand the class of curves $(φ_1(t),φ_2(t)),\ t\in[0,1]$ for which $\ell^2$ decoupling conjecture holds $2\leq p\leq 6$. Our includes all real-analytic regular with isolated points vanishing curvature and form $(t,t^{1+ν})$ $ν\in (0,\infty)$.

Abstract We prove a variable coefficient version of the square function estimate Guth–Wang–Zhang. By classical argument Mockenhaupt–Seeger–Sogge, it implies full range sharp local smoothing estimates for ‐dimensional Fourier integral operators satisfying cinematic curvature condition. In particular, conjecture wave equations on compact Riemannian surfaces is settled.

We study the generalization of Falconer distance problem to Riemannian setting. In particular, we extend result Guth--Iosevich--Ou--Wang for set in plane general surfaces. Key new ingredients include a family refined microlocal decoupling inequalities, which are related work Beltran--Hickman--Sogge on Wolff-type and an analog Orponen's radial projection lemma has proved quite useful recent sets.

We show that on compact Riemann surfaces of negative curvature, the generalized periods, i.e. $ν$-th order Fourier coefficient eigenfunctions $e_λ$ over a period geodesic $γ$ goes to 0 at rate $O((\logλ)^{-1/2})$, if $0

Abstract We prove an endpoint version of the uniform Sobolev inequalities in [C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 1987, 329–347]. It was known that strong type no longer hold at endpoints; however, we show restricted weak there, which imply earlier classical result by real interpolation. The key ingredient our proof is a interpolation first introduced Bourgain [J. Bourgain,...

We expand the class of curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis phi 1 left-parenthesis t right-parenthesis comma 2 element-of left-bracket 0 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>φ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>t</mml:mi> stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mtext> </mml:mtext>...

We use the Gauss-Bonnet theorem and triangle comparison theorems of Rauch Toponogov to show that on compact Riemann surfaces negative curvature period integrals eigenfunctions $e_\lambda$ over geodesics go zero at rate $O((\log\lambda)^{-1/2})$ if $\lambda$ are their frequencies. As discussed in \cite{CSPer}, no such result is possible constant case $\ge0$. Notwithstanding, we also these bounds for valid provided all geodesic balls radius $r\le 1$ pinched from above by $-\delta r^N$ some...

We study cospectral vertices on finite graphs in relation to the echolocation problem Riemannian manifolds. First, prove a computationally simple criterion determine whether two are cospectral. Then, we use this conjunction with computer search find minimal examples of various types which but non-similar exist, including walk-regular non-vertex-transitive graphs, turn out be non-planar. Moreover, as our main result, classify all planar proving that such must vertex-transitive.

Bounded holomorphic functions on the disk have radial limits in almost every direction, as follows from Fatou's theorem.Given a zero-measure set E torus T, we study of such that lim r→1 -f (r w) fails to exist for w ∈ (such were first constructed by Lusin).We show Lusin-type functions, fixed E, contains algebras algebraic dimension c (except zero function).When is countable, also several-variable case infinite dimensional Banach spaces and, moreover, plenty c-dimensional algebras.We address...

We study a variation of Kac's question, "Can one hear the shape drum?" if we allow ourselves access to some additional information. In particular, ``hear" local Weyl counting function at each point on manifold and ask this is enough uniquely recover Riemannian metric. This physically equivalent asking whether can determine drum allowed knock any place drum. show that answer question ``yes" provided Laplace-Beltrami spectrum simple. also provide counterexample illustrating why hypothesis necessary.

We refine the $L^p$ restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, G\'erard, and Tzvetkov. First, we establish of to arbitrary Borel sets following formulation Eswarathasan Pramanik. achieve this proving variable coefficient version weighted Fourier extension estimate Du Zhang. Our results naturally unify $L^p(M)$ Sogge $L^p(\gamma)$ bounds Tzvetkov, are sharp all $p \geq 2$, up $\lambda^\varepsilon$ loss. Second, derive tubular...

We study the generalization of Falconer distance problem to Riemannian setting. In particular, we extend result Guth-Iosevich-Ou-Wang for set in plane general surfaces. Key new ingredients include a family refined microlocal decoupling inequalities, which are related work Beltran-Hickman-Sogge on Wolff-type and an analog Orponen's radial projection lemma has proved quite useful recent sets.

We show that for any dimension $d\ge3$, one can obtain Wolff's $L^{(d+2)/2}$ bound on Kakeya-Nikodym maximal function in $\mathbb R^d$ $d\ge3$ without the induction scales argument. The key ingredient is to reduce a 2-dimensional $L^2$ estimate with an auxiliary function. also prove same holds Nikodym manifold $(M^d,g)$ constant curvature, which generalizes Sogge's results $d=3$ $d\ge3$. As 3-dimensional case, we handle manifolds of curvature due fact that, this two intersecting geodesics...

Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. 𝜈-th order Fourier coefficients eigenfunctions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>e</m:mi> <m:mi>λ</m:mi> </m:msub> </m:math> e_{\lambda} over a closed smooth curve 𝛾 which satisfies natural curvature condition, go to 0 at rate <m:mrow> <m:mi>O</m:mi> <m:mo>⁢</m:mo> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>log</m:mi> <m:mo>⁡</m:mo> </m:mrow>...

We show that for a smooth closed curve $\gamma$ on compact Riemannian surface without boundary, the inner product of two eigenfunctions $e_\lambda$ and $e_\mu$ restricted to $\gamma$, $|\int e_\lambda\overline{e_\mu}\,ds|$, is bounded by $\min\{\lambda^\frac12,\mu^\frac12\}$. Furthermore, given $0<c<1$, if $0<\mu<c\lambda$, we prove $\int e_\lambda\overline{e_\mu}\,ds=O(\mu^\frac14)$, which sharp sphere $S^2$. These bounds unify period integral estimates $L^2$-restriction in an explicit way....

We show that on compact Riemann surfaces of nonpositive curvature, the generalized periods, i.e. $\nu$-th order Fourier coefficients eigenfunctions $e_\lambda$ over a closed smooth curve $\gamma$ which satisfies natural curvature condition, go to 0 at rate $O((\log\lambda)^{-1/2})$, if $0<|\nu|/\lambda<1-\delta$, for any fixed $0<\delta<1$. Our result implies, instance, periods geodesic circles with would converge zero $O((\log\lambda)^{-1/2})$. A direct corollary our results and QER theorem...