This paper concerns an inverse boundary value problem for a semilinear wave equation on a globally hyperbolic Lorentzian manifold. We prove a Hölder stability result for recovering an unknown potential q of the nonlinear wave equation □gu+qum=0, m≥4, from the Dirichlet-to-Neumann map. Our proof is based on the recent higher-order linearization method and use of Gaussian beams. We also extend earlier uniqueness results by removing the assumptions of convex boundary and that pairs of light-like geodesics can intersect only once. For this, we construct special light-like geodesics and other general constructions in Lorentzian geometry. We expect these constructions to be applicable in studies of related problems as well.

We consider the Schrdinger equation in d , d 1, with a confining potential growing at most quadratically.Our main theorem characterizes open sets from which observability holds, provided they are sufficiently regular in a certain sense.The observability condition involves the Hamiltonian flow associated with the Schrdinger operator under consideration.It is obtained using semiclassical analysis techniques.It allows us to provide an accurate estimation of the optimal observation time.We illustrate this result with several examples.In the case of two-dimensional harmonic potentials, focusing on conical or rotationinvariant observation sets, we express our observability condition in terms of arithmetical properties of the characteristic frequencies of the oscillator.1. Introduction and main results 1147 2. Study of the classical dynamics 1171 3. Proof of the main theorem 1182 4. Proofs of observability results from conical sets 1187 5. Proofs of observability results from spherical sets 1201 Appendix A. Reduction to a weaker observability inequality 1219 Appendix B. Pseudodifferential operators 1222 Acknowledgments 1225 References 1226 T 0 e -it P u 2 L 2 () dt.Obs(, T ) When this property Obs(, T ) is true, we say that the Schrdinger equation (1-1) is observable from in time T, or that observes the Schrdinger equation.The question consists in finding conditions on

In this paper we consider Riemannian manifolds of dimension at least $3$, with nonnegative Ricci curvature and Euclidean Volume Growth. For every open bounded subset with smooth boundary we establish the validity of an optimal Minkowski Inequality. We also characterise the equality case, provided the domain is strictly outward minimising and strictly mean convex. Along with the proof, we establish in full generality sharp monotonicity formulas, holding along the level sets of $p$-capacitary potentials in $p$-nonparabolic manifolds with nonnegative Ricci curvature.

The relative heat content associated with a subset ⊂ M of a sub-Riemannian manifold is defined as the total amount of heat contained in at time t, with uniform initial condition on , allowing the heat to flow outside the domain. We obtain a fourth-order asymptotic expansion in the square root of t of the relative heat content associated with relatively compact noncharacteristic domains. Compared to the classical heat content that was studied by Rizzi and Rossi (J. Math. Pures Appl. ( 9) 148 (2021), 267-307), several difficulties emerge due to the absence of Dirichlet conditions at the boundary of the domain. To overcome this lack of information, we combine a rough asymptotics for the temperature function at the boundary, coupled with stochastic completeness of the heat semigroup. Our technique applies to any (possibly rank-varying) sub-Riemannian manifold that is globally doubling and satisfies a global weak Poincaré inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups. 1. Introduction 2997 2. Preliminaries 3004 3. Small-time asymptotics of u(t, x) at the boundary 3009 4. First-order asymptotic expansion of H (t) 3013 5. Higher-order asymptotic expansion of H (t) 3018 6. An alternative approach using the heat kernel asymptotics 3024 7. The noncompact case 3027 Appendix: Iterated Duhamel's principle for I φ(t, 0) 3030 Acknowledgments 3035 References 3036 MSC2020: 35R01, 53C17, 58J60.

The purpose of this paper is to introduce and study Poincaré-Steklov (PS) operators associated to the Dirac operator D m with the so-called MIT bag boundary condition. In a domain ⊂ ‫ޒ‬ 3 , for a complex number z and for U z a solution of (D m − z)U z = 0, the associated PS operator maps the value of − U zthe MIT bag boundary value of U z -to + U z , where ± are projections along the boundary ∂ and In the first part of this paper, we show that the PS operator is a zeroth-order pseudodifferential operator and give its principal symbol. In the second part, we study the PS operator when the mass m is large, we prove that it fits into the framework of 1/m-pseudodifferential operators, and we derive some important properties, especially its semiclassical principal symbol. Subsequently, we apply these results to establish a Krein-type resolvent formula for the Dirac operator H M = D m + Mβ1 ‫ޒ‬ 3 \ for large masses M > 0 in terms of the resolvent of the MIT bag operator on . With its help, the large coupling convergence with a convergence rate of O(M −1 ) is shown. 1. Introduction 2923 2. Preliminaries 2927 3. Basic properties of the MIT bag model 2933 4. Poincaré-Steklov operators as pseudodifferential operators 2939 5. Approximation of the Poincaré-Steklov operators for large masses 2945 6. Resolvent convergence to the MIT bag model 2955 Appendix: Dirac algebra and applications 2966 Acknowledgements 2968 References 2968

Beurling-Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups of Widom-type and the corona problem in quotient Banach algebras. After surveying these developments, we give a general definition of Beurling-Carleson sets and discuss some of their basic properties. We show that the Roberts decomposition characterizes measures that do not charge Beurling-Carleson sets. For a positive singular measure µ on the unit circle, let S µ denote the singular inner function with singular measure µ. In the second part of the paper, we use a corona-type decomposition to relate a number of properties of singular measures on the unit circle, such as membership of S ′ µ in the Nevanlinna class N , area conditions on level sets of S µ and wepability. It was known that each of these properties holds for measures concentrated on Beurling-Carleson sets. We show that each of these properties implies that µ lives on a countable union of Beurling-Carleson sets. We also describe partial relations involving the membership of S ′ µ in the Hardy space H p , membership of S µ in the Besov space B p and (1− p)-Beurling-Carleson sets and give a number of examples which show that our results are optimal. Finally, we show that measures that live on countable unions of α-Beurling-Carleson sets are almost in bijection with nearly maximal solutions of u = u p • χ u>0 when p > 3 and α = ( p − 3)/( p − 1).

Let P be a countable multiset of primes and let G = p∈P ‫/ޚ‬ p‫.ޚ‬ We study the universal characteristic factors associated with the Gowers-Host-Kra seminorms for the group G. We show that the universal characteristic factor of order < k + 1 is a factor of an inverse limit of finite-dimensional k-step nilpotent homogeneous spaces. The latter is a counterpart of a k-step nilsystem where the homogeneous group is not necessarily a Lie group. As an application of our structure theorem we derive an alternative proof for the L 2 -convergence of multiple ergodic averages associated with k-term arithmetic progressions in G and derive a formula for the limit in the special case where the underlying space is a nilpotent homogeneous system. Our results provide a counterpart of the structure theorem of Host and Kra (2005) and Ziegler (2007) concerning ‫-ޚ‬actions and generalize the results of Bergelson, Tao and Ziegler (2011, 2015) concerning ‫ކ‬ ω p -actions. This is also the first instance of studying the Host-Kra factors of nonfinitely generated groups of unbounded torsion. 1. Introduction 2379 2. Main results 2381 3. Conze-Lesigne-type equation 2389 4. Inverse limit of finite-dimensional systems 2397 5. Extension trick 2403 6. Proof of Theorem 2.12, part I 2411 7. Proof of Theorem 2.12, part II 2416 8. Proving the identification G(X )/ ∼ = X 2425 9. Limit formula and convergence result 2428 Appendix A. Survey of some notation and previous results 2437 Appendix B. Topological groups and measurable homomorphisms 2443 Appendix C. Some results about phase polynomials 2445 Acknowledgements 2447 References 2447 MSC2020: 37A05.

We revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity L 1 t W 1, p for all p < ∞ in space dimensions d ≥ 2 whose transport equations admit nonunique weak solutions belonging to L p t C k for all p < ∞ and k ∈ ‫.ގ‬ In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is essential. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions d ≥ 2.