Sharp Estimates Research Articles

Rings of differential operators on (k,s)-th Tjurina algebras of singularities

Abstract In this paper, we give a description of differential operators on tensor products A ⊗ 𝕂 B {A\otimes_{\mathbb{K}}B} , where A and B are finitely generated 𝕂 {\mathbb{K}} -algebras. We prove that any differential operator on A ⊗ 𝕂 B {A\otimes_{\mathbb{K}}B} can be written as a finite sum of D 1 ⊗ D 2 {D_{1}\otimes D_{2}} , where D 1 {D_{1}} and D 2 {D_{2}} are differential operators on A and B, respectively. Moreover, we introduce a series of new invariants, the ( k , s ) {(k,s)} -th Tjurina algebra A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} for an isolated hypersurface singularity ( V , 𝟎 ) = ( V ⁢ ( f ) , 𝟎 ) ⊆ ( ℂ r , 𝟎 ) {(V,\boldsymbol{0})=(V(f),\boldsymbol{0})\subseteq(\mathbb{C}^{r},\boldsymbol{% 0})} . We formulate a sharp upper estimate for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most 1, and we give lower and upper bounds for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most n.

Sharp PDE estimates for random two-dimensional bipartite matching with power cost function

Sharp PDE estimates for random two-dimensional bipartite matching with power cost function

Sharp estimates of solutions of free boundary problems with nonlocal diffusion

Sharp estimates of solutions of free boundary problems with nonlocal diffusion

Sharp estimates and non-degeneracy of low energy nodal solutions for the Lane–Emden equation in dimension two

Sharp estimates and non-degeneracy of low energy nodal solutions for the Lane–Emden equation in dimension two

Stability of Schur’s iterates and fast solution of the discrete integrable NLS

We prove a sharp stability estimate for Schur iterates of contractive analytic functions in the open unit disk. We then apply this result in the setting of the inverse scattering approach and obtain a fast algorithm for solving the discrete integrable nonlinear Schrödinger equation (Ablowitz–Ladik equation) on the integer lattice, \mathbb{Z} . We also give a self-contained introduction to the theory of the nonlinear Fourier transform from the perspective of Schur functions and orthogonal polynomials on the unit circle.

Two-dimensional gravity waves at low regularity I: Energy estimates

This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and develop the techniques to prove a new class of energy estimates, which we call balanced cubic estimates . This yields a key improvement over the earlier cubic estimates of Hunter–Ifrim–Tataru (2016), while preserving their scale-invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using any Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness, drastically improving earlier results obtained by Alazard–Burq–Zuily (2014, 2018), Hunter–Ifrim–Tataru (2016) and Ai (2019).

On products of inner radii of non-overlapping domains containing certain segment points

This paper is devoted to an open problem: find the maximum of the product ∏ k = 1 n r ( B k , a k ) ( r ( B , a ) is an inner radius of the domain B with respect to a point a) over all non-overlapping domains B k ⊂ C and points a k ∈ B k ∩ [ − 1 ; 1 ] , k = 1 , … , n , where n>4. Also we consider similar problems on extremal decomposition of the complex plane for the case when almost all points are on the real axis and one of the points is infinitely distant, the case when one of the points lies outside the straight line and the case when an inner radius of the domain containing one of the points of a certain segment is taken in some positive real power γ. The purpose of this paper is to obtain sharp estimates for these cases.

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at t=0 is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with 0<α<1) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in L2 norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is O(τtsα−1+τ2α+h12+h22). Numerical experiments are given to verify the theoretical analysis.

Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension

AbstractFor large classes of even‐dimensional Riemannian manifolds , we construct and analyze conformally invariant random fields. These centered Gaussian fields , called co‐polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: . They share a fundamental quasi‐invariance property under conformal transformations. In terms of the co‐polyharmonic Gaussian field , we define the Liouville Quantum Gravity measure, a random measure on , heuristically given as and rigorously obtained as almost sure weak limit of the right‐hand side with replaced by suitable regular approximations . In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's ‐curvature: we give a rigorous meaning to the Polyakov–Liouville measure and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2‐dimensional Riemannian manifolds, all compact non‐negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even‐dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds.

Non-radiating sources in elastodynamics and their applications in the exterior cloaking

In this work, we develop a mathematical framework on constructed general non-radiating sources of elastic waves governed by the Navier equation via the approach of Helmholtz decomposition and potential theory in elastodynamics. Our study offers a rather comprehensive analysis. We first provide a rigorous justification of the general non-radiating sources. Based on the complete destructive interference of external elastic fields generated by specific radiating sources, a general non-radiating elastic source is derived and shown to possess a hidden interior wave field. For an incident wave, targets remain invisible within non-radiating source regions, and the geometry and boundary conditions of obstacles can be very general, which holds significant practical implications. Moreover, we introduce an effective novel method for designing such generalized non-radiating sources. To avoid the complex structure, we propose to use radiating source overlay construction on specific nodes at the boundary of non-radiating regions construction and derive sharp error estimates to evaluate the cloaking performance. The proposed scheme is capable of nearly cloaking arbitrary obstacles with a high accuracy. Numerical verifications validate the precision of our analytical findings.

Sharp Trace and Korn Inequalities for Differential Operators

AbstractWe establish sharp trace and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to $$p=1$$ p = 1 , a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For $$p=1$$ p = 1 and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full k-th order gradient estimates. Moreover, for $$1&lt;p&lt;\infty $$ 1 &lt; p &lt; ∞ , where sharp trace estimates yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist – even though the reduction to global Calderón-Zygmund estimates by extension operators might not be possible.

On the Cauchy problem for a weakly coupled system of semi-linear σ-evolution equations with double dissipation

In this paper, we would like to consider the Cauchy problem for a multi-component weakly coupled system of semi-linear σ-evolution equations with double dissipation for any σ≥1. The first main purpose is to obtain the global (in time) existence of small data solutions in the supercritical condition by assuming additional L1 regularity for the initial data and using multi-loss of decay wisely. For the second main one, we are interested in establishing the blow-up results together with sharp estimates for lifespan of solutions in the subcritical case. The proof is based on a contradiction argument with the help of modified test functions to derive the upper bound estimates. Finally, we succeed in catching the lower bound estimate by constructing Sobolev spaces with the time-dependent weighted functions of polynomial type in their corresponding norms.

On sharp heat kernel estimates in the context of Fourier–Dini expansions

We prove sharp estimates of the heat kernel associated with Fourier–Dini expansions on (0,1) equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier–Dini semigroup.

Differential Subordination and Coefficient Functionals of Univalent Functions Related to $\cos z$

Differential subordination in the complex plane is the generalization of a differential inequality on the real line. In this paper, we consider two subclasses of univalent functions associated with the trigonometric function $\cos z$. Using some properties of the hypergeometric functions, we determine the sharp estimate on the parameter $\beta$ such that the analytic function $p(z)$ satisfying $p(0)=1$, is subordinate to $\cos z$ when the differential expression $p(z)+\beta z (dp(z)/dz)$ is subordinate to the Janowski function. We compute sharp bounds on coefficient functional Hermitian--Toeplitz determinants of the third and the fourth order with an invariance property for such functions. In addition, we estimate bound on Hankel determinants of the second and the third order.

Wiener densities for the Airy line ensemble

AbstractThe parabolic Airy line ensemble is a central limit object in the KPZ (Kardar–Parisi–Zhang) universality class and related areas. On any compact set , the law of the recentered ensemble has a density with respect to the law of independent Brownian motions. We show where is an explicit, tractable, non‐negative function of . We use this formula to show that is bounded above by a ‐dependent constant, give a sharp estimate on the size of the set where as , and prove a large deviation principle for . We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two‐point tail bounds that are stronger than those for Brownian motion. These estimates are a key input in the classification of geodesic networks in the directed landscape. The paper is essentially self‐contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.

The elliptic maximal function

We study the elliptic maximal functions defined by averages over ellipses and rotated ellipses which are multi-parametric variants of the circular maximal function. We prove that those maximal functions are bounded on Lp for some p≠∞. For this purpose, we obtain some sharp multi-parameter local smoothing estimates.

On the Fourier Coefficients of Powers of a Finite Blaschke Product

Abstract Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell ^{\infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty $. This norm decays as $n^{-1/N}$ for some $N\ge 3$. Furthermore, for every $N\ge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot \|T^{-1}\|\cdot \|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.

Heat Kernel Estimates of Fractional Schrödinger Operators with Hardy Potential on Half-line

AbstractWe provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by a Hardy potential.

Sharp and Weighted Boundedness for Multilinear Integral Operators

Abstract In this paper, the weighted boundedness for some multilinear operators related to the Littlewood-Paley operator and Marcinkiewicz operator on the generalized Morrey spaces are obtained by using the sharp estimates of the multilinear operators.

The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates

This is the second part of a two-part series devoted to an in depth understanding of the dynamical behaviour of the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary |x|=h(t). In Part 1 [19], we have shown that the long-time dynamics of this problem is characterised by a spreading-vanishing dichotomy, and there exists a threshold condition on the diffusion kernel J(|x|) such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed c0 is determined by an associated semi-wave problem established in [15]. In Part 2 here, we obtain more precise description of the spreading profile by focusing on some natural classes of kernel functions, including those satisfying J(|x|)∼|x|−β for |x|≫1 in RN. Our results for such kernels reveal a striking difference of behaviour from the pattern exhibited in the one dimension case [18] when β crosses the value N+2. More precisely, (a) when β∈(N,N+1], we show that for t≫1, h(t)∼t1/(β−N) if β∈(N,N+1), and h(t)∼tln⁡t if β=N+1, which is of the same pattern as in dimension one, namely we recover the result in [18] by letting N=1 in the above statements; (b) when β∈(N+1,N+2], the front has a finite spreading speed c0=c0(β) in the sense that limt→∞⁡h(t)/t=c0, and our results here on the order of shift c0t−h(t) are again of the same pattern as in dimension one; (c) when β>N+2, the front still has a finite spreading speed c0, but a significant change happens to the order of shift c0t−h(t) between N≥2 and N=1: for t≫1, c0t−h(t)∼ln⁡t when N≥2, but c0t−h(t)∼1 when N=1.