Sharp Estimates Research Articles

Refined blow-up asymptotics for a perturbed nonlinear heat equation with a gradient and a non-local term

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier works [1,2], we constructed a solution u for that equation such that u and ∇u both blow up at the origin and only there. We also gave the final blow-up profile. In this paper, we refine our construction method in order to get a sharper estimate on the gradient at blow-up.

Boundary Harnack principle for diffusion with jumps

Consider the operator Lb=L0+b1⋅∇+Sb2 on Rd, where L0 is a second order differential operator of non-divergence form, the drift function b1 belongs to some Kato class and Sb2f(x)≔∫Rdf(x+z)−f(x)−∇f(x)⋅z1{|z|≤1}b2(x,z)j0(z)dz,f∈Cb2(Rd).Here j0(z) is a nonnegative locally bounded function on Rd∖{0} satisfying that ∫Rd(1∧|z|2)j0(z)dz<∞ and that there are constants β∈(1,2) and c0>0 so that j0(z)≤c0|z|d+βfor|z|≤1,and b2(x,z) is a real-valued bounded function on Rd×Rd. There is conservative Feller process Xb associated with the non-local operator Lb. We derive sharp two-sided Green function estimates of Lb on bounded C1,1 domains, identify the Martin and minimal Martin boundary, and establish the Martin integral representation of Lb-harmonic functions on these domains. The latter in particular reveals how the process Xb exits a bounded C1,1 domain D, or equivalently, the structure of the harmonic measure of Lb on D, which consists of the continuously exiting term and the jump-off term. These results are then used to establish, under some mild conditions, Harnack principle and the boundary Harnack principle with explicit boundary decay rate for the operator Lb on C1,1 open sets.

Uniform Estimates for Concave Homogeneous Complex Degenerate Elliptic Equations Comparable to the Monge-Ampère Equation

We prove sharp uniform estimates for strong supersolutions of a large class of fully nonlinear degenerate elliptic complex equations. Our findings rely on ideas of Kuo and Trudinger who dealt with degenerate linear equations in the real setting. We also exploit the pluripotential theory for the complex Monge-Ampère operator as well as suitably tailored theory of Lp-viscosity subsolutions.

Sharp moment estimates for martingales with uniformly bounded square functions

We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on $\mathrm{BMO}([0,1])$. In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.

Sharp weak type estimates for a family of Córdoba bases

Sharp weak type estimates for a family of Córdoba bases

Sharp Bounds of Hankel Determinant on Logarithmic Coefficients for Functions of Bounded Turning Associated with Petal-Shaped Domain

The purpose of this article is to obtain the sharp estimates of the first four initial logarithmic coefficients for the class BTs of bounded turning functions associated with a petal-shaped domain. Further, we investigate the sharp estimate of Fekete-Szegö inequality, Zalcman inequality on the logarithmic coefficients and the Hankel determinant H2,1Ff/2 and H2,2Ff/2 for the class BTs with the determinant entry of logarithmic coefficients.

Maximal regularity for the Cauchy problem of the heat equation in BMO

AbstractWe consider maximal regularity for the Cauchy problem of the heat equation in a class of bounded mean oscillations (). Maximal regularity for non‐reflexive Banach spaces is not obtained by the established abstract theory. Based on the symmetric characterization of ‐expression, we obtain maximal regularity for the heat equation in and its sharp trace estimate. Our result shows that the homogeneous initial estimate obtained by Stein [50] and Koch–Tataru [32] can be strengthened up to the inhomogeneous estimate for the external forces and the obtained estimates can be applicable to quasilinear problems. Our method is based on integration by parts and can also be applicable to other type of parabolic problems.

Heat kernel estimates on spaces with varying dimension

We obtain sharp two-sided heat kernel estimates for some process whose regular Dirichlet form is strongly local on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume doubling property fails with respect to the measure induced by the associated Lebesgue measures. Thus the parabolic Harnack inequalities fail and the heat kernels do not enjoy Aronson type estimates. Our estimates show that the on-diagonal estimates are independent of the dimensions of the two parts of the space for small time, whereas they depend on their transience or recurrence for large time. These are multidimensional version of a space considered by Z.-Q. Chen and S. Lou (Ann. Probab. 2019), in which a 1-dimensional space and a 2-dimensional space are connected at one point.

On the 1/H-flow by p-Laplace approximation: New estimates via fake distances under Ricci lower bounds

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and for the mean curvature of its level sets, that are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the $p$-Laplace equation, and relies on new gradient and decay estimates for $p$-harmonic capacity potentials, notably for the kernel $\mathcal{G}_p$ of $\Delta_p$. These bounds, stable as $p \rightarrow 1$, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of $w$.

The High Dimensional Fisher-KPP Nonlocal Diffusion Equation with Free Boundary and Radial Symmetry, Part 1

This is Part 1 of a two-part series, where we study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and obtain a rather complete description of its long-time dynamical behavior, which reveals both similarities and fundamental differences to its one dimensional version considered in [J. Cao, Y. Du, F. Li, and W. T. Li, J. Funct. Anal., 277 (2019), pp. 2772--2814; Y. Du, F. Li, and M. Zhou, J. Math. Pure Appl., 154 (2021), pp. 30--66; Y. Du and W. Ni, Rate of Propagation for the Fisher-KPP Equation with Nonlocal Diffusion, and Free Boundaries, preprint, 2021] recently. Our goals on the long-time dynamics of the model include (a) find the threshold condition on the kernel function that governs the onset of accelerated spreading, (b) determine the spreading speed when it is finite, (c) obtain sharp estimates of the spreading profile for finite speed spreading as well as for accelerated spreading. The tasks in (a) and (b) are carried out here in Part 1, while (c) is done in a separate Part 2. This high dimensional problem poses considerable technical difficulties when the rate of spreading is considered, and we overcome that by introducing an intermediate kernel function which plays a crucial role both in determining the onset condition for accelerated spreading and in obtaining the spreading speed when it is finite.

Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems

In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincaré inequality is based on the work of Ledoux (In Séminaire de Probabilités, XXXV (2001) 167–194, Springer) and the uniform logarithmic Sobolev inequality is based on Zegarlinski’s theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from (J. Funct. Anal. 257 (2009) 4015–4033). The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKean–Vlasov equation with an explicit rate, We need here weaker conditions than the results of (Rev. Mat. Iberoam. 19 (2003) 971–1018) (by means of the displacement convexity approach), (Stochastic Process. Appl. 95 (2001) 109–132; Ann. Appl. Probab. 13 (2003) 540–560) (by Bakry–Emery’s technique) or the recent work (Arch. Ration. Mech. Anal. 208 (2013) 429–445) (by dissipation of the Wasserstein distance).

A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations

In this paper, a second-order scheme with nonuniform time meshes for Caputo–Hadamard fractional sub-diffusion equations with initial singularity is investigated. Firstly, a Taylor-like formula with integral remainder is proposed, which is crucial to studying the discrete convolution kernels and error estimate. Secondly, we come up with an error convolution structure (ECS) analysis for Llog,2−1σ interpolation approximation to the Caputo–Hadamard fractional derivative. The core result in this paper is an ECS bound and a global consistency analysis established at an offset point. By virtue of this result, we obtain a sharp L2-norm error estimate of a second-order Crank–Nicolson-like scheme for Caputo–Hadamard fractional differential equations. Ultimately, an example is presented to show the sharpness of our analysis.

Composition operators on Bloch and Hardy type spaces

The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open problem. Then some classes of composition operators on Bloch and Hardy type spaces will be investigated. The obtained results improve and extend some corresponding known results.

Frequency-explicit approximability estimates for time-harmonic Maxwell’s equations

We consider time-harmonic Maxwell’s equations set in a heterogeneous medium with perfectly conducting boundary conditions. Given a divergence-free right-hand side lying in $$L^2$$ , we provide a frequency-explicit approximability estimate measuring the difference between the corresponding solution and its best approximation by high-order Nédélec finite elements. Such an approximability estimate is crucial in both the a priori and a posteriori error analysis of finite element discretizations of Maxwell’s equations, but the derivation is not trivial. Indeed, it is hard to take advantage of high-order polynomials given that the right-hand side only exhibits $$L^2$$ regularity. We proceed in line with previously obtained results for the simpler setting of the scalar Helmholtz equation and propose a regularity splitting of the solution. In turn, this splitting yields sharp approximability estimates generalizing known results for the scalar Helmholtz equation and showing the interest of high-order methods.

Local Convergence of the FEM for the Integral Fractional Laplacian

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

AbstractConditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$ -matrices over $\mathbb{F}_p$ for primes $2 &lt; p \leq \exp(O(n^{1/4}))$ . Previously, such estimates were available only for $p = o(n^{1/8})$ . At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + \Omega(1/p^2)$ . Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C &gt; 1$ .

Note on the Fekete–Szegö Problem for Spirallike Mappings in Banach Spaces

In this note we present a remark on the paper “On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables” by Y. Lai and Q. Xu [10] published recently in the journal Results in Mathematics. We show that one of the theorems in [10] concerning the finite-dimensional space \({{\mathbb {C}}}^n\) is a direct consequence of another one, so it does not need an independent proof. Moreover, we prove that a sharp norm estimate on the Fekete–Szegö functional over spirallike mappings in a general Banach space can be deduced from a result in [10].

Functions of perturbed commuting dissipative operators

AbstractThe main objective of the paper is to obtain sharp Lipschitz type estimates for the norm of operator differences for pairs and of commuting maximal dissipative operators. To obtain such estimates, we use double operator integrals with respect to semi‐spectral measures associated with the pairs and . Note that the situation is considerably more complicated than in the case of functions of two commuting contractions and to overcome difficulties we had to elaborate new techniques. We deduce from the main result Hölder type estimates for operator differences as well as their estimates in Schatten–von Neumann norms.

Geometric estimates for doubly nonlinear parabolic PDEs

In this manuscript, we establish regularity estimates for bounded weak solutions of a certain class of doubly degenerate evolution PDEs, whose simplest model case is given by ∂u∂t−div(m|u|m−1|∇u|p−2∇u)=f(x,t)inΩT≔Ω×(0,T), where m ⩾ 1, p ⩾ 2 and f belongs to a suitable anisotropic Lebesgue space. Employing intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In this scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence of our findings and approach, we address a Liouville type result for entire weak solutions of a related homogeneous problem with frozen coefficients and asymptotic estimates under a certain approximating regime, which may have their own mathematical interest. We also present examples of degenerate PDEs where our results can be applied.

Interlacing and Scaling Exponents for the Geodesic Watermelon in Last Passage Percolation

In a discrete planar last passage percolation (LPP), random values are assigned independently to each vertex in $${\mathbb {Z}}^2$$ , and each finite upright path in $${\mathbb {Z}}^2$$ is ascribed the weight given by the sum of values attached to the vertices of the path. The weight of a collection of disjoint paths is the sum of its members’ weights. The notion of a geodesic, namely a path of maximum weight between two vertices, has a natural generalization concerning several disjoint paths. Indeed, a k-geodesic watermelon in $$[1,n]^2\cap {\mathbb {Z}}^2$$ is a collection of k disjoint upright paths contained in this square that has maximum weight among all such collections. While the weights of such collections are known to be important objects, the maximizing paths have remained largely unexplored beyond the $$k=1$$ case. For exactly solvable models, such as exponential and geometric LPP, it is well known that for $$k=1$$ the exponents that govern fluctuation in weight and transversal distance are 1/3 and 2/3; which is to say, the weight of the geodesic on the route $$(1,1) \rightarrow (n,n)$$ typically fluctuates around a dominant linear growth of the form $$\mu n$$ by the order of $$n^{1/3}$$ ; and the maximum Euclidean distance of the geodesic from the diagonal typically has order $$n^{2/3}$$ . Assuming a strong but local form of convexity and one-point moderate deviation estimates for the geodesic weight profile—which are available in all known exactly solvable models—we establish that, typically, the k-geodesic watermelon’s weight falls below $$\mu n k$$ by order $$k^{5/3}n^{1/3}$$ , and its transversal fluctuation is of order $$k^{1/3}n^{2/3}$$ . Our arguments crucially rely on, and develop, a remarkable deterministic interlacing property that the watermelons admit. Our methods also yield sharp rigidity estimates for naturally associated point processes. These bounds improve on estimates obtained by applying tools from the theory of determinantal point processes available in the integrable setting.