Sharp Estimates Research Articles

A scaling limit of the parabolic Anderson model with exclusion interaction

AbstractWe consider the (discrete) parabolic Anderson model , , , where the ξ‐field is ‐valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein–Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension upon considering a suitably renormalised version of the above equation, the sequence of solutions converges in law. As a by‐product of our main result we obtain precise asymptotics for the survival probability of a simple random walk that is killed at a scale dependent rate when meeting an exclusion particle. Our proof relies on the discrete theory of regularity structures of Erhard and Hairer and on novel sharp estimates of joint cumulants of arbitrary large order for the exclusion process. We think that the latter is of independent interest and may find applications elsewhere.

Schwarzian norm estimate for functions in Robertson class

Let A denote the class of analytic functions f in the unit disk D={z∈C:|z|<1} normalized by f(0)=0, f′(0)=1. For −π/2<α<π/2, let Sα be the subclass of A consisting of functions f that satisfy the relation Re{eiα(1+zf″(z)/f′(z))}>0 for z∈D. In the present article, we determine the sharp estimate of the pre-Schwarzian and Schwarzian norms for functions in the class Sα.

Stability of Hardy Littlewood Sobolev inequality under bubbling

In this note we will generalize the results deduced in Figalli and Glaudo (Arch Ration Mech Anal 237(1):201–258, 2020) and Deng et al. (Sharp quantitative estimates of Struwe’s Decomposition. Preprint http://arxiv.org/abs/2103.15360, 2021) to fractional Sobolev spaces. In particular we will show that for sin (0,1), n>2s and nu in mathbb {N} there exists constants delta = delta (n,s,nu )>0 and C=C(n,s,nu )>0 such that for any function uin dot{H}^s(mathbb {R}^n) satisfying,u-∑i=1νU~iH˙s≤δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| u-\\sum _{i=1}^{\ u } \ ilde{U}_{i}\\right\\| _{\\dot{H}^s} \\le \\delta \\end{aligned}$$\\end{document}where tilde{U}_{1}, tilde{U}_{2},ldots tilde{U}_{nu } is a delta -interacting family of Talenti bubbles, there exists a family of Talenti bubbles U_{1}, U_{2},ldots U_{nu } such that u-∑i=1νUiH˙s≤CΓif2s<n<6s,Γ|logΓ|12ifn=6s,Γp2ifn>6s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| u-\\sum _{i=1}^{\ u } U_{i}\\right\\| _{\\dot{H}^s} \\le C\\left\\{ \\begin{array}{ll} \\Gamma &{} \ ext{ if } 2s< n < 6s,\\\\ \\Gamma |\\log \\Gamma |^{\\frac{1}{2}} &{} \ ext{ if } n=6s, \\\\ \\Gamma ^{\\frac{p}{2}} &{} \ ext{ if } n > 6s \\end{array}\\right. \\end{aligned}$$\\end{document}for Gamma =left| Delta u+u|u|^{p-1}right| _{H^{-s}} and p=2^*-1=frac{n+2s}{n-2s}.

Investigation of the Hankel Determinant Sharp Bounds for a Specific Analytic Function Linked to a Cardioid-Shaped Domain

One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these coefficient-related problems, the problem of the third-order Hankel determinant sharp bound is the most difficult one. The aim of the present study is to determine the sharp bound of the Hankel determinant of third order by using the methodology of the aforementioned Carathéodory function family. Further, we also study some other coefficient-related problems, such as the Fekete–Szegő inequality and the second-order Hankel determinant. We examine these results for the family of bounded turning functions linked with a cardioid-shaped domain.

Approximation Characteristics of Gel’fand Type in Multivariate Sobolev Spaces with Mixed Derivative Equipped with Gaussian Measure

In this paper, we study the probabilistic Gel’fand N,δ-width of multivariate Sobolev spaces MW2rTd with mixed derivative that are equipped with Gaussian measure μ in LqTd. The sharp asymptotic estimates are determined by employing the discretization method.

Minimizations of positive periodic and Dirichlet eigenvalues for general indefinite Sturm-Liouville problems

The aim of this paper is to develop an analytical approach to obtain the sharp estimates for the lowest positive periodic eigenvalue and all Dirichlet eigenvalues of a general Sturm-Liouville problemy″=q(t)y+λm(t)y, where q is a nonnegative potential and another potential m admits to change sign. A typical example of such problems is the well-known Camassa-Holm equations with indefinite potentials, which corresponds to the case q(t)≡14. It is shown that the solution of the minimization problems of the lowest positive periodic eigenvalues and Dirichlet eigenvalues will lead to more general distributions of potentials which have no densities with respect to the Lebesgue measure. As a result, it is very natural to choose the general setting of the measure differential equationsdy•=y(t)dμ(t)+λydν(t), to understand the eigenvalues and their minimization, where μ and ν are two suitable measures. The variational characterization of lowest positive eigenvalues, together with a strong continuous dependence of eigenvalues on the potentials, will play crucial roles in our analysis. Different from the periodic case, we are able to obtain the optimal bounds for higher Dirichlet eigenvalues.

Off-diagonal bounds for the Dirichlet–to–Neumann operator on Lipschitz domains

Let Ω be a bounded domain of Rn+1 with n≥1. We assume that the boundary Γ of Ω is Lipschitz. Consider the Dirichlet–to–Neumann operator N0 associated with a system in divergence form of size m with real symmetric and Hölder continuous coefficients. We prove Lp(Γ)→Lq(Γ) off-diagonal bounds of the form‖1Fe−tN01Ef‖q≲(t∧1)nq−np(1+d(E,F)t)−1‖1Ef‖p for all measurable subsets E and F of Γ. If Γ is C1+κ for some κ>0 and m=1, we obtain a sharp estimate in the sense that (1+d(E,F)t)−1 can be replaced by (1+d(E,F)t)−(1+np−nq). Such bounds are also valid for complex time. For n=1, we apply our off-diagonal bounds to prove that the Dirichlet–to–Neumann operator associated with a system generates an analytic semigroup on Lp(Γ) for all p∈(1,∞). In addition, the corresponding evolution problem has Lq(Lp)-maximal regularity.

Hankel Determinant for a Subclass of Starlike Functions with Respect to Symmetric Points Subordinate to the Exponential Function

Let Ss*(ez) denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk U the condition 2zf′(z)f(z)−f(−z)≺ez(z∈U). We obtained the sharp estimate of the second-order Hankel determinants H2,3(f) and improved the estimate of the third-order H3,1(f) for this functions class Ss*(ez).

A sharpened energy‐Strichartz inequality for the wave equation

AbstractWe consider the sharp Strichartz estimate for the wave equation on in the energy space, due to Bez and Rogers. We show that it can be refined by adding a term proportional to the distance from the set of maximizers, in the spirit of the classical sharpened Sobolev estimate of Bianchi and Egnell.

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II: Asymptotic profiles of solutions and radial terrace solution

This paper is a continuation of our previous paper (Kaneko-Matsuzawa-Yamada, Discrete Contin. Dyn. Syst., 2022), where we have classified all large-time behaviors of radially symmetric solutions to a free boundary problem of reaction diffusion equation ut=Δu+f(u) with positive bistable nonlinearity f in high space dimensions. The positive bistable nonlinearity means that f(u)=0 has exactly two positive stable equilibria. Among the classified solutions, we are interested in a spreading solution, that is a solution (u(t,|x|),h(t)) for x∈RN with free boundary |x|=h(t) such that, as t→∞, |x|≤h(t) expands to the whole space RN and u(t,⋅) converges to a positive stable equilibrium for f(u) uniformly in any compact set of RN. When we discuss whole asymptotic profiles of spreading solutions, it has been known that they are generally described with use of a semi-wave obtained from the corresponding semi-wave problem.Our main purpose is to study precise asymptotic estimates for any spreading solution whose profile accompanies a propagating terrace with two different types of propagating speeds. Such a spreading phenomenon occurs when u(t,⋅) converges to the largest equilibrium of f and the related semi-wave problem does not have a solution. We will prove that the propagating terrace consists of two functions; one is a semi-wave corresponding to a smaller positive equilibrium of f and the other is a traveling wave connecting two positive equilibria of f. In order to give sharp estimates for a radial terrace solution with the above properties, we need so called logarithmic shiftings, which are revealed by Uchiyama (1983) and Du-Matsuzawa-Zhou (2015) in higher dimensional cases. Moreover, we will see that two types of the logarithmic shiftings appear in the estimates. The proof for the higher dimensional case is very different from the one-dimensional case established by Kaneko-Matsuzawa-Yamada (2020). To derive exact logarithmic terms, we have to construct a series of upper and lower solutions to make estimates more and more accurate.

Saturation phenomena of a nonlocal eigenvalue problem: the Riemannian case

ABSTRACT In this paper we investigate the Riemannian extensibility of saturation phenomena treated first in the Euclidean framework by Brandolini et al. [Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv Math (N Y). 2011;228(4):2352–2365.]. The saturation problem is formulated in terms of the first eigenvalue of the perturbation of the Laplace-Beltrami operator by the integral of the unknown function: the first eigenvalue increases with the weight affecting the integral up to a finite critical value and then remains constant, i.e. it saturates. Given a Riemannian manifold with certain curvature constraints, by using symmetrization arguments and sharp isoperimetric inequalities, we reduce the general problem to a variational one, formulated on either positively or negatively curved Riemannian model spaces; in addition, the possible scenarios for the optimal domains turn to be either geodesic balls or the union of two disjoint geodesic balls. We then explicitly compute the eigenvalues and eigenfunctions in terms of the radii, curvature and weight. A sufficient condition (incompatibility of a system of nonlinear equations involving special functions) is given that implies similar saturation phenomena to the Euclidean case. Due to its highly nonlinear character of the reduced problem (arising from the presence of curvature and special functions), we provide only partial answers to the original problem. However, both analytical computations and numerical tests suggest that the required incompatibility always persists. In addition, in the limit cases when the curvature tends to zero (for both positive an negative curvature), our results reduce to the Euclidean version.

Capacities, Green function, and Bergman functions

AbstractUsing the logarithmic capacity, we give quantitative estimates of the Green function, as well as precise lower bounds of the Bergman kernel for bounded pseudoconvex domains in and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies the sharp estimate if is a bounded pseudoconvex domain with ‐boundary.

Estimates for eigenvalues of the Neumann and Steklov problems

Abstract We prove Li-Yau-Kröger-type bounds for Neumann-type eigenvalues of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-type inequalities for the corresponding first nonzero eigenvalue.

Sharp endpoint estimates for Schrödinger groups on Hardy spaces

Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type with a dimension n. Suppose that the heat kernel of L satisfies the Davies-Gaffney estimates of order m≥2. Let HL1(X) be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrödinger group eitL associated with L such that‖(I+L)−n/2eitLf‖L1(X)+‖(I+L)−n/2eitLf‖HL1(X)≤C(1+|t|)n/2‖f‖HL1(X),t∈R for some constant C=C(n,m)>0 independent of t. We further apply our result to provide the sharp estimate for Schrödinger group of the Kohn Laplacian □b on polynomial model domains treated by Nagel–Stein [41], where e−t□b satisfies only the second order Davies-Gaffney estimates.Moreover, when the heat kernel of L satisfies a Gaussian upper bound, by a duality and interpolation argument, it gives a new proof of a recent result of [13] for sharp endpoint Lp-Sobolev bound for eitL:‖(I+L)−seitLf‖Lp(X)≤C(1+|t|)s‖f‖Lp(X),t∈R,s≥n|12−1p| for every 1<p<∞, which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space Rn.

ON CERTAIN CLOSE-TO-CONVEX FUNCTIONS

AbstractLet $\mathcal {K}_u$ denote the class of all analytic functions f in the unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z|&lt;1\}$ , normalised by $f(0)=f'(0)-1=0$ and satisfying $|zf'(z)/g(z)-1|&lt;1$ in $\mathbb {D}$ for some starlike function g. Allu, Sokól and Thomas [‘On a close-to-convex analogue of certain starlike functions’, Bull. Aust. Math. Soc.108 (2020), 268–281] obtained a partial solution for the Fekete–Szegö problem and initial coefficient estimates for functions in $\mathcal {K}_u$ , and posed a conjecture in this regard. We prove this conjecture regarding the sharp estimates of coefficients and solve the Fekete–Szegö problem completely for functions in the class $\mathcal {K}_u$ .

Multiple weighted estimates for bilinear Calderón-Zygmund operator and its commutator on non-homogeneous spaces

Let (X,d,μ) be a non-homogeneous metric measure space. In this setting, the author proves that the bilinear Calderón-Zygmund operator T˜ is bounded from the product of weighted Morrey spaces Lω1p1,κ(μ)×Lω2p2,κ(μ) into weighted weak Morrey spaces WLνω→p,κ(μ), and it is also bounded from the product of generalized weighted Morrey spaces Lω1p1,φ1(μ)×Lω2p2,φ2(μ) into generalized weighted weak Morrey spaces WLνω→p,φ(μ), where ω→=(ω1,ω2), ϱ∈[1,∞), ω→∈AP→ϱ(μ), P→=(p1,p2) satisfying 1p=1p1+1p2 for 1≤p1, p2<∞. Furthermore, via the sharp maximal estimate for the commutator T˜b1,b2 which is generated by b1, b2∈RBMO˜(μ) and T˜, the author shows that T˜b1,b2 is bounded from the product of spaces Lω1p1,κ(μ)×Lω2p2,κ(μ) into spaces WLνω→p,κ(μ), and it is also bounded from the product of spaces Lω1p1,φ1(μ)×Lω2p2,φ2(μ) into spaces WLνω→p,φ(μ).

Critical Points of Polynomials With Roots on the Unit Circle

Abstract Let ${\mathcal {L}}(u)$ be a polynomial with all its roots on the unit circle. We establish an “angle duality” theorem relating the roots and critical points of ${\mathcal {L}}(u)$. As an application, we prove that if there is a large gap between roots of ${\mathcal {L}}(u)$, then most critical points of ${\mathcal {L}}(u)$ lie close to the unit circle. We also use our result to strengthen Sendov’s well-known conjecture for these special polynomials. Finally, we prove a sharp estimate for the location of the critical point near a small gap between roots of ${\mathcal {L}}(u)$.

Compressive frequency-difference direction-of-arrival estimation

Direction-of-arrival estimation is difficult for signals spatially undersampled by more than half the wavelength. Frequency-difference beamforming [Abadi, Song, and Dowling (2012). J. Acoust. Soc. Am. 132, 3018–3029] offers an alternative approach to avoid such spatial aliasing by using multifrequency signals and processing them at a lower frequency, the difference-frequency. As with the conventional beamforming method, lowering the processing frequency sacrifices spatial resolution due to a beam broadening. Thus, unconventional beamforming is detrimental to the ability to distinguish between closely spaced targets. To overcome spatial resolution deterioration, we propose a simple yet effective method by formulating the frequency-difference beamforming as a sparse signal reconstruction problem. Similar to compressive beamforming, the improvement (compressive frequency-difference beamforming) promotes sparse nonzero elements to obtain a sharp estimate of the spatial direction-of-arrival spectrum. Analysis of the resolution limit demonstrates that the proposed method outperforms the conventional frequency-difference beamforming in terms of separation if the signal-to-noise ratio exceeds 4 dB. Ocean data from the FAF06 experiment support the validity.

Parametric estimation of a stability factor for certified reduced basis methods via adaptive Gaussian process

The a posteriori error analysis of a reduced basis (RB) method may certify the accuracy of an RB solution without the truth finite element (FE) solution. However, the error theory necessitates a numerically expensive stability factor such as coercivity or an inf–sup factor. Consequently, we perforce resort to the successive constraint method (SCM) to achieve a stability lower bound for a rigorous albeit rapid RB analysis. As an alternative, we propose an adaptive Gaussian process (AGP) to produce a stability lower bound for the rigorous, rapid, and sharp estimation of an RB solution error. We utilize linear elastostatic and Helmholtz acoustic problems with geometric parameterization to compare the AGP with the SCM for a numerical investigation. The results show that the AGP significantly surpasses the SCM. This is because the a posteriori error estimation of an RB solution obtained with the AGP is sharper and faster than that obtained with the SCM by factors of at least 103 and 10, respectively. With regard to a rigorous error estimation, the AGP succeeds for the linear elastostatic problem but fails for the Helmholtz acoustic problem when an RB size is one at parameter values incurring only resonance. Otherwise, it manages to attain a rigorous error estimation. To conclude, the AGP is observed to be heuristically more practical than the SCM for a certified RB analysis. This research may considerably aid a static condensation RB element (scRBE) method because the technique numerously repeats the a posteriori error estimation of a reduced bubble function for the a posteriori error estimation of an scRBE solution.

Problems Concerning Coefficients of Symmetric Starlike Functions Connected with the Sigmoid Function

In numerous geometric and physical applications of complex analysis, estimating the sharp bounds of coefficient-related problems of univalent functions is very important due to the fact that these coefficients describe the core inherent properties of conformal maps. The primary goal of this paper was to calculate the sharp estimates of the initial coefficients and some of their combinations (the Hankel determinants, Zalcman’s functional, etc.) for the class of symmetric starlike functions linked with the sigmoid function. Moreover, we also determined the bounds of second-order Hankel determinants containing coefficients of logarithmic and inverse functions of the same class.