Remainder Estimates Research Articles

Newton and Bouligand derivatives of the scalar play and stop operator

We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strengthened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable.

On Asymptotics of the Density of States for Hypoelliptic Almost Periodic Systems

In this paper, we find the asymptotics of integrated density of states with remainder estimate for hypoelliptic systems with almost periodic (a.p.) coefficients. We use the approximate spectral projector method for matrix a.p. operators with continuous spectrum.

Boundary Layer of Transport Equation with In-Flow Boundary

Consider the steady neutron transport equation in two dimensional convex domains with an in-flow boundary condition. We establish the diffusive limit while the boundary layers are present. Our contribution relies on a delicate decomposition of boundary data to separate the regular and singular boundary layers, novel weighted $$W^{1,\infty }$$ estimates for the Milne problem with geometric correction in convex domains, and an $$L^{2m}-L^{\infty }$$ framework which yields stronger remainder estimates.

Generalized Bergman kernels on symplectic manifolds of bounded geometry

We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtain the relation between the generalized Bergman kernel on a Galois covering of a compact symplectic manifold and the generalized Bergman kernel on the base. Then we state the full off-diagonal asymptotic expansion of the generalized Bergman kernel, improving the remainder estimate known in the compact case to an exponential decay. Finally, we establish the theory of Berezin-Toeplitz quantization on symplectic orbifolds associated with the renormalized Bochner-Laplacian.

Asymptotic expansions for the gamma function in terms of hyperbolic functions

The Smith's approximation formula for the gamma function is given byΓ(x+12)=2π(xe)x(2xtanh⁡12x)x/2(1+O(1x5)), as x→∞. In this paper, by means of a little known power series, we develop this formula to an asymptotic expansions:Γ(x+1/2)2π(x/e)x∼(2xtanh⁡12x)x/2exp⁡(∑n=3∞[(2n)!−(2n−1)22n−1](1−21−2n)2n(2n−1)(2n)!B2nx2n−1), as x→∞, and give an estimate of remainder in the above asymptotic series, where B2n is the Bernoulli number. Moreover, as relevant results, Ramanujan type asymptotic expansions for the gamma function Γ(x+1/2) are obtained; an asymptotic expansion for Wallis ratio and an estimate of the remainder are established.

Uniform Estimates of Remainders in Spectral Analysis of Linear Differential Systems

We study the problem of estimating the expression Υ(λ) = sup{|∫0xf(t)eiλω(t)dt|: x ∈ [0, 1]}, where the derivative of the function ω(t) is positive almost everywhere on [0, 1]. In particular, for f ∈ Lp[0, 1], p ∈ (1, 2], we prove the estimate ∥Υ(λ)∥ Lq(ℝ) ≤ C∥f∥Lp, where 1/p + 1/q = 1. The same estimate is obtained in the space Lq(dμ), where dμ is an arbitrary Carleson measure in the open upper half-plane ℂ+. In addition, we estimate more complicated expressions like Υ(λ) that arise when studying the asymptotics of fundamental solution systems for systems of the form y′ = λρ(x)By +A(x)y +C(x, λ)y as |λ| →∞in appropriate sectors of the complex plane.

Asymptotic analysis of unsteady neutron transport equation

Consider the unsteady neutron transport equation with diffusive boundary condition in 2D convex domains. We establish the diffusive limit with both initial layer and boundary layer corrections. The major difficulty is the lack of regularity in the boundary layer with geometric correction. Our contribution relies on a detailed analysis of asymptotic expansions inspired by the compatibility condition and an intricate L2m − L∞ framework, which yields stronger remainder estimates.

HIGHER ORDER DIFFERENTIABILITY OF OPERATOR FUNCTIONS IN SCHATTEN NORMS

We establish the following results on higher order ${\mathcal{S}}^{p}$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space:(i)$f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every bounded self-adjoint operator if and only if $f\in C^{n}(\mathbb{R})$;(ii)if $f^{\prime },\ldots ,f^{(n-1)}\in C_{b}(\mathbb{R})$ and $f^{(n)}\in C_{0}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every self-adjoint operator;(iii)if $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$, then $f$ is $n-1$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable and $n$ times Gâteaux ${\mathcal{S}}^{p}$-differentiable at every self-adjoint operator.We also prove that if $f\in B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{q}$-differentiable, $1\leqslant q<\infty$, at every self-adjoint operator. These results generalize and extend analogous results of Kissin et al. (Proc. Lond. Math. Soc. (3)108(3) (2014), 327–349) to arbitrary $n$ and unbounded operators as well as substantially extend the results of Azamov et al. (Canad. J. Math.61(2) (2009), 241–263); Coine et al. (J. Funct. Anal.; doi:10.1016/j.jfa.2018.09.005); Peller (J. Funct. Anal.233(2) (2006), 515–544) on higher order ${\mathcal{S}}^{p}$-differentiability of $f$ in a certain Wiener class, Gâteaux ${\mathcal{S}}^{2}$-differentiability of $f\in C^{n}(\mathbb{R})$ with $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$, and Gâteaux ${\mathcal{S}}^{q}$-differentiability of $f$ in the intersection of the Besov classes $B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$. As an application, we extend ${\mathcal{S}}^{p}$-estimates for operator Taylor remainders to a broad set of symbols. Finally, we establish explicit formulas for Fréchet differentials and Gâteaux derivatives.

The heat trace for the drifting Laplacian and Schrödinger operators on manifolds

We study the heat trace for both the drifting Laplacian as well as Schr\"odinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder estimate. We also demonstrate that the more precise asymptotic behavior of the remainder is determined by and conversely distinguishes higher (Sobolev) regularity on the potential or weight function. In the case of a smooth weight function, we determine the full asymptotic expansion of the heat trace for the drifting Laplacian for small times. We then use the heat trace to study the asymptotics of the eigenvalue counting function. In both cases the Weyl law coincides with the Weyl law for the Riemannian manifold with the standard Laplace-Beltrami operator. We conclude by demonstrating isospectrality results for the drifting Laplacian on compact manifolds.

A note on stability of Hardy inequalities

In this note we formulate recent stability results for Hardy inequalities in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for Rellich type inequalities on homogeneous groups. Main differences from the Euclidean results are that the obtained stability estimates hold for any homogeneous quasi-norm.

An Improved Remainder Estimate in the Weyl Formula for the Planar Disk

Y. Colin de Verdiere proved that the remainder term in the two-term Weyl formula for the eigenvalue counting function for the Dirichlet Laplacian associated with the planar disk is no more than $$O(\mu ^{2/3})$$ . In this paper, we study this problem from spectral geometry by using analysis and number theory. More precisely, by combining with the method of exponential sum estimation, we give a sharper remainder term estimate $$O(\mu ^{2/3-1/495})$$ .

Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for 𝐿^{𝑝}-weighted Hardy inequalities

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for 1 > p , q > ∞ 1>p,q>\infty , 0 > r > ∞ 0>r>\infty with p + q ≥ r p+q\geq r , δ ∈ [ 0 , 1 ] ∩ [ r − q r , p r ] \delta \in [0,1]\cap \left [\frac {r-q}{r},\frac {p}{r}\right ] with δ r p + ( 1 − δ ) r q = 1 \frac {\delta r}{p}+\frac {(1-\delta )r}{q}=1 and a a , b b , c ∈ R c\in \mathbb {R} with c = δ ( a − 1 ) + b ( 1 − δ ) c=\delta (a-1)+b(1-\delta ) , and for all functions f ∈ C 0 ∞ ( R n ∖ { 0 } ) f\in C_{0}^{\infty }(\mathbb {R}^{n}\backslash \{0\}) we have ‖ | x | c f ‖ L r ( R n ) ≤ | p n − p ( 1 − a ) | δ ‖ | x | a ∇ f ‖ L p ( R n ) δ ‖ | x | b f ‖ L q ( R n ) 1 − δ \begin{equation*} \||x|^{c}f\|_{L^{r}(\mathbb {R}^{n})} \leq \left |\frac {p}{n-p(1-a)}\right |^{\delta } \left \||x|^{a}\nabla f\right \|^{\delta }_{L^{p}(\mathbb {R}^{n})} \left \||x|^{b}f\right \|^{1-\delta }_{L^{q}(\mathbb {R}^{n})} \end{equation*} for n ≠ p ( 1 − a ) n\neq p(1-a) , where the constant | p n − p ( 1 − a ) | δ \left |\frac {p}{n-p(1-a)}\right |^{\delta } is sharp for p = q p=q with a − b = 1 a-b=1 or p ≠ q p\neq q with p ( 1 − a ) + b q ≠ 0 p(1-a)+bq\neq 0 . In the critical case n = p ( 1 − a ) n=p(1-a) we have ‖ | x | c f ‖ L r ( R n ) ≤ p δ ‖ | x | a log ⁡ | x | ∇ f ‖ L p ( R n ) δ ‖ | x | b f ‖ L q ( R n ) 1 − δ . \begin{equation*} \left \||x|^{c}f\right \|_{L^{r}(\mathbb {R}^{n})} \leq p^{\delta } \left \||x|^{a}\log |x|\nabla f\right \|^{\delta }_{L^{p}(\mathbb {R}^{n})} \left \||x|^{b}f\right \|^{1-\delta }_{L^{q}(\mathbb {R}^{n})}. \end{equation*} Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein’s homogeneous groups. Consequently, we obtain remainder estimates for L p L^{p} -weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of R n \mathbb {R}^{n} . The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of L p L^{p} -weighted Hardy inequalities involving a distance and stability estimate. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. We also establish sharp Hardy type inequalities in L p L^{p} , 1 > p > ∞ 1>p>\infty , with superweights, i.e., with the weights of the form ( a + b | x | α ) β p | x | m \frac {(a+b|x|^{\alpha })^{\frac {\beta }{p}}}{|x|^{m}} allowing for different choices of α \alpha and β \beta . There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters.

The equivariant spectral function of an invariant elliptic operator. Lp-bounds, caustics, and concentration of eigenfunctions

Let M be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a compact Lie group G, and P0 an invariant elliptic classical pseudodifferential operator on M. Using Fourier integral operator techniques, we prove a local Weyl law with remainder estimate for the equivariant (or reduced) spectral function of P0 for each isotypic component in the Peter–Weyl decomposition of L2(M), generalizing work of Avacumovič, Levitan, and Hörmander. From this we deduce a generalized Kuznecov sum formula for periods of G-orbits, and recover the local Weyl law for orbifolds shown by Stanhope and Uribe. Relying on recent results on singular equivariant asymptotics of oscillatory integrals, we further characterize the caustic behavior of the reduced spectral function near singular orbits, which allows us to give corresponding point-wise bounds for clusters of eigenfunctions in specific isotypic components. In case that G acts on M without singular orbits, we are able to deduce hybrid Lp-bounds for 2≤p≤∞ in the eigenvalue and isotypic aspect that improve on the classical estimates of Seeger and Sogge for generic eigenfunctions. Our results are sharp in the eigenvalue aspect, but not in the isotypic aspect, and reduce to the classical ones in the case G={e}.

Multilinear Schur multipliers and Schatten properties of operator Taylor remainders

We establish sharp conditions on scalar functions and perturbations that guarantee Schatten summability of nth order operator Taylor remainders. In the special case of dimension one, our estimates of these remainders deliver well known classical estimates of scalar Taylor remainders. We prove that if a scalar function f is in the set Cn and a perturbation is in the pth Schatten class Sp, p>n, then the respective nth order operator Taylor remainder is an element of Sp/n and has an estimate like the one in [16]. We construct examples of f∈Cn and perturbations in Sn such that the nth order Taylor remainder of the respective operator function is not in S1. Our construction relies, in particular, on novel dimension dependent estimates for Schatten norms of multilinear Schur multipliers from below that are of interest in their own right. Our results apply to both self-adjoint and unitary operators.

Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach

We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of $h$-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint Schrodinger operators with matrix-valued potentials. We give Weyl type semiclassical asymptotics with sharp remainder estimate for the spectral shift function, and, under the existence of a scalar escape function, a full asymptotic expansion in the strong sense for its derivative. A time-independent approach enables us to treat certain potentials with energy-level crossings.

Geometric Correction in Diffusive Limit of Neutron Transport Equation in 2D Convex Domains

Consider the steady neutron transport equation with diffusive boundary condition. In Wu and Guo (Commun Math Phys 336:1473–1553, 2015) and Wu et al. (J Stat Phys 165:585–644, 2016), it was discovered that geometric correction is necessary for the Milne problem of Knudsen-layer construction in a disk or annulus. In this paper, we establish the diffusive limit for a 2D convex domain. Our contribution relies on novel weighted $${W^{1,\infty}}$$ estimates for the Milne problem with geometric correction in the presence of a convex domain, as well as an $${L^{2m}-L^{\infty}}$$ framework which yields stronger remainder estimates.

Addendum to “Singular equivariant asymptotics and Weyl’s law”

Abstract Let M be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group G. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on T ∗ ⁢ M × G T^{\ast}M\times G with singular critical sets that were examined in [7] in order to determine the asymptotic distribution of eigenvalues of an invariant elliptic operator on M. As an immediate consequence, we deduce from this an asymptotic multiplicity formula for families of irreducible representations in L 2 ⁢ ( M ) \mathrm{L}^{2}(M) . The improved remainder is used in [4] to prove an equivariant semiclassical Weyl law and a corresponding equivariant quantum ergodicity theorem.

On the semiclassical functional calculus for h-dependent functions

We study the functional calculus for operators of the form \(f_h(P(h))\) within the theory of semiclassical pseudodifferential operators, where \(\{f_h\}_{h\in (0,1]}\subset \mathrm{C^\infty _c}({{\mathbb {R}}})\) denotes a family of h-dependent functions satisfying some regularity conditions, and P(h) is either an appropriate self-adjoint semiclassical pseudodifferential operator in \(\mathrm{L}^2({{\mathbb {R}}}^n)\) or a Schrodinger operator in \(\mathrm{L}^2(M), M\) being a closed Riemannian manifold of dimension n. The main result is an explicit semiclassical trace formula with remainder estimate that is well-suited for studying the spectrum of P(h) in spectral windows of width of order \(h^\delta \), where \(0\le \delta <\frac{1}{2}\).

REVISITING THE PLANE ELECTROMAGNETIC WAVE TRANSMISSION AND REFLECTION COEFFICIENTS FOR THE LAYER WITH AN ALTERNATING-SIGN DISTURBANCE OF RELATIVE DIELECTRIC PERMITTIVITY

In this paper, we consider the question of the plane electromagnetic wave incidence at the inhomogeneity with an arbitrary profile of the relative permittivity disturbance. Module estimation of Neumann series remainder is carried out for the field of the wave passing through the nonhomogeneous section. Based on that, the number of summands in the series, required to calculate with a given accuracy, the transmission and reflection coefficients have been determined.

Eigenvalue Problem in a Solid with Many Inclusions: Asymptotic Analysis

We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of the small inclusions is characterized by a small parameter which is much larger when compared with the nominal size of inclusions. Remainder estimates for the approximations to the first eigenvalue and associated eigenfield are presented. Numerical illustrations are given to demonstrate the efficiency of the asymptotic approach compared to conventional numerical techniques, such as the finite element method, for three-dimensional solids containing clusters of small inclusions.