Fu-Bo Li, Miao Li
Let $1< α<2$ and $A$ be the generator of an $α$-times resolvent family $\{S_α(t)\}_{t \ge 0}$ on a Banach space $X$. It is shown that the fractional Cauchy problem ${\bf D}_t^αu(t) = Au(t)+f(t)$, $t \in [0,r]$; $u(0), u'(0) \in D(A)$ has maximal regularity on $C([0,r];X)$ if and only if $S_α(\cdot)$ is of bounded semivariation on $[0,r]$.
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Continuous maximal regularity in locally convex spaces
本文研究局部凸空间上强连续半群的连续函数极大正则性，将 Travis 的 C-极大正则性刻画推广至一般情形，并在拓扑假设下证明其与容许算子等价。
Measure Theory in Convergence Spaces Part I: Sion-Carathéodory Extension in Net Convergence Groups
本文针对取值于一般分离阿贝尔网收敛群的预测度建立Carathéodory型扩张定理，构造正则布尔σ代数上的可数可加扩张，恢复相关经典扩张并处理有限半变差，给出测度理论的收敛理论解释。
Suppose that $λ- T$ is left-invertible in $L(H)$ for all $λ\in Ω$, where $Ω$ is an open subset of the complex plane. Then an operator-valued function $L(λ)$ is a left resolvent of $T$ in $Ω$ if and only if $T$ has an extension $\tilde{T}$, the resolvent of which is a dilation of $L(λ)$ of a particular form. Generalized resolvents exist on every open set $U$, with $\bar{U}$ included in the regular domain of $T$. This implies a formula for the maximal radius of regularity of $T$ in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by J. Zemánek is obt…
On Matrix Valued Schrödinger Operators on the Discrete Real Line: Resolvent Boundary Values, Limiting Absorption Principle, Hölder Regularity and Dispersive Estimates
本文研究了离散实线上矩阵值Schrödinger算子的谱理论和时间演化，通过构造Jost解并利用Volterra方程和转置算子，推导了解析算子的显式核表示，证明了边界值的Hölder连续性，并建立了色散估计。
We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on $L^p_σ (\mathbb{R}^d)$, $d \geq 2$, for $\lvert 1 / p - 1 / 2 \rvert < 1 / d$. These include optimal resolvent bounds and the property of maximal $L^q$-regularity. We further give regularity estimates on the gradient of the solution to the Stokes resolvent problem with bounded measurable coefficients. As a key to these results we establish the validity of a non-local Caccioppoli inequality to solutions of the Stokes resolvent problem.
Operators possessing analytic generalized inverses satisfying the resolvent identity are studied. Several characterizations and necessary conditions are obtained. The maximal radius of regularity for a Fredholm operator T is computed in terms of the spectral radius of a generalized inverse of T. This provides a partial answer to a conjecture of J. Zemánek.
Optimal regularity results for the Stokes--Dirichlet problem
本文研究了Stokes-Dirichlet问题在低正则性领域中的最优正则性结果，涵盖了Besov和Sobolev空间的完整图景，并获得了在半空间和有界域上的解析估计和显式描述。
Rational-Kernel Fractional Evolution Equations with Almost Sectorial Operators: A Resolvent Framework Unifying ABC and W Dynamics
本文提出了一种统一ABC和W动力学的解析框架，通过有理核时间算子研究分数演化方程，揭示了非奇异记忆对衰减和光滑性质的影响。
The Katznelson-Tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups. It states that for a power-bounded operator $T$ on a Banach space we have $||T^n(I-T)\|\to0$ if and only if $σ(T)\cap\mathbb{T}\subseteq\{1\}$. The main result of the present paper gives a sharp estimate for the rate at which this decay occurs for operators on Hilbert space, assuming the growth of the resolvent norms $\|R(e^{iθ},T)\|$ as $|θ|\to0$ satisfies a mild regularity condition. This significantly extends an earlier result by the second author, which covered the important case o…
A Discrete Resolvent Framework for Delay Differential Equations: Local Defects, Global Propagation, and Splitting Approximations
针对隐式欧拉和Lie-Trotter分裂逼近，建立基于离散传播子的预解框架，通过分数阶插值空间上的局部缺陷估计和更高正则尺度上的全局收敛恢复，获得有限时间收敛阶O(h^θ)。
The aim of this article is to deepen the understanding of the derivation of $\mathrm{L}^p$-estimates of non-local operators. We review the $\mathrm{L}^p$-extrapolation theorem of Shen which builds on a real variable argument of Caffarelli and Peral and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear for example in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire. As an applic…
It is shown that an elliptic scattering operator $A$ on a compact manifold with boundary with coefficients in the bounded operators of a bundle of Banach spaces of class (HT) and Pisier's property $(α)$ has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of $A$ on the scattering cotangent bundle of the manifold avoids the right half-plane. This is deduced directly from a Seeley theorem, i.e. the resolvent is represented in terms of pseudodifferential operators with R-bounded symbols, thus showing by an iteration argument the R-boundedness of $λ(A…
Maximal regularity for time-fractional Schrödinger equations and application to nonlinear equations
本文研究了抽象时间分数阶薛定谔方程的最大L^2正则性，利用 Mittag-Leffler 函数的虚数参数性质，避免了传统方法中的完全单调性证明，并推广到L^p正则性，用于证明拟线性和半线性时间分数阶薛定谔方程的局部解的存在性。
We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e. energy forms on finite weighted gra…
For $2a$-order strongly elliptic operators $P$ generalizing $(-Δ)^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $Ω\subset R^n$ by pseudodifferential methods, has been extended in a recent joint work with Helmut Abels to nonsmooth settings, showing regularity theorems in $L_q$-Sobolev spaces $H_q^s$ for $1<q<\infty $, when $Ω$ is $C^{τ+1}$ with a finite $τ>2a$. Presently, we study the $L_q$-Dirichlet realizations of $P$ and $P^*$, showing invertibility or Fredholmness, finding smoothness results for the kernels and cokernels, and establishing similar resu…
We give a rate of metastability for Halpern's iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth Banach spaces, extracted from a proof due to Xu. In Hilbert space, the latter is known, so we get an explicit rate of metastability. We also extract a rate of asymptotic regularity for general normed spaces. Such rates have already been extracted by Kohlenbach and Leuştean for different and incomparable conditions on $(λ_n)$. The proof analyzed in this paper is also more effective than the proofs treated by Kohlenbach and Leuştean in that i…
Maximal regularity for fractional difference equations with finite delay on UMD space
针对 UMD 空间中含有限时滞的分数阶差分方程，论文借助 α-预解序列给出解的显式表示，并运用 Blunck 算子值 Fourier 乘子定理完整刻画其 ℓ^p-极大正则性。
Maximal $L_p$-regularity for $x$-dependent fractional heat equations with Dirichlet conditions
本文研究了有限光滑有界区域上含非局部椭圆算子的线性抛物方程，利用极大 $L_p$-正则性与预解式 $\mathcal{R}$-有界性的联系，证明了最优正则性，并据此建立了一类非线性非局部抛物方程的时间局部强解存在性。
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