For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=g(z)f(z). An N-letter g-word is an operator of the form L=L1⋯LN, where each Lj is either Mg, Sg or Tg. It has been recently proved in [4], [5] that understanding the boundedness of a g-word on classical Hardy and Bergman spaces is a challenging problem due to the potential cancellations involved. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary g-word on a weighted Bergman space Aωp/2p, where ω=e−2φ is a smooth rapidly decreasing weight. In particular, it states that any N-letter g-word such that #{j:Lj=Tg}=n≥1 is bounded on Aωp/2p if and only if g satisfies the “fractional” Bloch-type condition‖g‖Bφss:=supz∈Ds|g(z)|s−1|g′(z)|1+φ′(|z|)<∞, where s=Nn, and ‖L‖Aωp/2p≃‖g‖BφsN.The class of smooth rapidly decreasing weights contains the radial weightsωn(z)=e−2expn(gα,c(|z|)),wheregα,c(r)=c(1−r2)α,for c,α>0, exp0(x)=x and expn(x)=eexpn−1(x), for n∈N. Therefore it contains weights which decrease arbitrarily rapidly to zero as |z|→1−.
For general hyperbolic system of two conservation laws, the unique method to obtain the a priori L∞ estimates is the invariant region theory introduced by Chueh, Conley and Smoller in 1977. However, this method is only valid when one could obtain the two Riemann invariants of a given system with explicit expressions. In this paper, the nonstrictly hyperbolic systems of conservation laws modelling polymer flooding in enhanced oil recovery are studied, where the first Riemann invariant T has an explicit expression, but the second explicit Riemann invariant is not available. With the help of the bounded total variation of T(x,t), the a priori L∞ estimates of S(x,t) are obtained. The innovation of the technology lies in the discovery of a special structural equation for the variable Tx(⋅,t). As a by-product, under some technical conditions on the nonlinear functions α(T) and β(T), the global existence of weak entropy solutions of the Cauchy problem is obtained by using the compensated compactness method. In particularly, when α(T)=0 and β(T)=0, the global existence results of weak solutions given in ‘J. Funct. Anal., 261 (2011), 2797-2815’ for the prototype function P(S)=1γSγ+A,γ>−1 are extended to the most difficult case γ<−1.
In this paper, we study the fundamental solution of the higher order Schr&odinger equation i partial derivative(t)u(x, t) = ((-Delta)(m) + V(x))u(x, t), t is an element of & Ropf;, x is an element of & Ropf;(n), for any odd dimension nand integer m > 1 satisfying n < 4m, where Vis a real-valued bounded potential with suitable decay. Let P-ac(H) denote the projection onto the absolutely continuous spectral subspace of H = (-Delta)(m) + V, and assume H has no positive embedded eigenvalues. Our main result says that the evolution operator e-itHPac(H) has an integral kernel K(t, x, y) satisfying the pointwise estimate |K(t, x, y)| <= C(1 + |t|)(-h)(1 + |t|(- n /2m) ) x (1 + |t|(- 1/ 2m) |x - y|)(- n(m-1) /2m-1), t not equal 0, x, y is an element of & Ropf;(n), where the exponent h depends on m, n, and the zero energy resonance structure of H. We also prove analogous estimates for smoothing operators of the form H alpha /2m e(-itH) P-ac(H). The key innovation of this paper is a unified approach to deriving asymptotic expansions of the perturbed resolvents around zero, which comprehensively addresses all possible resonance types. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This paper investigates Neumann initial-boundary value problems for nonlinear fractional partial differential equations in multidimensional, unbounded domains, with a focus on the fractional Ginzburg-Landau equation posed in hyperoctants. These problems are highly relevant in modeling nonlocal transport, anomalous diffusion, and interface dynamics in computational physics and engineering. The nonlocal nature of the fractional Laplacian, combined with the singular geometry of the hyperoctant and the intrinsic non-uniqueness of Neumann boundary conditions, presents deep analytical and numerical challenges. We establish the well-posedness of the problem and derive the necessary compatibility conditions between initial and boundary data to ensure solvability. Classical techniques-such as reflection principles and Laplace transforms-fail in this context due to the global influence of fractional operators and the complex boundary structure. To overcome these difficulties, we develop a novel analytical framework based on the Riemann-Hilbert method, which systematically captures nonlocal dynamics and boundary singularities in high-dimensional settings. The resulting theory provides sharp asymptotic estimates and explicit error bounds, offering a rigorous foundation for numerical simulation. These results have direct applications to the design, validation, and refinement of computational schemes for fractional PDEs in truncated domains, with implications for modeling time-dependent processes in superconductivity, nonlinear media, and diffusion-driven systems.
