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.
A Banach space X is said to have the Daugavet property if every rank-one operator T : X -> X satisfies parallel to I + T parallel to = 1 + parallel to T parallel to . We give a characterization of this property in the setting of L-embedded Banach spaces, solving a longstanding open problem in the field. For an L-embedded Banach space X (X & lowast;& lowast; = X (c) 1 N), the following assertions, are equivalent: (1) X & lowast; has the Daugavet property. (2) X has the Daugavet property. (3) The closed unit ball BN of Nis weak-star dense in BX & lowast;& lowast;. We also show that, if Xis a non-reflexive L-embedded Banach space, and Y is a Banach space with the metric approximation property, then (1) X((2n)(circle times) over cap Y-pi has octahedral norm for every even dual X((2)n of X. (2) X (circle times) over cap Y-pi has the Daugavet property whenever X has it. (c) 2026 The Authors. Published by Elsevier Inc.
Let Omega subset of C-n be a bounded symmetric domain and f Omega -> Omega ' subset of C-n be a proper holomorphic mapping which is factored by a finite complex reflection group G. We identify a family of reproducing kernel Hilbert spaces on Omega ' arising naturally from the isotypic decomposition of the regular representation of G on the Hardy space H-2(Omega). Each element of this family can be realized as a closed subspace of some L-2-space on the Silov boundary of Omega '. The reproducing kernel Hilbert space associated to the sign representation of G is the Hardy space H-2(Omega '). We establish a Brown-Halmos type characterization for the Toeplitz operators on H-2(Omega '), where Omega ' is the image of the open unit polydisc D-n in C-n under a proper holomorphic mapping factored by the finite complex reflection group G(m, p, n). Moreover, we prove various multiplicative properties of Toeplitz operators on H2(Q '), where Q ' is a proper holomorphic image of a bounded symmetric domain. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This paper is devoted to the study of quantitative weighted norm estimates for martingale square functions in both scalar-weighted and matrix-weighted settings. In particular, we introduce the martingale square functions SW via matrix weights W, and then use the matrix Ap condition, introduced in our previous work [9], to characterize the Lp estimate for SW. Our proof mainly relies on the idea of sparse domination, which gives explicit information about the characteristic of the matrix weight involved. For the range 1<p≤2, our result is sharp in terms of the characteristic of the matrix weight. With suitable modifications of the arguments, we can further improve the result in the scalar setting by obtaining the optimal exponent of the characteristic of the weight involved for all indices 1<p<∞, addressing a fundamental problem in classical martingale theory.
