Banach Journal of Mathematical Analysis最新文献

The concept of disjointness preserving mappings has proved to be a useful unifying idea in the study of Banach–Stone type theorems. In this paper, we examine disjointness preserving relations between sets of continuous functions (valued in general topological spaces). Under very mild assumptions, it is shown that a disjointness preserving relation is completely determined by a Boolean isomorphism between the Boolean algebras of regular open sets in the domain spaces. Building on this result, certain Banach–Stone type theorems are obtained for disjointness preserving relations. From these, we deduce a generalization of Kaplansky’s classical theorem concerning order isomorphisms to sets of continuous functions with values topological lattices. As another application, we prove some results on the characterization of nonvanishing preservers. Throughout, the domains of the function spaces need not be compact.

Abstract

This paper investigates the embedding relationships between Wiener amalgam spaces and classical spaces, including Sobolev spaces, local Hardy spaces, Besov spaces, and (alpha ) -modulation spaces. By establishing exact conditions, we provide a detailed characterization of the embeddings between Wiener amalgam spaces and these classical spaces, particularly the most general case when (alpha =0) , which extend the main results obtained by Guo–Wu–Yang–Zhao (J Funct Anal 273(1):404–443, 2017). Furthermore, we discuss the embedding relationship between Wiener amalgam spaces and Triebel–Lizorkin spaces (F_{p,r}^{s}) when (0<pleqslant 1) .

Abstract

In this paper, we focus on a class of fractional type integral operators that can be served as extensions of Riesz potential with kernels $$begin{aligned} K(x,y)=frac{Omega _1(x-A_1 y)}{|x-A_1 y |^{{n}/{q_1}}} cdots frac{Omega _m(x-A_m y)}{|x-A_m y |^{{n}/{q_m}}}, end{aligned}$$ where (alpha in [0,n)) , ( mgeqslant 1) , (sum limits _{i=1}^mfrac{n}{q_i}=n-alpha ) , ({A_i}^m_{i=1}) are invertible matrixes, (Omega _i) is homogeneous of degree 0 on (mathbb R^n) and (Omega _iin L^{p_i}(S^{n-1})) for some (p_iin [1,infty )) . Under appropriate assumptions, we obtain the weighted (L^p(mathbb R^n)-L^q(mathbb R^n)) estimates as well as weighted (H^p(mathbb R^n)-L^q(mathbb R^n)) estimates of the commutators for such operators with BMO-type function when (frac{1}{q}=frac{1}{p}-frac{alpha }{n}) . In addition, we acquire the boundedness of these operators and their commutators with a function in Campanato spaces on Orcliz–Morrey spaces as well as the compactness for such commutators in a special case: (m=1) and (A=I) .

In this article, the authors establish a non-smooth atomic decomposition of Triebel–Lizorkin-type spaces and, as a by-product, a non-smooth atomic decomposition of subspaces of BMO spaces is obtained. An application of this decomposition method to the boundedness of Marcinkiewicz integral operators is also presented.

In 2018, Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps, with a focus on the ({{textbf {n}}})-t-quasi-greedy property that is based on them. Building upon this foundation, our current work aims to further investigate these algorithms and bases while introducing new ideas for two primary purposes. First, we aim to prove that for ({{textbf {n}}}) with bounded quotient gaps, ({{textbf {n}}})-t-quasi-greedy bases are quasi-greedy bases. This generalization extends a previous result to the context of Markushevich bases and, also, completes the answer to a question by Oikhberg. The second objective is to extend certain approximation properties of the greedy algorithm to the context of sequences with gaps and study if there is a relationship between this new extension and the usual convergence.

We study the long-term behavior of solutions for stochastic delay p-Laplacian equation with multiplicative noise on unbounded thin domains. We first prove the existence and uniqueness of tempered random attractors for these equations defined on ((n+1))-dimensional unbounded thin domains. Then, the upper semicontinuity of these attractors when a family of ((n+1))-dimensional thin domains degenerates onto an n-dimensional domain as the thinness measure approaches zero is established.

The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be overdetermined or underdetermined. They may contain derivatives, of the unknown vector-valued function, whose integer or fractional orders exceed the order of the differential equation. Similar problems arise naturally in various applications. The theory introduces the notion of a rectangular number characteristic matrix of the problem. The index and Fredholm numbers of this matrix coincide, respectively, with the index and Fredholm numbers of the inhomogeneous boundary-value problem. Unlike the index, the Fredholm numbers (i.e., the dimensions of the problem kernel and co-kernel) are unstable even with respect to small (in the norm) finite-dimensional perturbations. We give examples in which the characteristic matrix can be explicitly found. We also prove a limit theorem for a sequence of characteristic matrices. Specifically, it follows from this theorem that the Fredholm numbers of the problems under investigation are semicontinuous in the strong operator topology. Such a property ceases to be valid in the general case.

Let T be a bounded linear operator on a complex Hilbert space (mathcal {H}). We present some necessary and sufficient conditions for T to be the generalized pencil (P + alpha Q +beta PQ) of a pair (P, Q) of projections at some point ((alpha , beta )in mathbb {C}^2). The range and kernel relations of the generalized pencil T are studied and comments on the additional properties of some special generalized pencil are given.

This paper presents refined estimates for functional dual affine quermassintegrals, building upon the estimates of Dann et al. To sharpen the inequality, Dann et al. (Proc. Lond. Math. Soc. (3) 113(2):140–162, 2016) incorporated an (L^infty)-weight into the integration. We further refine these estimates and extend the (L^infty)-weight estimates to include a wider range of (L^{lambda })-weights where (lambda >1.)

The notion of p-summing Bloch mapping from the complex unit open disc (mathbb {D}) into a complex Banach space X is introduced for any (1le ple infty .) It is shown that the linear space of such mappings, equipped with a natural seminorm (pi ^{mathcal {B}}_p,) is Möbius-invariant. Moreover, its subspace consisting of all those mappings which preserve the zero is an injective Banach ideal of normalized Bloch mappings. Bloch versions of the Pietsch’s domination/factorization Theorem and the Maurey’s extrapolation Theorem are presented. We also introduce the spaces of X-valued Bloch molecules on (mathbb {D}) and identify the spaces of normalized p-summing Bloch mappings from (mathbb {D}) into (X^*) under the norm (pi ^{mathcal {B}}_p) with the duals of such spaces of molecules under the Bloch version of the (p^*)-Chevet–Saphar tensor norms (d_{p^*}.)