Banach Journal of Mathematical Analysis最新文献

Let X be a separable rearrangement invariant space on ((0,infty )). If the intersection ((X cap L_{infty })(0,infty )) contains a complemented subspace isomorphic to ({ell }_2), then X contains a complemented sublattice lattice-isomorphic to ({ell }_2). Moreover, we prove that the space ((X+L_{infty })(0,infty )) cannot be isomorphically embedded into ((X cap L_{infty })(0,infty )) as a complemented subspace provided that X has nontrivial Rademacher cotype.

A reflexive Banach space with an unconditional basis admits an equivalent 1-unconditional 2R norm and embeds into a reflexive space with a 1-symmetric 2R norm. Partial results on 1-symmetric 2R renormings of spaces with a symmetric basis are obtained.

An abstract method is presented to show that upper semicontinuity of invariant measures of dissipative dynamical systems on thin domains. The abstract method presented can be used to many physical systems. As an example, we consider reaction-diffusion equations on thin domains. To this end, we first show the existence of invariant measures of the equations in a bounded domain in (mathbb {R}^{n+1}) which can be viewed as a perturbation of a bounded domain in (mathbb {R}^n). We then prove that any limit of invariant measures of the perturbed systems must be an invariant measure of the limiting system when the thin domains collapses.

Let X be a Borel metric measure space such that every closed ball is of positive and finite measure. In this paper, we give a sufficient condition and a necessary condition for averaging operators on a Banach function space E(X) on X to be compact. As a corollary, we show that the averaging operators on the Lorentz space (L^{p,q}(X)) of X are compact if and only if X is bounded, in the case where X is a doubling and Borel-regular metric measure space with some continuity between metric and measure.

Let ({mathfrak {A}}) be a maximal subdiagonal algebra with diagonal ({mathfrak {D}}) in a (sigma )-finite von Neumann algebra ({mathcal {M}}) with respect to a faithful normal conditional expectation (Phi ). We firstly give a type decomposition of an invariant subspace ({mathfrak {M}}) of ({mathfrak {A}}) in the acting Hilbert space. We then revisit certain useful properties of type 1 subdiagonal algebras. It is shown that a two-sided invariant subspace ({mathfrak {M}}) in the noncommutative (H^2) space has the form ({mathfrak {M}}=oplus _{nge 1}^{col}W_nH^2) for a family of partial isometries ({W_n:nge 1}) satisfying ( W_n^*W_m=0) when (nnot =m), (W_n^*W_nin {mathfrak {D}}) and (sum _{nge 1} W_nW_n^*=I) if ({mathfrak {D}}) is a factor. Furthermore, we give a noncommutative version of the Sarason’s generalized interpolation theorem for such a two-sided invariant subspace of a type 1 subdiagonal algebra.

In this paper we introduce a new class of sampling-type operators, named Steklov sampling operators. The idea is to consider a sampling series based on a kernel function that is a discrete approximate identity, and which constitutes a reconstruction process of a given signal f, based on a family of sample values which are Steklov integrals of order r evaluated at the nodes k/w, (k in {mathbb {Z}}), (w>0). The convergence properties of the introduced sampling operators in continuous functions spaces and in the (L^p)-setting have been studied. Moreover, the main properties of the Steklov-type functions have been exploited in order to establish results concerning the high order of approximation. Such results have been obtained in a quantitative version thanks to the use of the well-known modulus of smoothness of the approximated functions, and assuming suitable Strang-Fix type conditions, which are very typical assumptions in applications involving Fourier and Harmonic analysis. Concerning the quantitative estimates, we proposed two different approaches; the first one holds in the case of Steklov sampling operators defined with kernels with compact support, its proof is substantially based on the application of the generalized Minkowski inequality, and it is valid with respect to the p-norm, with (1 le p le +infty ). In the second case, the restriction on the support of the kernel is removed and the corresponding estimates are valid only for (1 < ple +infty ). Here, the key point of the proof is the application of the well-known Hardy–Littlewood maximal inequality. Finally, a deep comparison between the proposed Steklov sampling series and the already existing sampling-type operators has been given, in order to show the effectiveness of the proposed constructive method of approximation. Examples of kernel functions satisfying the required assumptions have been provided.

This work is concerned with the stochastic slow-fast evolutionary systems with white noises in Hilbert spaces. We first establish the stochastic Poincaré maps of the stochastic slow-fast systems in the neighborhood of the periodic orbit for the approximate systems with colored noises in distribution. Further employing the random slow manifold theory, we prove that the stochastic Poincaré maps of the stochastic slow-fast systems converge to the same fixed point of the stochastic Poincaré maps for the approximate systems in distribution as the colored noise parameter tends to zero. Moreover, we apply the moving orthonormal system to construct the exact portray of the stochastic Poincaré maps for the stochastic slow-fast systems in distribution. A concrete example is provided to illustrate the stochastic Poincaré maps.

In this paper we first characterize the Schatten p-class and Schatten h-class Hankel and Toeplitz operators on Bergman spaces (A_{omega _{1,2}}^2({mathbb {M}})) induced by regular weights (omega _{1,2}) of the annulus ({mathbb {M}}) with full range (pin (0,infty )) and h being a continuous increasing convex function on ((0,infty )). As an application, we then establish essential norm estimates for bounded Hankel operators from Bergman spaces (A_{omega _{1,2}}^p({mathbb {M}})) to Lebesgue spaces (L_{omega _{1,2}}^q({mathbb {M}})) for all possible (p,qin (1, infty )). Moreover, Schatten p-class properties and essential norm estimates for Hankel operators on Bergman spaces over the unit disk ({mathbb {D}}) induced by regular weights are also obtained, which can be viewed as a further application of boundedness and compactness of Hankel operators proved by Hu and Jin (J Geom Anal 29:3494–3519, 2019). To establish these desired characterizations, the diagonal and off-diagonal decompositions, various careful estimates for reproducing kernels, Berezin transforms, Carleson measures and the solution of ({bar{partial }})-equations are crucial tools in our proofs.

The main aim of this paper is to prove that the wave front set of (a^w(x,D)u), i.e. the action of the Weyl operator with symbol a on u, is contained in the wave front set of u and in the conic support of a in spaces of (omega )-tempered ultradistributions in the Beurling setting for adequate symbols of ultradifferentiable type. These symbols are not restricted to have order zero. To do so, we prove an almost diagonalization theorem on Weyl operators. Furthermore, an almost diagonalization theorem involving time-frequency analysis leads to additional applications, such as invertibility of pseudodifferential operators or boundedness of them in modulation spaces with exponential growth.

Some new classes of Kadison-Singer lattices (KS-lattices) and Kadison-Singer algebras (KS-algebras) are constructed. These KS-lattices are determined by a given KS-lattice, some discrete nest of projections and one special projection. Some quantities for these lattices are used to classify these KS-algebras. It is shown that these KS-algebras are isometrically isomorphic if and only if they are unitarily equivalent if and only if they have the same quantities.