Banach Journal of Mathematical Analysis最新文献

In this article, some new martingale inequalities in the framework of Orlicz–Karamata modular spaces are discussed. More precisely, we establish modular inequalities associated with Orlicz functions and slowly varying functions. The results obtained herein can weaken the restrictive condition that the slowly varying function b is nondecreasing in (Math Nachr 291(8–9):1450–1462, 2018).

We characterize the sequences of complex numbers ((z_{n})_{n in mathbb {N}}) and the locally complete (DF)-spaces E such that for each ((e_{n})_{n in mathbb {N}} in E^mathbb {N}) there exists an E-valued function (textbf{f}) on ((0,infty )) (satisfying a mild regularity condition) such that

where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution (textbf{f}) that is smooth on ((0,infty )) and satisfies certain optimal growth bounds near 0 and (infty ). The scalar-valued case ((E = mathbb {C})) was treated by Durán (Math Nachr 158:175–194, 1992). Our work is based upon his result.

We prove that for every countable ordinal (xi ), the Tsirelson’s space (T_xi ) of order (xi ), is naturally, i.e., via the identity, 3-isomorphic to its modified version. For the first step, we prove that the Schreier family (mathcal {S}_xi ) is the same as its modified version ( mathcal {S}^M_xi ), thus answering a question by Argyros and Tolias. As an application, we show that the algebra of linear bounded operators on (T_xi ) has (2^{{mathfrak {c}}}) closed ideals.

In this article, we discuss the applications of martingale Hardy Orlicz–Lorentz–Karamata spaces in Fourier analysis. More precisely, we show that the partial sums of the Walsh–Fourier series converge to the function in norm if (fin L_{Phi ,q,b}) with (1<p_-le p_+<infty ). The equivalence of maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces is presented. The Fejér summability method is also studied and it is proved that the maximal Fejér operator is bounded from martingale Hardy Orlicz–Lorentz–Karamata spaces to Orlicz–Lorentz–Karamata spaces. As a consequence, we obtain conclusions about almost everywhere and norm convergence of Fejér means.

In this article, we discuss the relationship between Birkhoff–James orthogonality of elementary tensors in the space (L^{p}(mu )otimes ^{Delta _{p}}X,; (1le p<infty )) with the individual elements in their respective spaces, where X is a Banach space whose norm is Fr(acute{e}chet) differentiable and (Delta _{p}) is the natural norm induced by (L^{p}(mu ,X)). In order to study the said relationship, we first provide some characterizations of Birkhoff–James orthogonality of elements in the Lebesgue-Bochner space (L^{p}(mu ,X)).

Let T be a multilinear Calderón–Zygmund operator of type (omega ). (T_{vec {b},S}) is the generalized commutator of T, which generalizes several commutators that already existed. It is shown in this paper that the weak and strong type quantitative weighted estimates for (T_{vec {b},S}) when (vec {b}={b_i}_{i=1}^{infty }) belongs to exponential oscillation spaces and Lipschitz spaces, respectively. As applications, we obtain the multiple weighted norm inequalities for the generalized commutators of bilinear pseudo-differential operators and paraproducts with mild regularity.

In this paper, we consider a class of generalized closed linear manifolds in a nonseparable Hilbert space H, which is closely related to the generalized Fredholm theory. We first investigate properties of the set ({mathcal {B}}_{vartriangleleft }={Tin {mathcal {M}}:overline{T(H)}subset A(H)) for some (Ain {mathcal {B}}},) where ({mathcal {B}}) is a (C^*)-subalgebra of a von Neumann algebra ({mathcal {M}}). It is proved that a selfadjoint ({mathcal {B}}_{vartriangleleft }) is always an ideal in ({mathcal {M}}). In a type (textrm{II}_infty ) factor, we show that there exists a tracial weight (whose range containing infinite cardinals) such that two projections are equivalent if and only if they have the same tracial weight, which leads to a complete characterization of such selfadjoint ({mathcal {B}}_{vartriangleleft }) when ({mathcal {M}}) is a factor. Then we introduce the concept of closed manifolds with respect to a pair of C*-algebras and study some properties. Finally, when m is an infinite cardinal, as a special important case we focus on m-closed subspaces and operators which preserve m-closed subspaces. It is proved that these operators are either of rank less than m, or the generalized left semi-Fredholm operators.

Let (mathcal {B(H)}) be the collection of bounded linear operators on a complex separable Hilbert space (mathcal {H}). For (Tin mathcal {B(H)}), its numerical range and maximal numerical range are denoted by W(T) and (W_0(T)), respectively. First, we give in this paper a characterization of the maximal numerical range and, as applications, we determine maximal numerical ranges of weighted shifts, partial isometries, the Volterra integral operator and classical Toeplitz operators. Second, we study the universality of maximal numerical ranges, showing that any nonempty bounded convex closed subset of (mathbb {C}) is the maximal numerical range of some operator. Finally, we discuss the relations among the numerical range, the maximal numerical range and the spectrum. It is shown that the collection of those operators T with (W_0(T)cap sigma (T)=emptyset ) is a nonempty open subset of (mathcal {B(H)}) precisely when (dim mathcal {H}>1), and is dense precisely when (1<dim mathcal {H}<infty ). We also show that those operators T with (W_0(T)= W(T)) constitute a nowhere dense subset of (mathcal {B(H)}) precisely when (dim mathcal {H}>1)

First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of d-(sigma )-stability in a random metric space can be regarded as a special case of the notion of (sigma )-stability in a random normed module; as another application we give the final version of the characterization for a d-(sigma )-stable random metric space to be stably compact. Second, we prove that an (L^{p})-normed (L^{infty })-module is exactly generated by a complete random normed module so that the gluing property of an (L^{p})-normed (L^{infty })-module can be derived from the (sigma )-stability of the generating random normed module, as applications the direct relation between module duals and random conjugate spaces are given. Third, we prove that a random normed space is order complete iff it is ((varepsilon ,lambda ))-complete, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-(sigma )-stability. Finally, we prove that an equivalence relation on the product space of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric on X, as an application it is proved that a nonempty subset of a Boolean set (X, d) is universally complete iff it is a B-stable set defined by a regular equivalence relation.

Let G be a locally compact group and let (A_Phi (G)) be the Orlicz version of the Figà–Talamanca Herz algebra of G associated with a Young function (Phi .) We show that if (A_Phi (G)) is Arens regular, then G is discrete. We further explore the Arens regularity of (A_Phi (G)) when the underlying group G is discrete. In the running, we also show that (A_Phi (G)) is finite dimensional if and only if G is finite. Further, for amenable groups, we show that (A_Phi (G)) is reflexive if and only if G is finite, under the assumption that the associated Young function (Phi ) satisfies the MA condition.