Banach Journal of Mathematical Analysis最新文献

The main aim of the paper is to investigate some sufficient conditions which guarantee the convergence of sequences of positive linear operators towards composition operators within the framework of function spaces defined on a metric space. Among other things, the adopted approach allows to obtain a unifying reassessment of two milestones of the approximation theory by positive linear operators, namely, Korovkin’s theorem and Feller’s theorem together with some new extensions of them to the more general case where the limit operator is a composition operator. Some applications are shown and, among them, the convergence of Bernstein–Schnabl operator is enlightened in the framework of Banach spaces.

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.