We introduce new function spaces to handle the Fourier transform on Morrey spaces and investigate fundamental properties of the spaces. As an application, we generalize the Stein–Tomas Strichartz estimate to our spaces. The geometric property of Morrey spaces and related function spaces will improve some well-known estimates.

We completely characterize those continuous operators on the space of real analytic functions on the real line for which the associated matrix is Toeplitz (that is, we describe Toeplitz operators on this space). We also prove a necessary and sufficient condition for such operators to be Fredholm operators. While the space of real analytic functions is neither Banach space nor has a basis which makes available methods completely different from classical cases of Hardy spaces or Bergman spaces, nevertheless the results themselves show surprisingly strong similarity to the classical Hardy-space theory.

We give a characterization of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$-embedded space with another nonzero Banach space. The former characterization also motivates the introduction and short study of two indices related to the Daugavet property.

In this article, for the unimodular multipliers $e^{i\mu \left(D\right)}$, we establish two Hörmander-type multiplier theorems by assuming conditions on their phase functions $\mu$. As applications, we obtain two multiplier theorems particularly fitting for the modulation spaces, thus allowing us to extend and improve some known results.

In this article, we show that there exist a function $g\in L^1[0,1]$ and a weight function $0<\mu \left(x\right)\le 1$ so that $g$ is universal for each class $L_{\mu }^p[0,1]$, $p\ge 1$, with respect to signs-subseries of its Fourier–Walsh series.

Let $X$ be a finite CW-complex, and let $\mathcal{A}$ be a unital separable simple finite $\mathcal{Z}$-stable C∗-algebra with real rank $0$. We prove an approximate uniqueness theorem for almost multiplicative contractive completely positive linear maps from $C\left(X\right)$ into $\mathcal{A}$. We also give conditions for when such a map can, within a certain “error,” be approximated by a finite-dimensional ∗-homomorphism.

The first aim of this article is to survey and revisit some uncertainty principles for the Hankel transform by means of the Hankel multiplier. Then we define the wavelet Hankel multiplier and study its boundedness and Schatten-class properties. Finally, we prove that the wavelet Hankel multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator, for which we deduce a trace formula.

We present solutions to local connectivity problems in matrix representations of the form $C\left(\right[-1,1{]}^N)\to C^{\ast }(u_{\epsilon },v_{\epsilon })$, with $C_{\epsilon }\left({\mathbb{T}}^2\right)\twoheadrightarrow C^{\ast }(u_{\epsilon },v_{\epsilon })$ for any $\epsilon \in [0,2]$ and any integer $n\ge 1$, where $C^{\ast }(u_{\epsilon },v_{\epsilon })\subseteq M_n$ is an arbitrary matrix representation of the universal $C^{\ast }$-algebra $C_{\epsilon }\left({\mathbb{T}}^2\right)$ that denotes the soft torus. We solve the connectivity problems by introducing the so-called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in differential algebraic topology.

To deal with the locality constraints, we have combined some techniques introduced in this article with some techniques from matrix geometry, combinatorial optimization, and classification and representation theory of $C^{\ast }$-algebras.

We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and $f$-algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean $f$-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean $\Phi$-algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean $\Phi$-algebras is obtained.

In this paper, we investigate the g-frame and Bessel g-sequence related to a linear bounded operator $K$ in Hilbert space, which we call a $K$-g-frame and a $K$-dual Bessel g-sequence, respectively. Since the frame operator for a $K$-g-frame may not be invertible, there is no classical canonical dual for a $K$-g-frame. So we characterize the concept of a canonical $K$-dual Bessel g-sequence of a $K$-g-frame that generalizes the classical dual of a g-frame. Moreover, we use a family of linear operators to characterize atomic systems. We also consider the construction of new atomic systems from given ones and bounded operators.

We prove that, if Banach spaces $X$ and $Y$ are $\delta$-average rough, then their direct sum with respect to an absolute norm $N$ is $\delta /N(1,1)$-average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty )$, the space $X{\oplus }_{p}Y$ is $2^{1-1/p}$-average rough, which is in general optimal. Another consequence is that for any $\delta$ in $(1,2]$ there is a Banach space which is exactly $\delta$-average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diametral strong diameter 2 property is stable only for 1- and $\infty$-sums.

Given a pointed metric space $X$ and a weight $v$ on $\tilde{X}$ (the complement of the diagonal set in $X\times X$), let ${\mathrm{Lip}}_v\left(X\right)$ and ${\mathrm{lip}}_v\left(X\right)$ denote the Banach spaces of all scalar-valued Lipschitz functions $f$ on $X$ vanishing at the basepoint such that $v\Phi \left(f\right)$ is bounded and $v\Phi \left(f\right)$ vanishes at infinity on $\tilde{X}$, respectively, where $\Phi \left(f\right)$ is the de Leeuw’s map of $f$ on $\tilde{X}$, under the weighted Lipschitz norm. The space ${\mathrm{Lip}}_v\left(X\right)$ has an isometric predual ${\mathcal{F}}_v\left(X\right)$ and it is proved that $\left({\mathrm{Lip}}_v\right(X),{\tau }_{{bw}^{\ast }})=\left({\mathcal{F}}_v\right(X{)}^{\ast },{\tau }_c)$ and ${\mathcal{F}}_v\left(X\right)=\left(\right({\mathrm{Lip}}_v\left(X\right),{\tau }_{{bw}^{\ast }})\text{'},{\tau }_c)$, where ${\tau }_{{bw}^{\ast }}$ denotes the bounded weak∗ topology and ${\tau }_c$ the topology of uniform convergence on compact sets. The linearization of the elements of ${\mathrm{Lip}}_v\left(X\right)$ is also tackled. Assuming that $X$ is compact, we address the question as to when ${\mathrm{Lip}}_v\left(X\right)$ is canonically isometrically isomorphic to ${\mathrm{lip}}_v(X{)}^{\ast \ast }$, and we show that this is the case whenever ${\mathrm{lip}}_v\left(X\right)$ is an M-ideal in ${\mathrm{Lip}}_v\left(X\right)$ and the so-called associated weights ${\tilde{v}}_L$ and ${\tilde{v}}_l$ coincide.