Related papers: Multi-normed spaces, based on non-discrete measure…

We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact $T1$ theorem for multilinear Calder\'{o}n--Zygmund operators on product spaces. More…

Inequalities for product operators on mixed norm Lebesgue spaces and permuted mixed norm Lebesgue spaces are established. They depend only on inequalities for the factors and on the Lebesgue indices involved. Inequalities for the bivariate…

We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration…

We consider $\ell_p$-direct sums ($1\leq p<\infty$) and $c_0$-direct sums of countably many normed spaces and find the duals of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize…

We consider weighted anchored and ANOVA spaces of functions with first order mixed derivatives bounded in $L_p$. Recently, Hefter, Ritter and Wasilkowski established conditions on the weights in the cases $p=1$ and $p=\infty$ which ensure…

Let $p\in(0,1]$, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. The authors introduce the anisotropic Hardy-Lorentz space $H^{p,q}_A(\mathbb{R}^n)$ associated with $A$ via the non-tangential grand maximal function…

This expository article explores the vital role of interpolation theory and Lorentz spaces in the rigorous analysis of partial differential equations (PDEs). While classical Lebesgue spaces ($L_{p}$) successfully measure the magnitude of…

An L-embedded Banach spaace is a Banach space which is complemented in its bidual such that the norm is additive between the two complementary parts. On such spaces we define a topology, called an abstract measure topology, which by known…

This is an exposition of the known techniques for constructing $\Cal L_p$-spaces for $p\in (1,\infty)\setminus \{2\}$, including some unpublished work of Alspach. Isomorphic and complemented embedding relations between these spaces are also…

Let $A^p_\omega$ denote the Bergman space in the unit disc induced by a radial weight~$\omega$ with the doubling property $\int_{r}^1\omega(s)\,ds\le C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$. The positive Borel measures such that the…

A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence…

Increasingly in recent years, probabilistic computation has been investigated through the lenses of categorical algebra, especially via string diagrammatic calculi. Whereas categories of discrete and Gaussian probabilistic processes have…

We introduce new entanglement measures on the set of density operators on tensor product Hilbert spaces. These measures are based on the greatest cross norm on the tensor product of the sets of trace class operators on Hilbert space. We…

Let $N^p$ $(1<p<\infty)$ be the Privalov class $N^p$ of holomorphic functions on the open unit disk $\Bbb D$ in the complex plane. In 1977 M. Stoll proved that the class $N^p$ equipped with the topology given by the metric $\lambda_p$…

We introduce vector space norms associated to the Mahler measure by using the L^p norm versions of the Weil height recently introduced by Allcock and Vaaler. In order to do this, we determine orthogonal decompositions of the space of…

We construct the extension of the curvilinear summation for bounded Borel measurable sets to the $L_p$ space for multiple power parameter $\bar{\alpha}=(\alpha_1, \cdots, \alpha_{n+1})$ when $p>0$. Based on this…

G\"ahler ([4],[5]) introduced and investigated the notion of 2-metric spaces and 2-normed spaces in sixties. These concepts are inspired by the notion of area in two dimensional Euclidean space. In this paper, we choose a fundamentally…

Let $\M$ be a semi-finite von Neumann algebra equipped with a faithful normal trace $\tau$. We study the subspace structures of non-commutative Lorentz spaces $L_{p,q}(\M, \tau)$, extending results of Carothers and Dilworth to the…

A proto-quantum space is a (general) matricially normed space in the sense of Effros and Ruan presented in a `matrix-free' language. We show that these spaces have a special (projective) tensor product possessing the universal property with…

Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting operator…