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Register a new userIsomorphic classification of $L_{p,q}$-spaces, II
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This is a continuation of the papers [Kuryakov-Sukochev, JFA, 2015] and [Sadovskaya-Sukochev, PAMS, 2018], in which the isomorphic classification of $L_{p,q}$, for $1< p<infty$, $1le q<infty$, $p e q $, on resonant measure spaces, has been obtained. The aim of this paper is to give a complete isomorphic classification of $L_{p,q}$-spaces on general $sigma$-finite measure spaces. Towards this end, several new subspaces of $L_{p,q}(0,1)$ and $L_{p,q}(0,infty)$ are identified and studied.
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Let $mathcal{M}(Omega, mu)$ denote the algebra of all scalar-valued measurable functions on a measure space $(Omega, mu)$. Let $B subset mathcal{M}(Omega, mu)$ be a set of finitely supported measurable functions such that the essential range of each $f in B$ is a subset of ${ 0,1 }$. The main result of this paper shows that for any $p in (0, infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(Omega, mu)$ if and only if $B$ is an affinely independent subset of $mathcal{M}(Omega, mu)$ (when $mathcal{M}(Omega, mu)$ is considered as a real vector space). It follows that every two-valued (Schauder) basis of $L_{p}(Omega, mu)$ has strict $p$-negative type. For instance, for each $p in (0, infty)$, the system of Walsh functions in $L_{p}[0,1]$ is seen to have strict $p$-negative type. The techniques developed in this paper also provide a systematic way to construct, for any $p in (2, infty)$, subsets of $L_{p}(Omega, mu)$ that have $p$-negative type but not $q$-negative type for any $q > p$. Such sets preclude the existence of certain types of isometry into $L_{p}$-spaces.
Suppose $0 < p leq 2$ and that $(Omega, mu)$ is a measure space for which $L_{p}(Omega, mu)$ is at least two-dimensional. The central results of this paper provide a complete description of the subsets of $L_{p}(Omega, mu)$ that have strict $p$-negative type. In order to do this we study non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$. These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form begin{eqnarray*} sumlimits_{j, i = 1}^{n} alpha_{j} alpha_{i} {| z_{j} - z_{i} |}_{p}^{p} & = & 0 end{eqnarray*} where ${ z_{1}, ldots, z_{n} }$ is a subset of $L_{p}(Omega, mu)$ and $alpha_{1}, ldots, alpha_{n}$ are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$. The cases $p < 2$ and $p = 2$ are substantially different and are treated separately. The case $p = 1$ generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin. Another reason for studying non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$ is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if $(X,d)$ is a metric space that has strict $q$-negative type for some $q geq p$, then: (1) $(X,d)$ is not isometric to any linear subspace $W$ of $L_{p}(Omega, mu)$ that contains a pair of disjointly supported non-zero vectors, and (2) $(X,d)$ is not isometric to any subset of $L_{p}(Omega, mu)$ that has non-empty interior. Furthermore, in the case $p = 2$, it also follows that $(X,d)$ is not isometric to any affinely dependent subset of $L_{2}(Omega, mu)$.
In this note we continue giving the characterisation of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. This is a continuation of our paper [M. Ruzhansky and D. Verma. Hardy inequalities on metric measure spaces, Proc. R. Soc. A., 475(2223):20180310, 2018] where we treated the case $pleq q$. Here the remaining range $p>q$ is considered, namely, $0<q<p$, $1<p<infty.$ We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds. We note that doubling conditions are not required for our analysis.
We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use $ell_{Psi}$-Hilbertian and $infty$-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition in a Hilbert space and introduce the class of spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and $M$-cotype of a Banach space and study the properties of unconditional Schauder decompositions in spaces possessing certain geometric structure.
The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space T[(S_n,theta_{n})_{n=1}^{infty}] and its modified version T_{M}[(S_{n},theta_{n})_{n=1}^{infty}]. It is shown that these spaces are not isomorphic for a large class of parameters (theta_{n}).
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