Poincaré type inequalities for group measure spaces and related transportation cost inequalities

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H  . We prove _Lp$L_p$ Poincaré inequalities for the group measure space _L∞(_ΩH,γ)?G$L_{\infty }({\Omega }_H,\gamma )?G$, where both the group action and the Gaussian measure space (_ΩH,γ)$({\Omega }_H,\gamma )$ are associated with the representation α  . The idea of proof comes from Pisier?s method on the boundedness of Riesz transform and Lust-Piquard?s work on spin systems. Then we deduce a transportation type inequality from the _Lp$L_p$ Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffel?s compact quantum metric spaces.     

Orlicz–John ellipsoids

The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved _Lp$L_p$ John ellipsoids. The analog of Ball's volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.     

Compact covers and function spaces

For a Tychonoff space X  , we denote by _Cp(X)$C_p(X)$ and _Cc(X)$C_c(X)$ the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X   in terms of _Cp(X)$C_p(X)$, we extend Okunev?s results by showing that if there exists a surjection from _Cp(X)$C_p(X)$ onto _Cp(Y)$C_p(Y)$ (resp. from _Lp(X)$L_p(X)$ onto _Lp(Y)$L_p(Y)$) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen?s theorem, we extend Pelant?s result by proving that if X is a separable completely metrizable space and Y   is first countable, and there is a quotient linear map from _Cc(X)$C_c(X)$ onto _Cc(Y)$C_c(Y)$, then Y   is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by _?p${?}_p$-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X   is completely metrizable and _?p${?}_p$-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.     

A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

For a locally compact group G   and 1<p<∞$1<p<\infty$ let _Ap(G)$A_p(G)$ be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G  , A(G)$A(G)$ (p=2$p=2$). It is shown that for any amenable group H  , a proper affine map α:Y⊂H→G$\alpha :Y\subset H\to G$ induces a p  -completely contractive algebra homomorphism _?α:_Ap(G)→_Ap(H)${?}_{\alpha }:A_p(G)\to A_p(H)$ by setting _?α(u)=u°α${?}_{\alpha }(u)=u°\alpha$ on Y   and _?α(u)=0${?}_{\alpha }(u)=0$ off of Y. Moreover, we show that if both G and H are amenable then any p  -completely contractive algebra homomorphism ?:_Ap(G)→_Ap(H)$?:A_p(G)\to A_p(H)$ is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2$p=2$) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.     

Error estimates in Sobolev spaces for interpolating thin plate splines under tension

This paper discusses _Lp$L_p$-error estimates for interpolation by thin plate spline under tension of a function in the classical Sobolev space on an open bounded set with a Lipschitz-continuous boundary. A property of convergence is also given when the set of interpolating points becomes more and more dense.     

The Orlicz Brunn–Minkowski inequality

The Orlicz Brunn–Minkowski theory originated with the work of Lutwak, Yang, and Zhang in 2010. In this paper, we first introduce the Orlicz addition of convex bodies containing the origin in their interiors, and then extend the _Lp$L_p$ Brunn–Minkowski inequality to the Orlicz Brunn–Minkowski inequality. Furthermore, we extend the _Lp$L_p$ Minkowski mixed volume inequality to the Orlicz mixed volume inequality by using the Orlicz Brunn–Minkowski inequality.     

A condition-based replacement and spare provisioning policy for deteriorating systems with uncertain deterioration to failure

A new policy, referred to as the condition-based replacement and spare provisioning policy, is presented for deteriorating systems with a number of identical units. It combines the condition-based replacement policy with periodical inspections and the (S,s$S\text{,}s$) type inventory policy, noted as the (T,S,s,_Lp$T\text{,}S\text{,}s\text{,}{L}_{\text{p}}$) policy, where T is the inspection interval, S is the maximum stock level, s   is the reorder level, and _Lp$L_{\text{p}}$ is the preventive replacement threshold for the deterioration levels of units. The deterioration level of each unit in the system can be described by a scalar random variable, which is continuous and increasing monotonically. Furthermore, the deterioration level just when the unit failure occurs, termed deterioration to failure, is uncertain. Therefore, the condition-based reliability is proposed in order to characterize various and uncertain deterioration levels when unit failure occurs. A simulation model is developed for the system operation under the proposed condition-based replacement and spare provisioning policy. Thus, via the simulation method and the genetic algorithm, the decision variables T, S, s  , and _Lp$L_{\text{p}}$ can be jointly optimized for minimizing the cost rate. A case study is given, showing the procedure of applying the proposed policy and the condition-based reliability methodology to optimizing the maintenance scheme of haul truck motors at a mine site based on oil inspections, and proving beneficial for plant maintenance managers to reduce maintenance cost.     

A monotone version of the Sokolov property and monotone retractability in function spaces

We introduce the monotone Sokolov property and show that it is dual to monotone retractability in the sense that X   is monotonically retractable if and only if _Cp(X)$C_p(X)$ is monotonically Sokolov. Besides, a space X   is monotonically Sokolov if and only if _Cp(X)$C_p(X)$ is monotonically retractable. Monotone retractability and monotone Sokolov property are shown to be preserved by R$\mathbb{R}$-quotient images and _Fσ$F_{\sigma }$-subspaces. Furthermore, every monotonically retractable space is Sokolov so it is collectionwise normal and has countable extent. We also establish that if X   and _Cp(X)$C_p(X)$ are Lindelöf Σ-spaces then they are both monotonically retractable and have the monotone Sokolov property. An example is given of a space X   such that _Cp(X)$C_p(X)$ has the Lindelöf Σ-property but neither X   nor _Cp(X)$C_p(X)$ is monotonically retractable. We also establish that every Lindelöf Σ-space with a unique non-isolated point is monotonically retractable. On the other hand, each Lindelöf space with a unique non-isolated point is monotonically Sokolov.     

The distribution of ITRM-recognizable reals

Infinite Time Register Machines (ITRM's) are a well-established machine model for infinitary computations. Their computational strength relative to oracles is understood, see e.g. ,  and . We consider the notion of recognizability, which was first formulated for Infinite Time Turing Machines in [6] and applied to ITRM's in [3]. A real x is ITRM-recognizable iff there is an ITRM-program P   such that P^y$P^y$ stops with output 1 iff y=x$y=x$, and otherwise stops with output 0. In [3], it is shown that the recognizable reals are not contained in the ITRM-computable reals. Here, we investigate in detail how the ITRM  -recognizable reals are distributed along the canonical well-ordering _<L${<}_L$ of Gödel's constructible hierarchy L  . In particular, we prove that the recognizable reals have gaps in _<L${<}_L$, that there is no universal ITRM in terms of recognizability and consider a relativized notion of recognizability.     

Korn type inequalities in Orlicz spaces

We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the L^B$L^B$ norm of the gradient of vector-valued functions and the L^A$L^A$ norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in L^p$L^p$, with 1<p<∞$1<p<\infty$, and an Orlicz version involving a Young function A   satisfying both the _Δ2${\mathrm{\Delta }}_2$ and the _∇2${\mathrm{\nabla }}_2$ condition.     

The Calderón–Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds

We introduce the concept of Calderón–Zygmund inequalities on Riemannian manifolds. For 1<p<∞$1<p<\infty$, these are inequalities of the form

_{‖Hess(u)‖Lp}≤_C1‖u‖Lp+_{C2‖Δu‖Lp},${\Vert \mathrm{Hess}(u)\Vert }_{{\mathsf{L}}^p}\le C_1{\Vert u\Vert }_{{\mathsf{L}}^p}+C_2{\Vert \mathrm{\Delta }u\Vert }_{{\mathsf{L}}^p},$

valid a priori for all smooth functions u   with compact support, and constants _C1≥0$C_1\ge 0$, _C2>0$C_2>0$. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calderón–Zygmund inequality (like new denseness results for second order L^p${\mathsf{L}}^p$-Sobolev spaces and gradient estimates), we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact Riemannian manifolds. These results in particular apply to many noncompact hypersurfaces of constant mean curvature.     

Weitzenböck derivations of free metabelian Lie algebras

A nonzero locally nilpotent linear derivation δ   of the polynomial algebra K[_Xd]=K[_x1,…,_xd]$K[X_d]=K[x_1\text{,}\dots \text{,}{x}_d]$ in several variables over a field K   of characteristic 0 is called a Weitzenböck derivation. The classical theorem of Weitzenböck states that the algebra of constants K[_Xd]^δ$K{[X_d]}^{\delta }$ (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation δ of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K  . Now the algebra of constants is usually not finitely generated. Except for some trivial cases this holds for the algebra of constants (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ of the free metabelian Lie algebra _Ld/_Ld″$L_d/L_d″$ with d   generators. We show that the vector space of the constants (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ in the commutator ideal _Ld′/_Ld″$L_d\prime /L_d″$ is a finitely generated K[_Xd]^δ$K{[X_d]}^{\delta }$-module. For small d  , we calculate the Hilbert series of (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ and find the generators of the K[_Xd]^δ$K{[X_d]}^{\delta }$-module (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$. This gives also an (infinite) set of generators of the algebra (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$.     

Canonical variation of a Lorentzian metric

Given a Lorentzian manifold (M,_gL)$(M,g_L)$ and a timelike unitary vector field E  , we can construct the Riemannian metric _gR=_gL+2ω⊗ω$g_R=g_L+2\omega \otimes \omega$, ω being the metrically equivalent one form to E. We relate the curvature of both metrics, especially in the case of E   being Killing or closed, and we use the relations obtained to give some results about (M,_gL)$(M,g_L)$.     

Compact and weakly compact composition operators from the Bloch space into Möbius invariant spaces

We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like BMOA  , _Qα$Q_{\alpha }$, and analytic Besov spaces B^p$B^p$. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding “small” spaces we also characterize the boundedness and show that it is equivalent to compactness.     

BMO from dyadic BMO via expectations on product spaces of homogeneous type

Using the random dyadic lattices developed by Hytönen and Kairema, we build up a bridge between BMO and dyadic BMO, and hence one between VMO and dyadic VMO, via expectations over dyadic lattices on spaces of homogeneous type, including both the one-parameter and product cases. We also obtain a similar relationship between _Ap$A_p$ and dyadic _Ap$A_p$, as well as one between the reverse Hölder class _RHp${\mathit{RH}}_p$ and dyadic _RHp${\mathit{RH}}_p$, via geometric–arithmetic expectations. These results extend the earlier theory along this line, developed by Garnett, Jones, Pipher, Ward, Xiao and Treil, to the more general setting of spaces of homogeneous type in the sense of Coifman and Weiss.