V Kadets, B. Shumyatskiy, R. Shvidkoy, L. Tseytlin, K. Zheltukhin
The article presents a new method of integration of functions with values in Banach spaces. This integral and related notions prove to be a useful tool in the study of Banach space geomtry.
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L-valued integration
本文发展了取值于Dedekind完备单位$f$-代数$\mathbb{L}$的函数关于$\mathbb{L}$值测度的积分理论，并证明了$\mathbb{L}$值$L^p$空间的完备性。
We discuss relationships between the McShane, Pettis, Talagrand and Bochner integrals. A large number of different methods of integration of Banach-space-valued functions have been introduced, based on the various possible constructions of the Lebesgue integral. They commonly run fairly closely together when the range space is separable (or has w^*-separable dual) and diverge more or less sharply for general range spaces. The McShane integral as described by [Go] is derived from the `gauge-limit' integral of [McS]. Here we give both positive and negative results concerning it and the other thr…
Gel'fand Integration of B(E, F*)-Valued Functions With Emphasis on (q, p)-Summing Operators
本文推广了Gel'fand积分在Banach空间上的结果，探讨了(q, p)-求和算子的充分条件，并应用新方法回答了相关问题，还给出了正算子值函数的应用。
We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for $p$-Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are $p$-convex for all $0<p<1$, so it is not always possible to choose an optimal $p$ to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of ga…
A "Bochner-type" integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures is presented with respect to abstract convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion.
Cylindrical Martingale-Valued Measures, Stochastic Integration and SPDEs
本研究将二次变差概念拓展至可分巴拿赫空间中带不连续路径的柱鞅值测度情形，建立了对应的希尔伯特空间值随机积分理论，并将其用于解决希尔伯特空间中SPDEs解的存在唯一性问题。
On Monadic Vector-Valued Integration
本文在hk-空间范畴上构造交换概率单子，发展配对线性hk-空间理论，并通过自由-遗忘伴随自然导出单子向量值积分理论。
Analytic integration of metric-valued functions in Lipschitz free spaces
该研究为度量值函数发展积分理论，通过博赫纳型构造定义自由积分，建立其性质，讨论度量空间为巴拿赫空间时的情况，还将自由积分与$\mathcal{F}(M)$几何联系起来，并给出例子说明框架，为相关研究提供了新方法和理论基础。
We remark that a dyadic version of the Carleson embedding theorem for the Bergman space extends to vector-valued functions and operator-valued measures. This is in contrast to a result by Nazarov, Treil, Volberg in the context of the Hardy space. We also discuss some embeddings for analytic vector-valued functions.
Some Remarks on the Vector-Valued Variable Exponent Lebesgue Spaces $\ell^{q(\cdot)} (L^{p(\cdot)})$
本文研究了变指数混合Lebesgue序列空间 $\ell^{q(\cdot)} (L^{p(\cdot)})$ 的几何性质，证明了其在特定参数下的严格凸与一致凸性，并揭示了范数收敛与依测度收敛的关系及逼近恒等式的成立条件。
Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the "positive multifunctions". Among them an investigation of multifunctions determined by vector-valued functions is presented. Finally, decomposition results are obtained for scalarly and gauge-defined integrals of multifunctions and a full description of McShane integrability in terms of Henstock and Pettis integrability is given.
We define a monad M on a category of measurable bornological sets, and we show how this monad gives rise to a theory of vector-valued integration that is related to the notion of Pettis integral. We show that an algebra X of this monad is a bornological locally convex vector space endowed with operations which associate vectors \int f dm in X to incoming maps f:T --> X and measures m on T. We prove that a Banach space is an M-algebra as soon as it has a Pettis integral for each incoming bounded weakly-measurable function. It follows that all separable Banach spaces, and all reflexive Banach sp…
On the Set-Valued Katugampola Fractional Integral: Properties and Regular Selections
本文建立了统一两类经典分数积分的集值Katugampola分数积分理论，证明其保留集值映射的多种分析特性，且正则选择存在性可传递，拓展了分数积分在集值映射领域的应用。
Towards a Theory of Dobrakov-Sobolev Spaces
本文基于Dobrakov积分引入Sobolev空间的推广，针对有限维空间上Banach值函数与Fomin可微算子值测度，建立可微实值测度理论的类比结果，并嵌入向量值分布理论框架。
In this paper we consider the problem of optimization of approximate integration of set-valued functions from the class defined by given majorant of their moduli of continuity, using values of the functions at $n$ fixed or free points of their domain. We consider the cases of exact information and information with error.
We develop aspects of functional analysis in an abstract axiomatic setting, through monoidal and enriched category theory. We work in a given closed category, whose objects we call spaces, and we study R-module objects therein (or algebras of a commutative monad), which we call linear spaces. Building on ideas of Lawvere and Kock, we study functionals on the space of scalar-valued maps, including compactly-supported Radon measures and Schwartz distributions. We develop an abstract theory of vector-valued integration with respect to these scalar functionals and their relatives. We study three a…
Multidimensional vector-valued $Z$-transform and its applications
本文提出多维向量值Z变换的理论框架及其在抽象Volterra差分方程中的应用。
This is the second installment of a series of papers aimed at developing a theory of Hrushovski-Kazhdan style motivic integration for certain types of nonarchimedean $o$-minimal fields, namely power-bounded $T$-convex valued fields, and closely related structures. The main result in the first installment is a canonical isomorphism between the Grothendieck rings of certain categories of definable sets, which is understood as a universal additive invariant or a generalized Euler characteristic because the categories do not carry volume forms. Here we introduce two types of volume forms into each…
In this paper, we introduce the paths space $\mathcal C_0^{\mathrm{gBm}}$ which is consists of generalized Brownian motion path-valued continuous functions on $[0,T]$. We next present several relevant examples of the paths space integral. We then discuss the concept of the analytic Feynman integration theory for functionals $F$ on the paths space $\mathcal C_0^{\mathrm{gBm}}$.
We prove new upper and lower bounds for Banach space-valued stochastic integrals with respect to a compensated Poisson random measure. Our estimates apply to Banach spaces with non-trivial martingale (co)type and extend various results in the literature. We also develop a Malliavin framework to interpret Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a Clark-Ocone representation formula.
A vector-valued version of the Girsanov theorem is presented, for a scalar process with respect to a Banach-valued measure. Previously, a short discussion about the Birkhoff-type integration is outlined, as for example integration by substitution, in order to fix the measure-theoretic tools needed for the main result, Theorem 6, where a martingale equivalent to the underlying vector probability has been obtained in order to represent the modified process as a martingale with the same marginals as the original one.
The Dunford-Choquet integral
本文定义了Banach值函数的Dunford-Choquet积分，并研究了其基本性质及相伴的不定容度。
We prove a uniformly continuous linear extension principle in topological vector spaces from which we derive a very short and canonical construction of the Lebesgue integral of Banach space valued maps on a finite measure space. The Vitali Convergence Theorem and the Riesz-Fischer Theorem are shown to follow as easy consequences from our construction.
In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on a locally convex space $Φ$. Our construction of the stochastic integral is based on the theory of tensor products of topological vector spaces and the property of good integrators of real-valued semimartingales. This theory is further developed in the case where $Φ$ is a complete, barrelled, nuclear space, where we obtain a complete description of the class of integrands as $Φ$-valued locally bounded and weakly predictable processes. Several other properties of the stocha…
Using a multiplicative structure (for example that of a Banach algebra) and a partial order we construct a weak version of a Banach space valued stochastic integral with respect to square integrable martingales.
The Real-Valued Bochner integral and the Modern Real-Valued Measurable function on $\mathbb{R}$
本文证明了在有限测度下，实值可测函数的类 Lebesgue 积分（RVM-MI）与 Bochner 积分在实数域上等价，并指出该等式可应用于 Banach 空间的弱极限研究。
Banach spaces that are complemented in the second dual are characterised precisely as those spaces $X$ which enjoy the property that for every amenable semigroup $S$ there exists an $X$-valued analogue of an invariant mean defined on the Banach space of all bounded $X$-valued functions on $S$. This was first observed by Bustos Domecq (J. Math. Anal. Appl., 2002), however the original proof was slightly flawed as remarked by Lipecki. The primary aim of this note is to present a corrected version of the proof. We also demonstrate that universally separably injective spaces always admit invariant…
Properties of a maximal function for vector-valued martingales were studied by the author in an earlier paper. Restricting here to the dyadic setting, we prove the equivalence between (weighted) L^p inequalities and weak type estimates, and discuss an extension to the case of locally finite Borel measures on R^n. In addition, to compensate for the lack of an L^\infty inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.
The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical ``$T(1)\in BMO$'' assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical $T(1)$ criterion still guarantees their $L^2$-boundedness on Hilbert space …
A result of G. Pisier says that a convolution operator $\star f : M(G) \to C(G),$ where $G$ is a compact Abelian group, can be factored through a Hilbert space if and only if $f$ has the absolutely summable set of Fourier coefficients. P. Saab (2010) generalized this result in some directions in the vector-valued cases. We give some further generalizations of the results of G. Pisier and P. Saab, considering, in particular, the factorizations of the operators through the operators of Schatten classes in Hilbert spaces. Also, some related theorem on the factorization of operators through the op…
In this work we establish a Stokes-type integral equality for scalarly essentially integrable forms on an orientable smooth manifold with values in the locally convex linear space $\langle B(G),σ(B(G),\mathcal{N})\rangle$, where $G$ is a complex Banach space and $\mathcal{N}$ is a suitable linear subspace of the norm dual of $B(G)$. This result widely extends the Newton-Leibnitz-type equality stated in one of our previous articles. To obtain our equality we generalize the main result of that article, and employ the Stokes theorem for smooth locally convex vector valued forms established in a p…
We present a natural and simple proof of the Radon - Nikodym theorem for measures with values in the space of bounded linear operators on a separable Hilbert space. This space is not separable, that is why it is essential to assume in the theorem that the range of the measure is separable. We also discuss distinctions between uniform and strong operator measures (Remark 3.3 to Lemma 3.2, Theorem 3.5 and its Corollary 3.6). Proof of this version of the Radon - Nikodym theorem either is not presented in known literature, or given for special cases and with some inaccuracies (see Comment at the e…
This paper presents a brief survey of the theory of stochastic integration in Banach spaces. Expositions of the stochastic integrals in martingale type 2 spaces and UMD spaces are presented, as well as some applications of the latter to vector-valued Malliavin calculus and the stochastic maximal regularity problem. A new proof of the stochastic maximal regularity theorem is included.
In this paper we study the forward integral of operator-valued processes with respect to a cylindrical Brownian motion. In particular, we provide conditions under which the approximating sequence of processes of the forward integral, converges to the stochastic integral process with respect to Sobolev norms of smoothness alpha < 1/2. This result will be used to derive a new integration by parts formula for the forward integral.
This paper presents Stieltjes-type integration for operator-valued functions with respect to spectral families. The relation between Riemann-Stieltjes integrals associated with some classes of spectral families including, in particular, those that arise in the context of the Stone representation theorem for Banach spaces is established. The developed approach is applied to the structural analysis of the operators commuting with a one-parameter representation satisfying the Generalized Stone theorem.
The Wodzicki residue and the cut-off integral extend to classical symbol-valued forms. We show that they obey a Stokes' type property and that the extended Wodzicki residue can be interpreted as a complex residue like the ordinary one. In the case of cut-off integrals, Stokes' property (i.e. vanishing on exact forms) only holds for non-integer order symbol-valued forms and leads to an integration by parts formula and translation invariance for cut-off integrals on non-integer order classical symbols. The extended Wodzicki residue yields an even residue cycle on classical symbols and an odd coc…
It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions o…
We apply a recent result of Borichev-Golinskii-Kupin on the Blaschke-type conditions for zeros of analytic functions on the complex plane with a cut along the positive semi-axis to the problem of the eigenvalues distribution of the Fredholm-type analytic operator-valued functions.
A Bochner-type integration theory for random normed modules
本文发展了随机范数模的测度与积分理论，研究了值域为L^0(m)的测度，并建立了针对完整随机范数模的Bochner型积分理论，证明了Radon-Nikodým定理和Riesz-Markov-Kakutani表示定理的版本，并探讨了其应用。
Convex body domination is an important elaboration of the technique of sparse domination that has seen significant development and applications over the past ten years. In this paper, we present an abstract framework for convex body domination, which also applies to Banach space -valued functions, and yields matrix-weighted norm inequalities in this setting. We explore applications to "generalised commutators", obtaining new examples of bounded operators among linear combinations of compositions of pointwise multipliers and a singular integral operator.
We study the inequalities of the type $|\int_{\mathbb{R}^d} Φ(K*f)| \lesssim \|f\|_{L_1(\mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $α- d$ and possibly vector-valued, the function $Φ$ is positively $p$-homogeneous, and $p = d/(d-α)$. Under mild regularity assumptions on $K$ and $Φ$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.
Remarks on positive definite functions on a group
本文利用正分块矩阵研究群上算子值正定函数，给出Brehmer正性的新证明，分类有限群上的交换酉表示，并刻画幂保持正定或酉性的条件。
In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holo…
In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the $γ$-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gund…
This expository paper contains a concise introduction to some significant works concerning the Karush-Kuhn-Tucker condition, a necessary condition for a solution in local optimality in problems with equality and inequality constraints. The study of this optimality condition has a long history and culminated in the appearance of subdifferentials. The 1970s and early 1980s were important periods for new developments and various generalizations of subdifferentials were introduced, including the Clarke subdifferential and Demyanov-Rubinov quasidifferential. In this paper, we mainly present four ge…
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