We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian spaces. For a monotone family, we show the existence of weak derivative, obtain an isomorphism to the standard Sobolev space, and provide some scalar characteristics.
In this work we are concerned with maximality of monotone operators representable by certain convex functions in non-reflexive Banach spaces. We also prove that these maximal monotone operators satisfy a Bronsted-Rockafellar type property. We show that if a function in XxX^* and its conjugate are above the duality product in their respective domains, then this function represents a maximal monotone operator.
A reflexive Banach space $X$ with a basis $(e_i)$ is constructed having the property that every monotone basis is block finitely representable in each block basis of $X$.
It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone op…
Our goal is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their domains.
In this paper, we deal with the existence of $ω$-periodic mild solutions for the abstract evolution equation with delay in an ordered Banach space $E$ $$u'(t)+Au(t)=F(t,u(t),u(t-τ)),\ \ \ \ t\in\R,$$ where $A:D(A)\subset E\rightarrow E$ is a closed linear operator and $-A$ generates a positive $C_{0}$-semigroup $T(t)(t\geq0)$, $F:\R\times E\times E\rightarrow E$ is a continuous mapping which is $ω$-periodic in $t$, and $τ\geq0$ is a constant. Under some weaker assumptions, we construct monotone iterative method for the delayed evolution equation periodic problems, and obtain the existence of m…
In a real Banach space, we first prove that the sum of a monotone operator of type (FPV) and maximal monotone operator Rockafellar's constraint qualification is maximal. This prove leads to the solution of most interesting long-time outstanding problem in monotone operator theory is the sum problem.
It is known that for Banach valued functions there are several approaches to define a Sobolev class. We compare the usual definition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. As well we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.
Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : E\rightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)\neq \emptyset$. Inspired by Alber [2], we introduce Lyapunov functions and use the new geometric properties of Banach spaces to show the strong convergence of an iterative algorithm to the solution of $Ax=0$.
It was proved by Argyros and Dodos that, for many classes $ C $ of separable Banach spaces which share some property $ P $, there exists an isomorphically universal space that satisfies $ P $ as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that $ C $ consists of spaces with a monotone Schauder basis. For example, we prove that if $ C $ is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every $ X \in C $ is reflexive and has a monotone Schauder basis, then there exists a…
We give, for general Banach spaces, a characterization of the sequential lower limit of maximal monotone operators of type (D) and prove its representability. As a consequence, using a recent extension of the Moreau-Yosida regularization for type (D) operators, we extend to general Banach spaces the definitions of the variational sum of monotone operators and the variational composition of monotone operators with continuous linear mappings, and we prove that both operators are representable.
It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.
We obtain a pointwise description of functions belonging to function spaces with the lattice property. In particular, it is valid for Banach function spaces provided that the Hardy-Littlewood maximal operator is bounded. We also study weighted grand Sobolev spaces, defined on arbitrary open sets (of finite or infinite measure), and provide pointwise description of them.
In this paper we provide some extension results for n-cyclically monotone operators in reflexive Banach spaces by making use of the Fenchel duality. In this way we give a positive answer to a question posed by Bauschke and Wang in [4].
The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to H.Lebesgue. It was introduced, in the setting of Sobolev spaces, by J.Manfredi, and thoroughly investigated in the more general framework of Orlicz-Sobolev spaces by diverse authors, including T.Iwaniec, J.Kauhanen, P.Koskela, J.Maly, J.Onninen, X.Zhong. The present paper complements and augments the available theory of pointwise regularity properties of weakly monotone functions in Orlicz-Sobolev spaces. In particular, a variant is proposed in a customary condition ensurin…
The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also establishe…
We augment the dimension of the Euclidean space by one and the Picard iteration of a contraction by a simple iteration on the real line such that the resulting iteration becomes monotone increasing and bounded with respect to the order defined by the Lorentz cone of the augmented space. This provides a different way of showing the convergence of the Picard iteration of a contraction, exhibiting the strong relationship between the Banach fixed point principle and the ordering structure of the Euclidean space ordered by the Lorentz cone.
Some new examples of K-monotone couples of the type (X, X(w)), where X is a symmetric space on [0, 1] and w is a weight on [0, 1], are presented. Based on the property of the w-decomposability of a symmetric space we show that, if a weight w changes sufficiently fast, all symmetric spaces X with non-trivial Boyd indices such that the Banach couple (X, X(w)) is K-monotone belong to the class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental function of X is t^{1/p} for some p \in [1, \infty], then X = L_p. At the same time a Banach couple (X, X(w)) may be K-monotone for some non…
In this paper, we unify the theory of SSD spaces and the theory of strongly representable sets, and we apply our results to the theory of the various classes of maximally monotone sets. In particular, we prove that type (ED), dense type, type (D), type (NI) and strongly representable are equivalent concepts and, consequently, that the known properties of strongly representable sets follow from known properties of sets of type (ED).
In complete metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality, we investigate when a given Banach-valued Sobolev function defined on a subset satisfying a measure-density condition is the restriction of a Banach-valued Sobolev function defined on the whole space. We investigate the problem for Hajłasz- and Newton-Sobolev spaces, respectively. First, we show that Hajłasz-Sobolev extendability is independent of the target Banach spaces. We also show that every $c_0$-valued Newton-Sobolev extension set is a Banach-valued Newton-Sobolev extension set …
We are concerned with surjectivity of perturbations of maximal monotone operators in non-reflexive Banach spaces. While in a reflexive setting, a classical surjectivity result due to Rockafellar gives a necessary and sufficient condition to maximal monotonicity, in a non-reflexive space we characterize maximality using a ``enlarged'' version of the duality mapping, introduced previously by Gossez.
In this paper, we construct maximally monotone operators that are not of Gossez's dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC--functions will not always be a BC--function. This provides a negative answer to a challenging question posed by Stephen Simons. Among other consequences, we deduce that every Banach space which contains an isomorphic copy of the James space $\mathbf{J}$ or its dual $\mathbf{J}^*$, or $c_0$ or its dual $\ell^1$, a…
We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concepts of "$r_L$-density" and its specialization "quasidensity": the closed quasidense monotone multifunctions from a Banach space into its dual form a (generally) strict subset of the maximally monotone ones, though all surjective maximally monotone and all maximally monotone multifunctions on a reflexive space are quasidense. W…
Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space $W^{s,p}(\mathcal{M}, \mathcal{N})$ imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach--Steinhaus uniform boundedness principle in linear B…
In this paper, we study properties of ultramaximally monotone operators. We characterize the interior and the closure of the range of an ultramaximally monotone operator. We establish the Brezis--Haraux condition in the setting of a general Banach space. Moreover, we show that every ultramaximally monotone operator is of type (NA). We also provide some sufficient conditions for a Banach space to be reflexive by a linear continuous and ultramaximally monotone operator.
This is a 30 page set of lecture notes, in Plain TeX, which were prepared for and presented as a series of lectures (10 1/2 hours over two weeks) at the 2nd Summer School on Banach Spaces, Related Areas and Applications in Prague and Paseky, Czech Republic, during August, 1993. They consist of a largely self-contained exposition of both classical and recent basic facts about maximal monotone operators on Banach spaces, motivated in part by the goal of highlighting several fundamental properties of such operators which remain open questions in nonreflexive Banach spaces.
Maximal monotone operators on a Banach space into its dual can be represented by convex functions bounded below by the duality product. It is natural to ask under which conditions a convex function represents a maximal monotone operator. A satisfactory answer, in the context of reflexive Banach spaces, has been obtained some years ago. Recently, a partial result on non-reflexive Banach spaces was obtained. In this work we study some others conditions which guarantee that a convex function represents a maximal monotone operator in non-reflexive Banach spaces.
The main result of this paper is a fixed point result relating the spreading model structure of Banach spaces and Schauder basis with not too large basis constant. As a striking consequence, we deduce that every super-reflexive space has the fixed point property thus solving a long-standing open question in metric fixed point theory.
We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space $X$. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincaré inequalities, and thanks to these inequalities we deduce the spectral properties of the Ornstein-Uhlenbeck operator.
Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$. The associated Cameron--Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $ν=e^{-U}μ$ and $Ω=G^{-1}(-\infty,0]$. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \begin{align}\label{Probelma in abstract} λu-L_{ν,Ω} u=f, \end{align} where $λ>0$, $f\in L^2(Ω,ν)$ and $L_{ν,Ω}$ is the self-adjoint operator associated with the quadratic fo…
Banach Hardy-Sobolev Spaces on the Upper Half-plane and Operator Theory
College of Mathematics and Informatics, South China Agricultural University（华南农业大学数学与信息学院）；Faculty of Innovation Engineering, Macau University of Science and Technology（澳门科技大学创新工程学院）
Division of Science, NYU Abu Dhabi（纽约大学阿布扎比分校科学系）；Center for Interdisciplinary Data Science and AI, NYUAD Research Institute（NYUAD研究院跨学科数据科学与人工智能中心）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
We show that Sobolev maps with values in a dual Banach space can be characterized in terms of weak derivatives in a weak* sense. Since every metric space embeds isometrically into a dual Banach space, this implies a characterization of metric space valued Sobolev maps in terms of such derivatives. Furthermore, we investigate for which target spaces Sobolev maps are weak* differentiable almost everywhere.
We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev--Bochner spaces. An Aubin--Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we…
We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a Köthe function space $E$ is upper locally uniformly monotone if and only if its $p$-conv…
Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the s…
We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $Ω\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(Ω, V)$ with the so-called Sobolev-Reshetnyak space $R^{1,p}(Ω, V)$. We see that, in general, $W^{1,p}(Ω, V)$ is a closed subspace of $R^{1,p}(Ω, V)$. As a main result, we obtain that $W^{1,p}(Ω, V)=R^{1,p}(Ω, V)$ if, and only if, the Banach space $V$ has the Radon-Nikodým property
We study in this short preprint the theory of trigonometric approximation in the so-called Banach functional rearrangement invariant Sobolev-Grand Lebesgue Spaces.
By using quasi-Banach techniques as key ingredient we prove Poincaré- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of Błocki's integrability conjecture for $m$-subharmonic functions.
Recently it was introduced the so-called Bishop-Phelps-Bollob{á}s property for positive operators between Banach lattices. In this paper we prove that the pair $(C_0(L), Y) $ has the Bishop-Phelps--Bollob{á}s property for positive operators, for any locally compact Hausdorff topological space $L$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. In case that the space $C_0(L)$ is separable, the same statement holds for any uniformly monotone Banach lattice $Y .$ We also show the following partial converse of the main result. In case that $Y$ is a strictly monotone Banach l…
This paper is primarily concerned with the problem of maximality for the sum $A+B$ and composition $L^{*}ML$ in non-reflexive Banach space settings under qualifications constraints involving the domains of $A,B,M$. Here $X$, $Y$ are Banach spaces with duals $X^{*}$, $Y^{*}$, $A,B:X\rightrightarrows X^{*}$, $M:Y\rightrightarrows Y^{*}$ are multi-valued maximal monotone operators, and $L:X\to Y$ is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results …
We consider Banach spaces equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. Using these semigroups we introduce an analog of a modulus of continuity and define analogs of Besov norms. A generalization of a classical interpolation theorem is proven in which the role of Sobolev spaces is played by subspaces defined in terms of infinitesimal operators of these semigroups. We show that our assumptions about a given set of semigroups are satisfied in the case of a strongly continuous bounded representation of a Lie group. In the case of a unitary represent…
In this paper, we unify the theory of SSD spaces, part of the theory of strongly representable multifunctions, and the theory of the equivalence of various classes of maximally monotone multifunctions.
In the context of general Banach spaces characterizations for the maximal monotonicity of operators with non-empty domain interior as well as stronger continuity properties for such operators are provided.
During the 1970s Brézis and Browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces. In this paper, we extend and refine their result to a general Banach space.
Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on uniformly Banach spaces with uniformly convex duals, or their domains have nonempty interiors on reflexive real Banach spaces. Faces and support functions are characterized by the limit values of the minimal-norm selections of maximal monotone operators in the first case while in the second case they are represented by the limit values of any selection of maximal monotone operators. These obtained formulas are applied to study the…
We prove that the class of positive operators from $L_\infty (μ)$ to $Y$ has the Bishop-Phelps-Bollobás property for any positive measure $μ$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. The same result also holds for the pair $(c_0, Y)$ for any uniformly monotone Banach lattice $Y.$ Further we show that these results are optimal in case that $Y$ is strictly monotone.
