Z. M. Balogh, A. Calogero, R. Pini
In this paper we prove that maximal H-monotone operators $T:H^n\rightrightarrows V_1$ whose domain is all the Heisenberg group $H^n$ are locally bounded. This implies that they are upper semicontinuous. As a consequence, maximal H-monotonicity of an operator on $H^n$ can be characterized by a suitable version of Minty's type theorem.
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In the present paper, we shall give a characterization for weak/strong Adams-type boundedness of the fractional maximal operator on generalized Orlicz-Morrey spaces.
We prove an uniform boundedness principle for the Lipschitz seminorm of continuous, monotone, positively homogeneous and subadditive mappings on suitable cones of functions. The result is applicable to several classes of classically non-linear operators.
We define a family of linear type (D) operators for which the inverse of their maximal monotone extensions to the bidual are not of type (D) and provide an example of an operator in this family.
Here, question raised by Borwein and Yao has been settled by establishing that the sum of two maximal monotone operators A and B is maximal monotone with the condition that A is of type (FPV) and satisfies Rockafellar's constraints qualification. Also we have proved that A+B is of type (FPV) without assuming convexity on the domain of A.
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.
In this paper the boundedness of the weighted iterated Hardy-type operators $T_{u,b}$ and $T_{u,b}^*$ involving suprema from weighted Lebesgue space $L_p(v)$ into weighted Cesàro function spaces ${\operatorname{Ces}}_{q}(w,a)$ are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator $R_u$ from $L^p(v)$ into ${\operatorname{Ces}}_{q}(w,a)$ on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator $P_{u,b }$ from $L^p(v)$ into ${\oper…
In this paper, we prove strong type, weak type inequalities of Hardy-Littlewood maximal operator and fractional Hardy-Littlewood maximal operator on variable sequence spaces lp(Z). This is achieved using Calderon-Zygmund decomposition for sequences, properties of modular functional and Log Holder continuity.
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.
The local boundedness of classes of operators is analyzed on different subsets directly related to their Fitzpatrick functions and characterizations of the topological vector spaces for which that local boundedness holds is given in terms of the uniform boundedness principle. For example the local boundedness of a maximal monotone operator on the algebraic interior of its domain convex hull is a characteristic of barreled locally convex spaces.
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
In this note we use recent results concerning the sum theorem for maximal monotone multifunctions in general Banach spaces to find new characterizations and properties of regular maximal monotone multifunctions and then use these to describe the domain of certain enlargements.
In the first part of the note we prove that a sufficient condition (due to Simons) for the convexity of the closure of the domain/range of a monotone operator is also necessary when the operator has bounded domain and is maximal. Simons' condition is closely related to the notion of regular maximal monotone operator. In the second part of the note we give several characterizations for the regularity of a maximal monotone operator, show that a maximal monotone operator of type (FPV) is regular and improve a previous sum theorem type result.
Let $\mathcal{M}$ be the bilinear Hardy-Littlewood maximal function and $\vec{b}=(b,b)$ be a collection of locally integrable functions. In this paper, the authors establish characterizations of the weighted {\rm BMO} space in terms of several different commutators of bilinear Hardy-Littlewood maximal function, respectively; these commutators include the maximal iterated commutator $\mathcal{M}_{Π\vec{b}}$, the maximal linear commutator $\mathcal{M}_{Σ\vec{b}}$, the iterated commutator $[Π\vec{b},\mathcal{M}]$ and the linear commutator $[Σ\vec{b},\mathcal{M}]$.
Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop Phelps and sum representability results are presented in the last part.
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.
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds, which is called the "sum problem". In this paper, we establish the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximally monotone operators such that $\dom A\cap\inte\dom B\neq\varnothing$, and $A+N_{\overline{\dom A}}$ is of type (FPV). This generalizes various current results and also gives an affirmative answer to a problem posed by Borwein and Yao. Moreover, we p…
We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces. To do this we prove the weak-weak type modular inequality of the Hardy-Littlewood maximal operator with respect to the Young function. Orlicz-Morrey spaces contain $L^p$ spaces ($1\le p\le\infty$), Orlicz spaces and generalized Morrey spaces as special cases. Hence we get necessary and sufficient conditions on these function spaces as corollaries.
We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.
We study the relations between some geometric properties of maximal monotone operators and generic geometric and analytical properties of the functions on the associate Fitzpatrick family of convex representations. We also investigate under which conditions a convex function represents a maximal monotone operator with bounded range and provide an example of a non type (D) operator on this class.
Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a $p$-Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among others, that a suitable maximal operator is locally weakly bounded. In particular, various suff…
In the present paper, we will characterize the boundedness of the generalized fractional integral operators $I_ρ$ and the generalized fractional maximal operators $M_ρ$ on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators $M_ρ$ and $I_ρ$ on generalized Orlicz--Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators $M_ρ$ and $I_ρ$ on generalized Orlicz--Morrey spaces.
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.
We study sharp growth conditions for the boundedness of the Hardy-Littlewood maximal function in the generalized Orlicz spaces. We assume that the generalized Orlicz function $ϕ(x, t)$ satisfies the standard continuity properties (A0), (A1) and (A2). We show that if the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space to itself then $ϕ(x,t)/ t^p$ is almost increasing for large $t$ for some $p>1$. Moreover we show that the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space $L^ϕ(\mathbb{R}^n)$ to itself if and only if $ϕ$ is weakly equiva…
The goal of this article is to give a positive answer to Rockafellar's maximality of the sum conjecture in the linear multi-valued operator case.
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds. In this paper, we establish the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximally monotone operators such that $\sta(\dom A)\cap\inte\dom B\neq\varnothing$, and $A$ is of type (FPV). We show that when also $\dom A$ is convex, the sum operator: $A+B$ is also of type (FPV). Our result generalizes and unifies several recent sum theorems.
Sufficient condition for boundedness of maximal operator on weighted generalized Orlicz spaces
本文给出加权广义 Orlicz 空间中 Hardy-Littlewood 极大算子有界的充分条件，即在标准条件外要求权重满足 $A_p$ 且相应函数近乎递增。
We provide a concise analysis about what is known regarding when the closure of the domain of a maximally monotone operator on an arbitrary real Banach space is convex. In doing so, we also provide an affirmative answer to a problem posed by Simons.
We consider generalized Orlicz-Morrey spaces $M_{Φ,φ}(\mathbb{R}^{n})$ including their weak versions $WM_{Φ,φ}(\mathbb{R}^{n})$. We find the sufficient conditions on the pairs $(φ_{1},φ_{2})$ and $(Φ, Ψ)$ which ensures the boundedness of the fractional maximal operator $M_α$ from $M_{Φ,φ_1}(\mathbb{R}^{n})$ to $M_{Ψ,φ_2}(\mathbb{R}^{n})$ and from $M_{Φ,φ_1}(\mathbb{R}^{n})$ to $WM_{Ψ,φ_2}(\mathbb{R}^{n})$. As applications of those results, the boundedness of the commutators of the fractional maximal operator $M_{b,α}$ with $b \in BMO(\mathbb{R}^{n})$ on the spaces $M_{Φ,φ}(\mathbb{R}^{n})$ is …
Recently in [1] a new class of maximal monotone operators has been introduced. In this note we study domain range properties as well as connections with other classes and calculus rules for these operators we called strongly-representable. While not every maximal monotone operator is strongly-representable, every maximal monotone NI operator is strongly-representable, and every strongly representable operator is locally maximal monotone, maximal monotone locally, and ANA. As a consequence the conjugate of the Fitzpatrick function of a maximal monotone operator is not necessarily a representati…
We show that every maximally monotone operator of Fitzpatrick-Phelps type defined on a real Banach space must be of dense type. This provides an affirmative answer to a question posed by Stephen Simons in 2001 and implies that various important notions of monotonicity coincide.
The generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping. In this article, by assuming that the operators are maximal monotone of Gossez type (D), we provide sufficient conditions of both interiority- and closedness-type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of Gossez type (D), too. This result will follow as a particular instance of a more gene…
We consider the global Morrey-type spaces with variable exponents and general function defining these spaces. In the case of unbounded sets, we prove boundedness of the Hardy-Littlewood maximal operator, potential type operator in these spaces.
The local equicontinuity of an operator $T:X\rightrightarrows X^{*}$ with proper Fitzpatrick function $φ_{T}$ and defined in a barreled locally convex space $X$ has been shown to hold on the algebraic interior of $\operatorname*{Pr}\,_{X}(\operatorname*{dom}φ_{T})$). The current note presents direct consequences of the aforementioned result with regard to the local equicontinuity of a maximal monotone operator defined in a barreled locally convex space.
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximal monotone operators such that $\dom A\cap\inte\dom B\neq\varnothing$, $A+N_{\overline{\dom B}}$ is of type (FPV), and $\dom A\cap\overline{\dom B}\subseteq\dom B$. The proof utilizes the Fitzpatrick function in an essential way.
Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exist a convex representation of the operator which is a fixed point of this conjugation.
In this paper, we first give some new characterizations of Muckenhoupt type weights through establishing the boundedness of maximal operators on the weighted Lorentz and Morrey spaces. Secondly, we establish the boundedness of sublinear operators including many interesting in harmonic analysis and its commutators on the weighted Morrey spaces. Finally, as an application, the boundedness of strongly singular integral operators and commutators with symbols in BMO space are also given.
We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of t…
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 …
The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+B$ provided that $A, B$ are maximally monotone and $A$ is a linear relation, as soon as Rockafellar's constraint qualification holds: $\dom A\cap\inte\dom B\neq\varnothing$. Moreover, $A+B$ is of type (FPV).
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.
In this paper, we show the strong and weak type boundedness of $T_{Ω,α}^A$ and $M_{Ω,α}^A$, the multilinear fractional integral operators and the corresponding fractional maximal operators, on the two weights weighted Morrey space. Furthermore, we also obtain the mulitilinear singular integral operators $T_Ω^A$ and the corresponding maximal operators $M_Ω^A$ are strong bounded operators on the same spaces.
We show a surprising connexion between a property of the inf convolution of a family of convex lower semicontinuous functions and the fact that the intersection of maximal cyclically monotone graphs is the critical set of a bipotential. We then extend the results from arXiv:math/0608424v4 to bipotentials convex covers, generalizing the notion of a bi-implicitly convex lagrangian cover. As an application we prove that the bipotential related to Coulomb's friction law is related to a specific bipotential convex cover with the property that any graph of the cover is non maximal cyclically monoton…
In this paper we prove that the Hardy-Littlewood maximal operator is bounded on Morrey spaces $\mathcal{M}_{1,λ}(\rn)$, $0 \le \la < n$ for radial, decreasing functions on $\rn$
We study new weighted estimates for the 2-fold product of Hardy-Littlewood maximal operators defined by $M^{\otimes}(f,g):= MfMg$. This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón-Zygmund operators, the bilinear Hardy-Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures $$ \Vert fg \Vert_{L^{p,\infty}\left(w_1^{p/p_1} w_2^{p/p_2}\right)} \le C \Vert f \Vert_{L^{p_1,\infty}(w_1 )} \Vert g \Vert_{L^{p_2,\…
We present a counterexample showing that the graphical limit of maximally monotone operators might not be maximally monotone. We also characterize the directional differentiability of the resolvent of an operator $B$ in terms of existence and maximal monotonicity of the proto-derivative of $B$.
We discuss the Hardy-Littlewood maximal operator on discrete Morrey spaces of arbitrary dimension. In particular, we obtain its boundedness on the discrete Morrey spaces using a discrete version of the Fefferman-Stein inequality. As a corollary, we also obtain the boundedness of some Riesz potentials on discrete Morrey spaces.
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+\partial f$ provided that $A$ is a maximally monotone linear relation, and $f$ is a proper lower semicontinuous convex function satisfying $\dom A\cap\inte\dom \partial f\neq\varnothing$. Moreover, $A+\partial f$ is of type (FPV). The maximal monotonicity of $A+\partial f$ when $\intdom A\cap\dom \partial f\neq\varnothing$ follows from a resu…
Boundedness of the Hardy-Littlewood maximal operator on generalized fofana spaces
本文引入了广义Fofana空间并研究了其基本性质，这些空间是广义Morrey空间的推广，进而证明了Hardy-Littlewood极大算子在这些新空间上的有界性。
Cones of monotone functions generated by a generalized fractional maximal function
本文研究广义分数次极大函数，引入相关函数空间与三类单调函数锥，给出锥的等价刻画与互覆盖条件，还构建了对应空间嵌入重排不变空间的准则并确定最优RIS。
We study nearly equal and nearly convex sets, ranges of maximally monotone operators, and ranges and fixed points of convex combinations of firmly nonexpansive mappings. The main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings. A striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular. Throughout, examples are provided to illustrate the theory. We also obtain detailed information on the domain and range of the resolven…
On the Maximal Monotone Operators in Hadamard Spaces
本文在Hadamard空间中引入$p$-Fenchel共轭并证明Fenchel-Young不等式，研究$p$-Fitzpatrick变换性质，并建立极大单调算子与特定凸函数类间的联系。
In this paper, new equivalence theorems for the boundedness of the composition of a quasilinear operator $T$ with the Hardy and Copson operators in weighted Lebesgue spaces are proved. The usefulness of the obtained results is illustrated in the case of weighted Hardy-type and weighted iterated Hardy-type inequalities.
A Beginner-Friendly Note on Maximal Monotone Operators
本文为初学者提供了一个关于最大单调算子的入门概述，介绍了相关泛函分析工具及Brézis定理的核心内容。
Maximally monotone operators play a key role in modern optimization and variational analysis. Two useful subclasses are rectangular (also known as star monotone) and paramonotone operators, which were introduced by Brezis and Haraux, and by Censor, Iusem and Zenios, respectively. The former class has useful range properties while the latter class is of importance for interior point methods and duality theory. Both notions are automatic for subdifferential operators and known to coincide for certain matrices; however, more precise relationships between rectangularity and paramonotonicity were n…
In this paper we derive the maximal subspace of positive numbers, for which the restricted maximal operator of Fejér means in this subspace is bounded from the Hardy space $H_{p}$ to the space $L_{p}$ for all $0<p\leq 1/2.$ Moreover, we prove that the result is in a sense sharp.
Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p is greater than 2, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Muller on this subject
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…
Let $M$ be a manifold with ends constructed in \cite{GS} and $Δ$ be the Laplace-Beltrami operator on $M$. In this note, we show the weak type $(1,1)$ and $L^p$ boundedness of the Hardy-Littlewood maximal function and of the maximal function associated with the heat semigroup $\M_Δf(x)=\sup_{t> 0} |\exp (-tΔ)f(x)| $ on $L^p(M)$ for $1 < p \le \infty$. The significance of these results comes from the fact that $M$ does not satisfies the doubling condition.
In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator $S$ on $\ell^{2}$ is skew. We show its domain is a proper subset of the domain of its adjoint $S^*$, and $-S^*$ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator $T$ on $L^{2}[0,1]$. We compare the domain of $T$ with the domain of its adjoint $T^*$ and show that the skew part of $T$ admits two distinct linear maximal monot…
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