Displaying similar documents to “Solvability of the functional equation f = (T-I)h for vector-valued functions”

Invariance of the Gibbs measure for the Benjamin–Ono equation

Yu Deng (2015)

Journal of the European Mathematical Society

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In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than L 2 , extending the L 2 well-posedness result of Molinet [20].

On Ordinary and Standard Lebesgue Measures on

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on for α = (1,1,...).

Unique Bernoulli g -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

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We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a g -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique g -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the g -measure.

Algebraic genericity of strict-order integrability

Luis Bernal-González (2010)

Studia Mathematica

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We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space L p ( μ , X ) (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of L p ( μ , X ) (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...

The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

Mrinal Kanti Roychowdhury, Daniel J. Rudolph (2008)

Fundamenta Mathematicae

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Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) ϕ | X is continuous in the relative topology on X₀ and ϕ - 1 | Y is continuous in the relative topology on Y₀, (2) ϕ ( O r b T ( x ) ) = O r b S ( ϕ ( x ) ) for μ-a.e. x ∈ X. (X,,μ,T)...

Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar (2008)

Annales Polonici Mathematici

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Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure μ S : = a S ( 1 - | a | ² ) δ a is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure μ S bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual...

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets ν ( X ) and ν ( X ) of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace 𝒦 = 𝒦 T ( B ) affiliated with ℳ, such that the Brown measure of T | 𝒦 is concentrated...

Extensions of generic measure-preserving actions

Julien Melleray (2014)

Annales de l’institut Fourier

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We show that, whenever Γ is a countable abelian group and Δ is a finitely-generated subgroup of Γ , a generic measure-preserving action of Δ on a standard atomless probability space ( X , μ ) extends to a free measure-preserving action of Γ on ( X , μ ) . This extends a result of Ageev, corresponding to the case when Δ is infinite cyclic.

On Beurling measure algebras

Ross Stokke (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show how the measure theory of regular compacted-Borel measures defined on the δ -ring of compacted-Borel subsets of a weighted locally compact group ( G , ω ) provides a compatible framework for defining the corresponding Beurling measure algebra ( G , ω ) , thus filling a gap in the literature.

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set E with C 1 boundary and the spherical Hausdorff measure S Q 1 restricted to E , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and Q denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of E is less than or equal to S Q 1 ( E ) up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Relations between Shy Sets and Sets of ν p -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a non-zero non-atomic translation-invariant Borel measure ν p on the Banach space p ( 1 p ) is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " ν p -almost every element of p has a property P" implies that “almost every” element of p (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

Asymptotic nature of higher Mahler measure

(2014)

Acta Arithmetica

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We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure m k ( P ) of a polynomial P , where m k ( P ) is the integral of l o g k | P | over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding m k ( P ) , in particular | m k ( P ) | / k ! 1 / π as k → ∞.

Lattice copies of c₀ and in spaces of integrable functions for a vector measure

S. Okada, W. J. Ricker, E. A. Sánchez Pérez

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The spaces L¹(m) of all m-integrable (resp. L ¹ w ( m ) of all scalarly m-integrable) functions for a vector measure m, taking values in a complex locally convex Hausdorff space X (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, L ¹ w ( m ) is always a complex vector lattice; this is not necessarily so for L¹(m). To identify precisely when L¹(m) is also a complex vector lattice is one of our central aims. Whenever X is sequentially complete, then this is the case. If,...

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a new theorem on the differential inequality u ( m ) w ( u ) . Next, we apply this result to obtain existence theorems for the equation x ( m ) = f ( t , x ) .

On an Invariant Borel Measure in Hilbert Space

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold μ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift μ ( a ) of μ by a vector a μ are neither equivalent nor orthogonal. This extends a result established in [7].

Exact covering maps of the circle without (weak) limit measure

Roland Zweimüller (2002)

Colloquium Mathematicae

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We construct maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ( n - 1 k = 0 n - 1 ν T - k ) n 1 of arithmetical averages of image measures does not converge weakly.

On inhomogeneous self-similar measures and their L q spectra

Przemysław Liszka (2013)

Annales Polonici Mathematici

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Let S i : d d for i = 1,..., N be contracting similarities, let ( p , . . . , p N , p ) be a probability vector and let ν be a probability measure on d with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on d such that μ = i = 1 N p i μ S i - 1 + p ν . We give satisfactory estimates for the lower and upper bounds of the L q spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...

Less than 2 ω many translates of a compact nullset may cover the real line

Márton Elekes, Juris Steprāns (2004)

Fundamenta Mathematicae

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We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from c o f ( ) < 2 ω ) that less than 2 ω many translates of a compact set of measure zero can cover ℝ.

Pisier's inequality revisited

Tuomas Hytönen, Assaf Naor (2013)

Studia Mathematica

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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies - 1 , 1 | | j = 1 n j f j ( ε ) | | p d μ ( ε ) p - 1 , 1 - 1 , 1 | | j = 1 n δ j Δ f j ( ε ) | | p d μ ( ε ) d μ ( δ ) , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and j j = 1 n and Δ = j = 1 n j are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by p ( X ) , we show that p ( X ) k = 1 n 1 / k for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

Quantization Dimension Estimate of Inhomogeneous Self-Similar Measures

Mrinal Kanti Roychowdhury (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider an inhomogeneous measure μ with the inhomogeneous part a self-similar measure ν, and show that for a given r ∈ (0,∞) the lower and the upper quantization dimensions of order r of μ are bounded below by the quantization dimension D r ( ν ) of ν and bounded above by a unique number κ r ( 0 , ) , related to the temperature function of the thermodynamic formalism that arises in the multifractal analysis of μ.

The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions

Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into d . The paper deals with Y-weak cluster points ϕ̅ of the sequence ϕ ( · , z j ( · ) ) in X, where z j : Ω m is measurable for j ∈ ℕ and ϕ : Ω × m d is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set A ϕ , the integral I ( ϕ , ν x ) : = m ϕ ( x , λ ) d ν x ( λ ) exists for x Ω A ϕ and ϕ ̅ ( x ) = I ( ϕ , ν x ) on Ω A ϕ , where ν = ν x x Ω is a measurable-dependent family of Radon probability measures on m .

Generalized gradients for locally Lipschitz integral functionals on non- L p -type spaces of measurable functions

Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)

Banach Center Publications

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Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, E * ω * be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put G ( x ) : = Ω g ( s , x ( s ) ) d μ ( s ) . Consider the integral functional G defined on some non- L p -type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued...

Duality of measures of non-𝒜-compactness

Juan Manuel Delgado, Cándido Piñeiro (2015)

Studia Mathematica

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Let be a Banach operator ideal. Based on the notion of -compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non–compactness of an operator. We consider a map χ (respectively, n ) acting on the operators of the surjective (respectively, injective) hull of such that χ ( T ) = 0 (respectively, n ( T ) = 0 ) if and only if the operator T is -compact (respectively, injectively -compact). Under certain conditions on the ideal , we prove an equivalence inequality involving...

Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak (2005)

Banach Center Publications

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Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space ( X , | | · | | X ) let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function | | f ( · ) | | X belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let D u ( = f E ( X ) : | | f ( · ) | | X u ) stand for the order interval in E(X). For a real Banach space ( Y , | | · | | Y ) a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...

An M q ( ) -functional calculus for power-bounded operators on certain UMD spaces

Earl Berkson, T. A. Gillespie (2005)

Studia Mathematica

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For 1 ≤ q < ∞, let q ( ) denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes q ( ) as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction...

Sequentially Right Banach spaces of order p

Mahdi Dehghani, Mohammad B. Dehghani, Mohammad S. Moshtaghioun (2020)

Commentationes Mathematicae Universitatis Carolinae

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We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order p , and those defined by the dual property, the sequentially Right * Banach spaces of order p for 1 p . These classes of Banach spaces are characterized by the notions of L p -limited sets in the corresponding dual space and R p * subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space X and a reflexive Banach...

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let Ω be a countable infinite product Ω of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have E * [ X ] l i m i n f S / n l i m s u p S / n E * [ X ] , where E * and E* are the lower and upper expectations....