Rearrangement Inequalities

1972 ◽  
Vol 24 (5) ◽  
pp. 930-943 ◽  
Author(s):  
Peter W. Day
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Rearrangement Inequalities

In recent years a number of inequalities have appeared which involve rearrangements of vectors in Rn and of measurable functions on a finite measure space. These inequalities are not only interesting in themselves, but also are important in investigations involving rearrangement invariant Banach function spaces and interpolation theorems for these spaces [2; 8; 9].

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

2021 ◽  
Vol 40 (3) ◽  
pp. 5517-5526
Author(s):  
Ömer Kişi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ideal Convergence ◽  
Convergence In Measure ◽  
Ideal Form ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Convergence Of Sequences

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

scholarly journals Ultrabornological Bochner integrable function spaces

1993 ◽  
Vol 47 (1) ◽  
pp. 119-126
Author(s):  
J.C. Ferrando
Keyword(s):  
Normed Space ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Function Spaces ◽  
Nikodym Property ◽  
Finite Measure Space ◽  
Bochner Integrable Function

If (Ω, Σ, μ) is a finite measure space and X is a normed space such that X* has the Radon-Nikodym property with respect to μ, we show first that each space Lp(μ, x), 1 < p < ∞, is ultrabornological whenever μ is atomless. When μ is arbitrary, we prove later on that the space Lp(μ, X) is ultrabornological if X* has the Radon-Nikodym property with respect to μ and X is itself an ultrabornological space.

scholarly journals Spaces of measurable functions

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Piotr Niemiec
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Finite Measure ◽  
Metrizable Space ◽  
Equivalence Classes ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Completely Metrizable

AbstractFor a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

Nonexistence of Maxima for Perturbations of Some Inequalities with Critical Growth

1996 ◽  
Vol 39 (2) ◽  
pp. 227-237 ◽  
Author(s):  
Alexander R. Pruss
Keyword(s):  
Continuous Function ◽  
Measure Space ◽  
Finite Measure ◽  
Critical Growth ◽  
Extremal Functions ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Nonnegative Continuous Function ◽  
Image Position

AbstractWe study the question of nonexistence of extremal functions for perturbations of some sharp inequalities such as those of Moser-Trudinger (1971) and Chang- Marshall (1985). We shall show that for each critically sharp (in a sense that will be precisely defined) inequality of the formwhere is a collection of measurable functions on a finite measure space (I, μ) and O a nonnegative continuous function on [0, ∞), we have a continuous Ψ on [0, ∞) with 0 ≤ Ψ ≤ Φ, but withnot being attained even if the supremum in (1) is attained. We then apply our results to the Moser-Trudinger and Chang-Marshall inequalities. Our result is to be contrasted with the fact shown by Matheson and Pruss (1994) that if Ψ(t) = o(Φ(t) as t —> ∞ then the supremum in (2) is attained. In the present paper, we also give a converse to that fact.

L1-Sequential Convergence on Stone Spaces and Stone's Theorem on Unitary Semigroups

1975 ◽  
Vol 18 (2) ◽  
pp. 191-193
Author(s):  
J. B. Cooper
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Sequential Convergence ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Stone Spaces ◽  
Sequence Of Functions ◽  
Stone’S Theorem

The following theorem is a well-known tool in the study of measurable functions:Theorem. Let (M; μ) be a finite measure space and let (xn) be a sequence of functions in L1(M; μ) so that xn→0 in the norm of L1(M; μ). Then there is a sub-sequence so that xnk →0 pointwise almost everywhere on M.

scholarly journals The continuity principle in exponential type Orlicz spaces

1995 ◽  
Vol 137 ◽  
pp. 55-75 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peškir ◽  
M. Weber
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Linear Space ◽  
Exponential Type ◽  
Measure Space ◽  
Orlicz Spaces ◽  
Finite Measure ◽  
Continuity Principle ◽  
Measurable Functions ◽  
Finite Measure Space

In this section we shall present some facts in the background of the problem under our next consideration. Let (X, A, μ) be a finite measure space, and let B be a Banach space with a norm ‖ • ‖. Let M(μ) denote the linear space of all μ-measurable functions from X into R, and let T be a linear operator from B into M (μ).

scholarly journals Construction of Weights for Positive Integral Operators

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1004 ◽  
Author(s):  
Ron Kerman
Keyword(s):  
Weight Function ◽  
Measure Space ◽  
Integral Operators ◽  
Finite Measure ◽  
Fundamental Result ◽  
Averaging Operators ◽  
Measurable Functions ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Schur's Lemma

Let ( X , M , μ ) be a σ -finite measure space and denote by P ( X ) the μ -measurable functions f : X → [ 0 , ∞ ] , f < ∞ μ ae. Suppose K : X × X → [ 0 , ∞ ) is μ × μ -measurable and define the mutually transposed operators T and T ′ on P ( X ) by ( T f ) ( x ) = ∫ X K ( x , y ) f ( y ) d μ ( y ) and ( T ′ g ) ( y ) = ∫ X K ( x , y ) g ( x ) d μ ( x ) , f , g ∈ P ( X ) , x , y ∈ X . Our interest is in inequalities involving a fixed (weight) function w ∈ P ( X ) and an index p ∈ ( 1 , ∞ ) such that: (*): ∫ X [ w ( x ) ( T f ) ( x ) ] p d μ ( x ) ≲ C ∫ X [ w ( y ) f ( y ) ] p d μ ( y ) . The constant C > 1 is to be independent of f ∈ P ( X ) . We wish to construct all w for which (*) holds. Considerations concerning Schur’s Lemma ensure that every such w is within constant multiples of expressions of the form ϕ 1 1 / p − 1 ϕ 2 1 / p , where ϕ 1 , ϕ 2 ∈ P ( X ) satisfy T ϕ 1 ≤ C 1 ϕ 1 and T ′ ϕ 2 ≤ C 2 ϕ 2 . Our fundamental result shows that the ϕ 1 and ϕ 2 above are within constant multiples of (**): ψ 1 + ∑ j = 1 ∞ E − j T ( j ) ψ 1 and ψ 2 + ∑ j = 1 ∞ E − j T ′ ( j ) ψ 2 respectively; here ψ 1 , ψ 2 ∈ P ( X ) , E > 1 and T ( j ) , T ′ ( j ) are the jth iterates of T and T ′ . This result is explored in the context of Poisson, Bessel and Gauss–Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are defined through symmetric kernels K ( x , y ) = K ( y , x ) , so that T ′ = T . This means that only the first series in (**) needs to be studied.

scholarly journals Identification of Fully Measurable Grand Lebesgue Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-3 ◽  
Author(s):  
Giuseppina Anatriello ◽  
Ralph Chill ◽  
Alberto Fiorenza
Keyword(s):  
Lebesgue Space ◽  
Measure Space ◽  
Function Spaces ◽  
Lebesgue Spaces ◽  
Banach Function Spaces ◽  
Measurable Functions ◽  
Grand Lebesgue Spaces ◽  
Almost Everywhere ◽  
Function Norm

We consider the Banach function spaces, called fully measurable grand Lebesgue spaces, associated with the function norm ρ(f)=ess supx∈X⁡δ(x)ρp(x)(f), where ρp(x) denotes the norm of the Lebesgue space of exponent p(x), and p(·) and δ(·) are measurable functions over a measure space (X,ν), p(x)∈[1,∞], and δ(x)∈(0,1] almost everywhere. We prove that every such space can be expressed equivalently replacing p(·) and δ(·) with functions defined everywhere on the interval (0,1), decreasing and increasing, respectively (hence the full measurability assumption in the definition does not give an effective generalization with respect to the pointwise monotone assumption and the essential supremum can be replaced with the simple supremum). In particular, we show that, in the case of bounded p(·), the class of fully measurable Lebesgue spaces coincides with the class of generalized grand Lebesgue spaces introduced by Capone, Formica, and Giova.

scholarly journals Some Properties Related With L^0 (Ω,F,μ) Space

2020 ◽  
Vol 25 (2) ◽  
pp. 22-26
Author(s):  
Asawer Jabar ◽  
Noori Al-Mayahi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Metric Functions

The purpose of this paper is to investigate elementary properties of these measure and relation between these measure and we define a metric functions on the space of measurable functions and defined on finite measure space and we call the topology induced by p the topology of convergence in measure and we investigate now the connection between the convergence in metric of X and convergence in measure .    

scholarly journals Isometries of measurable functions

1981 ◽  
Vol 24 (1) ◽  
pp. 13-26 ◽  
Author(s):  
Michael Cambern
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Finite Measure ◽  
Measurable Operator ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Linear Isometries

Let (X, Σ, μ) be a σ-finite measure space and denote by L∞(X, K) the Banach space of essentially bounded, measurable functions F defined on X and taking values in a separable Hilbert space K. In this article a characterization is given of the linear isometries of L∞(X, K) onto itself. It is shown that if T is such an isometry then T is of the form (T(F))(x) = U(x)(φ(F))(x), where φ is a set isomorphism of Σ onto itself, and U is a measurable operator-valued function such that U(x) is almost everywhere an isometry of K onto itself. It is a consequence of the proof given here that every isometry of L∞(X, K) is the adjoint of an isometry of L1(x, K).

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