Characterization of associate spaces of weighted Lorentz spaces with applications

Гогатишвили, Амиран; Pick, Luboš; Soudský, Filip

doi:10.4064/sm224-1-1

Cited by 14 publications

(18 citation statements)

References 26 publications

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“…Remark 3.3(i)) points x k , y k ∈ (0, t k ) such that Define t k−1 := min{x k , y k }. Then (6) and (7) are both satisfied and one of the identities (8) and (9) holds true as well. One continues by induction, replacing k with k − 1 and repeating Step 2.…”

Section: Discretization Of the Copson-lorentz Functionalmentioning

confidence: 86%

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Embeddings and associated spaces of Copson—Lorentz spaces

Křepela

2020

JAMA

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Let m, p, q ∈ (0, ∞) and let u, v, w be nonnegative weights. We characterize validity of the inequalityfor all t > 0.By the characterization of linearity of Λ spaces [4, Theorem 1.4], CL m,p (u, v) is not a linear set. The Λ spaces are not always linear sets either, as seen above. The Γ spaces are at least linear spaces thanks to sublinearity of the mapping f → f * * , the functional · Γ p (v) however does not have to be a norm (see [10,15]). Nevertheless, the term "space" is used to describe all these structures, for the sake of simplicity.Suppose that · X : M → [0, ∞] is a functional such that λf X = |λ| f X and f X ≤ g X for all λ ∈ [0, ∞) and f, g ∈ M such that |f | ≤ |g| a.e. on R n . Let · Y be another functional with the same properties. Let X, Y be two function "spaces" given by (1) holds for all f ∈ M is called the optimal constant. Hence, if X is not embedded in Y , the optimal constant in (1) is infinite. Assume that X is moreover rearrangement-invariant, i.e., that f X = h X whenever f, h ∈ M are such that f * = h * on (0, ∞). Then the associated space of X, denoted X ′ , is defined asIf X is a Banach function space (see [2]), then X ′ is a Banach function space as well. However, to define X ′ in the way described above is possible even for a more general X, though without claiming space-like properties of X ′ . For more details, see [2].As it can be observed from the previous definition, embeddings into Λ spaces play a rather significant role since the associate "norm" g X of a function g ∈ M is equal to the optimal

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Section: Discretization Of the Copson-lorentz Functionalmentioning

confidence: 86%

“…One of the few examples of rearrangement-invariant spaces whose embedding into Λ was successfully and satisfactorily characterized are the Γ spaces, including their generalized variants. The results of this type were obtained in [6,7] by a method of discretization. Various steps in this direction were made even earlier, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Embeddings and associated spaces of Copson—Lorentz spaces

Křepela

2020

JAMA

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“…Using the assumption of integrability at the origin of u, one may show then by integration by parts that the above expression is equivalent to A (16) . While handling the second term in the sum, one also needs to use Proposition 2.1.…”

Section: Consider Now the Case (I) It Holdsmentioning

confidence: 99%

“…The "K-spaces" were defined in [18], where they appeared as optimal spaces in certain Youngtype convolution inequalities. Besides that, in [16] it was shown that the associate space to the generalized Γ space is also a "K-space". Now, let us briefly present some background to the problems we are about to investigate.…”

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confidence: 99%

Bilinear weighted Hardy inequality for nonincreasing functions

Křepela

2017

Publicacions Matemàtiques

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“…Then by[1] Theorem 7, we have GΓ(p, r; w) = (L (p,β , L (p,α ) θ,r .Thus, using duality relation of real interpolation spaces (see, for instance,[5] Theorem 3.7.1), we get[GΓ(p, r; w)] ′ = (L p ′ ),β , L p ′ ),α ) θ,r ′ = (L p ′ ),α , L p ′ ),β ) 1−θ,r ′ ,finally, an application of Theorem 3.6 completes the proof. In view of Lemma 2.4 , we get the following equivalent norm on [GΓ(p, r; w)] ′ :f [GΓ(p,r;w)] ′ ≈   Log t) − r ′ δ rwhich is apparently simpler than the one which follows from[14] Theorem 1.1 (vi).…”

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confidence: 99%

Some new results related to Lorentz GΓ-spaces and interpolation

Ahmed

Fiorenza

Formica

et al. 2020

Journal of Mathematical Analysis and Applications

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We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of [12]. This computation allows to determine the interpolation space in the sense of Peetre for such couple. It happens that the result is always a GΓ-space, since this last space covers many spaces. The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain Ω with data in the space of the integrable function with respect to the distance function to the boundary of Ω.

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Characterization of associate spaces of weighted Lorentz spaces with applications

Abstract: We characterize the associate spaces of weighted Lorentz spaces GΓ(p, m, w) and present some applications of this result including necessary and sufficient conditions for a Sobolev-type embedding into L ∞ .

Cited by 14 publications

References 26 publications

Embeddings and associated spaces of Copson—Lorentz spaces

Embeddings and associated spaces of Copson—Lorentz spaces

Bilinear weighted Hardy inequality for nonincreasing functions

Some new results related to Lorentz GΓ-spaces and interpolation

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