Real interpolation with logarithmic functors

We present reiteration formulae with limiting values θ = 0 and θ = 1 for a real interpolation method involving slowly varying functions. Applications to the Lorentz-Karamata spaces, the Fourier transform and the Riesz potential are given. In particular, our results yield improvements of limiting Sobolev-type embeddings due to Trudinger, Hansson, Brézis and Wainger, Edmunds, Gurka and Opic, Fusco, Lions and Sbordone, et al., and they are related to those of Edmunds, Kerman and Pick, or Pustylnik.

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“…We omit the proof since it can be done as in [16,Theorems 6.3,6.5, and 6.6], see also [20]. Now, we are able to consider the reiteration problem.…”

mentioning

confidence: 99%

Limiting reiteration for real interpolation with slowly varying functions

Гогатишвили¹,

Opic²,

Trebels

2004

Mathematische Nachrichten

124

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“…were established in [6,9,[11][12][13]15,17,21] for the parameters 0 ≤ θ 0 < θ 1 ≤ 1, and 0 ≤ θ ≤ 1; similar theorems were proven in [18,20] for the spaces (X θ 0 ,b 0 ,E 0 , X θ 1 ,b 1 ,E 1 ) θ,b,E . In [1] sharp reiteration theorems were proven when 0 ≤ θ 0 < θ 1 ≤ 1 and θ ∈ {0, 1}.…”

Section: Introductionmentioning

confidence: 89%

Limiting reiteration for real interpolation with logarithmic functions

Doktorski

2016

Bol. Soc. Mat. Mex.

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The real interpolation method X¯¯¯¯θ,q,b=(X0,X1)θ,q,b involving iterated logarithms with any number of iterations is considered. Reiteration relations of the types (X0,X¯¯¯¯0,q,a)θ,r,b and (X¯¯¯¯1,q,aX1)θ,r,b (0≤θ≤1) are investigated. Using any number of iterations allows in particular obtaining effects where the resulting space includes three iterated logarithms although the initial scale includes only the uniterated logarithm. Application to Lorentz–Zygmund spaces is given

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“…However, to get a meaningful definition they need to insert powers of the function g(t) = 1 + | log t| (see, for example, [18] or [19]) or, more generally, certain slowly varying function (see [1], [22]). We do not use here any weight of these kinds but the suitable term with the supremum which corresponds to the norm of the Gagliardo completion of A j in A 0 + A 1 (j = 0, 1) (see [3]).…”

Section: Limiting K-spacesmentioning

confidence: 99%

New limiting real interpolation methods and their connection with the methods associated to the unit square

Cobos¹,

Fernández─Cabrera²,

Silvestre³

2012

Mathematische Nachrichten

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over the unit square MSC (2010) Primary: 46B70; Secondary: 46E30Dedicated to Professor Hans Triebel on the occasion of his 75th birthday.We study limit K-spaces for general Banach couples, not necessarily ordered. They correspond to the extreme choice θ = 0, 1 in the realization of the real method as a K-space. We also show the connection of these limit spaces with interpolation methods defined by the unit square.

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Real interpolation with logarithmic functors

Cited by 47 publications

References 1 publication

Limiting reiteration for real interpolation with slowly varying functions

Limiting reiteration for real interpolation with slowly varying functions

Limiting reiteration for real interpolation with logarithmic functions

New limiting real interpolation methods and their connection with the methods associated to the unit square

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