Hardy-Littlewood maximal operator in weighted Lorentz spaces Elona Agora IAM-CONICET Based on joint works with: J. Antezana, M. J. Carro and J. Soria Function Spaces, Differential Operators and Nonlinear Analysis Prague (Czech Republic), 4-9 July, 2016
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Hardy-Littlewood maximal operator in weighted Lorentz spacesfsdona.karlin.mff.cuni.cz/Talks/Agora.pdf · 2016. 7. 25. · Hardy-Littlewood maximal operator in weighted Lorentz spaces
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General case: Let S ⊂ I ⊂ R. Construct fS,I , so that for every λ ∈ [|S|/|I|, 1]
so that |S| = λ|Eλ|, where Eλ = {x : fS,I(x) > λ}. Then, for s = |I||S|
||MfS,I ||pΛp,∞u (w)
≤ C ||fS,I ||pΛpu(w)⇒ W(u(I))
W(u(S))≤ C (1 + log s)1−psp,
which “implies” that M : Λpu(w)→ Λp
u(w).
bg=whiteWeak-type boundedness of M
If S = S1 ∪ S2 ⊂ I ⊂ R, the function fS,I can be:
Several dimensions: We consider the function
χ{MχS>λ}MχS,
for any measurable set S ⊂ Rd.
bg=whitePossible problem to study
Recall that the Riesz transforms are given by
Rj f (x) =Γ( d+1
2 )
πd+1
2
limε→0
∫Rd\Bε(x)
xj − yj
|x− y|d+1 f (y) dy,
for j = 1, . . . , d, whenever they are well defined. Study theboundedness of the Riesz transforms on weighted Lorentz spaces.
bg=whiteSome references
1. E. Agora, J. Antezana, and M. J. Carro, The complete solution to theweak-type boundedness of Hardy-Littlewood maximal operator onweighted Lorentz spaces, To appear in J. Fourier Anal. Appl. (2016)
2. E. Agora, J. Antezana, M. J. Carro and J. Soria, Lorentz-Shimogaki andBoyd Theorems for weighted Lorentz spaces, J. London Math. Soc. 89(2014) 321-336.
3. E. Agora, M. J. Carro and J. Soria, Complete characterization of theweak-type boundedness of the Hilbert transform on weighted Lorentzspaces, J. Fourier Anal. Appl. 19 (2013) 712-730.
4. M. J. Carro, J. A. Raposo and J. Soria, Recent developments in thetheory of Lorentz Spaces and Weighted Inequalities, Mem. Amer.Math. Soc. 187 (2007) no. 877, xii+128.
5. G. Lorentz, Some new functional spaces, Ann. of Math. 51 (1950) no 2,37–55.