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Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces

Sergio Antonio Tozoni (2004)

Studia Mathematica

Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis ( e j ) j 1 . Given an operator T from L c ( X ) to L¹(X), we consider the vector-valued extension T̃ of T given by T ̃ ( j f j e j ) = j T ( f j ) e j . We prove a weighted integral inequality for the vector-valued extension of the Hardy-Littlewood maximal operator and a weighted Fefferman-Stein inequality between the vector-valued extensions of the Hardy-Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give sufficient...

Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator

S. Bloom, R. Kerman (1994)

Studia Mathematica

Necessary and sufficient conditions are given for the Hardy-Littlewood maximal operator to be bounded on a weighted Orlicz space when the complementary Young function satisfies Δ 2 . Such a growth condition is shown to be necessary for any weighted integral inequality to occur. Weak-type conditions are also investigated.

Weighted Sobolev-Lieb-Thirring inequalities.

Kazuya Tachizawa (2005)

Revista Matemática Iberoamericana

We give a weighted version of the Sobolev-Lieb-Thirring inequality for suborthonormal functions. In the proof of our result we use phi-transform of Frazier-Jawerth.

Weighted weak type inequalities for certain maximal functions

Hugo Aimar, Liliana Forzani (1991)

Studia Mathematica

We give an A_p type characterization for the pairs of weights (w,v) for which the maximal operator Mf(y) = sup 1/(b-a) ʃ_a^b |f(x)|dx, where the supremum is taken over all intervals [a,b] such that 0 ≤ a ≤ y ≤ b/ψ(b-a), is of weak type (p,p) with weights (w,v). Here ψ is a nonincreasing function such that ψ(0) = 1 and ψ(∞) = 0.

Zygmund's program: some partial solutions

Alexander Stokolos (2005)

Annales de l’institut Fourier

We present a simple criterion to decide whether the maximal function associated with a translation invariant basis of multidimensional intervals satisfies a weak type ( 1 , 1 ) estimate. This allows us to complete Zygmund’s program of the description of the translation invariant bases of multidimensional intervals in the particular case of products of two cubic intervals. As a conjecture, we suggest a more precise version of Zygmund’s program.

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