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Eigenfunctions of the Hardy-Littlewood maximal operator

Leonardo Colzani, Javier Pérez Lázaro (2010)

Colloquium Mathematicae

We prove that peak shaped eigenfunctions of the one-dimensional uncentered Hardy-Littlewood maximal operator are symmetric and homogeneous. This implies that the norms of the maximal operator on L(p) spaces are not attained.

Elementary operators on Banach algebras and Fourier transform

Miloš Arsenović, Dragoljub Kečkić (2006)

Studia Mathematica

We consider elementary operators x j = 1 n a j x b j , acting on a unital Banach algebra, where a j and b j are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families a j and b j , i.e. a j = a j ' + i a j ' ' ( b j = b j ' + i b j ' ' ), where all a j ' and a j ' ' ( b j ' and b j ' ' ) commute. The main tool is an L¹ estimate of the Fourier transform of a certain class of C c p t functions...

Elliptic functions, area integrals and the exponential square class on B₁(0) ⊆ ℝⁿ, n > 2

Caroline Sweezy (2004)

Studia Mathematica

For two strictly elliptic operators L₀ and L₁ on the unit ball in ℝⁿ, whose coefficients have a difference function that satisfies a Carleson-type condition, it is shown that a pointwise comparison concerning Lusin area integrals is valid. This result is used to prove that if L₁u₁ = 0 in B₁(0) and S u L ( S n - 1 ) then u | S n - 1 = f lies in the exponential square class whenever L₀ is an operator so that L₀u₀ = 0 and S u L implies u | S n - 1 is in the exponential square class; here S is the Lusin area integral. The exponential square theorem,...

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