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BMO and Lipschitz approximation by solutions of elliptic equations

Joan Mateu, Yuri Netrusov, Joan Orobitg, Joan Verdera (1996)

Annales de l'institut Fourier

We consider the problem of qualitative approximation by solutions of a constant coefficients homogeneous elliptic equation in the Lipschitz and BMO norms. Our method of proof is well-known: we find a sufficient condition for the approximation reducing matters to a weak * spectral synthesis problem in an appropriate Lizorkin-Triebel space. A couple of examples, evolving from one due to Hedberg, show that our conditions are sharp.

Bohr Cluster Points of Sidon Sets

L. Ramsey (1995)

Colloquium Mathematicae

It is a long standing open problem whether Sidon subsets of ℤ can be dense in the Bohr compactification of ℤ ([LR]). Yitzhak Katznelson came closest to resolving the issue with a random process in which almost all sets were Sidon and and almost all sets failed to be dense in the Bohr compactification [K]. This note, which does not resolve this open problem, supplies additional evidence that the problem is delicate: it is proved here that if one has a Sidon set which clusters at even one member of...

Boundedness from H 1 to L 1 of Riesz transforms on a Lie group of exponential growth

Peter Sjögren, Maria Vallarino (2008)

Annales de l’institut Fourier

Let G be the Lie group 2 + endowed with the Riemannian symmetric space structure. Let X 0 , X 1 , X 2 be a distinguished basis of left-invariant vector fields of the Lie algebra of G and define the Laplacian Δ = - ( X 0 2 + X 1 2 + X 2 2 ) . In this paper we consider the first order Riesz transforms R i = X i Δ - 1 / 2 and S i = Δ - 1 / 2 X i , for i = 0 , 1 , 2 . We prove that the operators R i , but not the S i , are bounded from the Hardy space H 1 to L 1 . We also show that the second-order Riesz transforms T i j = X i Δ - 1 X j are bounded from H 1 to L 1 , while the transforms S i j = Δ - 1 X i X j and R i j = X i X j Δ - 1 , for i , j = 0 , 1 , 2 , are not.

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