Bi-fonctions moment.
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.
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...
Let be the Lie group endowed with the Riemannian symmetric space structure. Let be a distinguished basis of left-invariant vector fields of the Lie algebra of and define the Laplacian . In this paper we consider the first order Riesz transforms and , for . We prove that the operators , but not the , are bounded from the Hardy space to . We also show that the second-order Riesz transforms are bounded from to , while the transforms and , for , are not.