Comparison of the classical BMO with the BMO spaces associated with operators and applications

Deng , Donggao ; Duong , Xuan Thinh ; Sikora , Adam ; Yan , Lixin

Rev. Mat. Iberoamericana, Tome 24 (2008) no. 2, p. 267-296 / Harvested from Project Euclid

Résumé

Let $L$ be a generator of a semigroup satisfying the Gaussian upper bounds. A new ${\rm BMO}_L$ space associated with $L$ was recently introduced in [Duong, X. T. and Yan, L.: {New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications}. \textit{Comm. Pure Appl. Math.} {\bf 58} (2005), 1375-1420] and [Duong, X. T. and Yan, L.: {Duality of Hardy and BMO spaces associated with operators with heat kernels bounds}. \textit{J. Amer. Math. Soc.} {\bf 18} (2005), 943-973]. We discuss applications of the new ${\rm BMO}_L$ spaces in the theory of singular integration. For example we obtain ${\rm BMO}_L$ estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space ${\rm BMO}_L$ might coincide with or might be essentially different from the classical BMO space.

Détail

Publié le : 2008-04-15
Classification:  BMO space,  Hardy space,  Dirichlet and Neumann Laplacians,  semigroup,  Gaussian bounds,  fractional powers,  purely imaginary powers,  spectral multiplier,  42B35,  42B25,  47B38