Details

Autor(en) / Beteiligte

• Ortiz-Caraballo, Carmen

Titel

Quadratic A1Bounds for Commutators of Singular Integrals with BMO Functions

Ist Teil von

• Indiana University mathematics journal, 2011-01, Vol.60 (6), p.2107-2129

Ort / Verlag

Department of Mathematics of Indiana University

Erscheinungsjahr

2011

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• For any Calderón-Zygmund operator T and any BMO function b we prove the following quadratic estimate: ${\Vert [\mathrm{b},\mathrm{T}]\Vert }_{{\mathrm{L}}^{\mathrm{p}}\left(\mathrm{w}\right)}\le \mathrm{c}{\Vert \mathrm{b}\Vert }_{\mathrm{B}\mathrm{M}\mathrm{O}}{(\mathrm{p}\mathrm{p}\prime )}^{2}{\left[\mathrm{w}\right]}_{{\mathrm{A}}_{1}}^{2}, \quad 1<\mathrm{p}<\mathrm{\infty },\mathrm{w}\in {\mathrm{A}}_{1}$, with constant c = c(n, T) being the estimate optimal on p and the exponent of the weight constant. As an endpoint estimate we prove $\begin{array}{c}\mathrm{w}\left(\{\mathrm{x}\in {\mathrm{\mathbb{R}}}^{\mathrm{n}}:\left|[\mathrm{b},\mathrm{T}]\mathrm{f}\left(\mathrm{x}\right)\right|>\mathrm{\lambda }\}\right)\\ \le \mathrm{c}\mathrm{\Phi }([\mathrm{w}{]}_{{\mathrm{A}}_{1}}{)}^{2}{\int }_{{\mathrm{\mathbb{R}}}^{\mathrm{n}}}\mathrm{\Phi }\left(\frac{\left|\mathrm{f}\left(\mathrm{x}\right)\right|}{\mathrm{\lambda }}\right)\mathrm{w}\left(\mathrm{x}\right)\mathrm{d}\mathrm{x}\end{array}$, where Φ(t) = t(1 + log+ t) and constant c = c(n, T, ∥b∥BMO).