Exponential decay estimates for Singular Integral operators
arXiv:1204.1666
Abstract
The following subexponential estimate for commutators is proved |[|\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\leq c\,e^{-\sqrt{α\, t\|b\|_{BMO}}}\, |Q|, \qquad t>0.\] where and are absolute constants, is a Calderón--Zygmund operator, is the Hardy Littlewood maximal function and is any function supported on the cube . It is also obtained \[|\{x\in Q: |f(x)-m_f(Q)|>tM_{1/4;Q}^#(f)(x) \}|\le c\, e^{-α\,t}|Q|,\qquad t>0,\] where is the median value of on the cube and $M_{1/4;Q}^#$ is Strömberg's local sharp maximal function. As a consequence it is derived Karagulyan's estimate \[|\{x\in Q: |Tf(x)|> tMf(x)\}|\le c\, e^{-c\, t}\,|Q|\qquad t>0,\] improving Buckley's theorem. A completely different approach is used based on a combination of "Lerner's formula" with some special weighted estimates of Coifman-Fefferman obtained via Rubio de Francia's algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calderón--Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calderón--Zygmund operators. On each case, will be replaced by a suitable maximal operator.
To appear in Mathematische Annalen