Exponential decay estimates for Singular Integral operators

Abstract: The following subexponential estimate for commutators is proved |[|{x\in Q: |[b,T]f(x)|>tM^2f(x)}|\leq c\,e^{-\sqrt{\alpha\, t|b|{BMO}}}\, |Q|, \qquad t>0.] where $c$ and $\alpha$ are absolute constants, $T$ is a Calder\'on--Zygmund operator, $M$ is the Hardy Littlewood maximal function and $f$ is any function supported on the cube $Q$. It is also obtained [|{x\in Q: |f(x)-m_f(Q)|>tM{1/4;Q}^#(f)(x) }|\le c\, e^{-\alpha\,t}|Q|,\qquad t>0,] where $m_f(Q)$ is the median value of $f$ on the cube $Q$ and $M_{1/4;Q}^#$ is Str\"omberg'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 Calder\'on--Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calder\'on--Zygmund operators. On each case, $M$ will be replaced by a suitable maximal operator.