Exponential decay estimates for singular integral operators

Abstract

The following subexponential estimate for commutators is proved |{x ∈ Q : |[b, T]f(x)| > tM2 f(x)}| ≤ c e− √ α tkbkBMO |Q|, t > 0. where c and α 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 ⊂ Rn. We also obtain that |{x ∈ Q : |f(x) − mf (Q)| > tM# λn;Q(f)(x)}| ≤ c e−α t|Q|, t > 0, where mf (Q) is the median value of f on the cube Q and M# λn;Q is Str¨omberg’s local sharp maximal function with λn = 2−n−2 . As a consequence we derive Karagulyan’s estimate: |{x ∈ Q : |T f(x)| > tM f(x)}| ≤ c e−c t |Q| t > 0, from 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 type 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. In each case, M will be replaced by a suitable maximal operator.