Singular integral operators on tent spaces

We extend the recent results concerning boundedness of the maximal regularity operator on tent spaces. This leads us to develop a singular integral operator theory on tent spaces. Such operators have operator-valued kernels. A seemingly appropriate condition on the kernel is time-space decay measured by off-diagonal estimates with various exponents.Comment: modification of the introduction and references added as suggested by the refere

BCR algorithm and the T(b)T(b) theorem

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on $L^p$, $1<p<\infty$, and of a perfect dyadic singular integral operator. This allows to deduce a local $T(b)$ theorem for singular integral operators from the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Thiele, Tao and the first author.Comment: Change of title. New abstract and new introductio

Rank one perturbations and singular integral operators

We consider rank one perturbations $A_\alpha=A+\alpha(\cdot,\varphi)\varphi$ of a self-adjoint operator $A$ with cyclic vector $\varphi\in\mathcal H_{-1}(A)$ on a Hilbert space $\mathcal H$. The spectral representation of the perturbed operator $A_\alpha$ is given by a singular integral operator of special form. Such operators exhibit what we call 'rigidity' and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms $T_\varepsilon$ are uniformly (in $\varepsilon$) bounded operators from $L^2(\mu)$ to $L^2(\mu_\alpha)$, where $\mu$ and $\mu_\alpha$ are the spectral measures of $A$ and $A_\alpha$, respectively. As an application, a sufficient condition for $A_\alpha$ to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of $A$ with respect to $\varphi$. Some examples, like Jacobi matrices and Schr\"odinger operators with $L^2$ potentials are considered.Comment: 24 page

Exponential decay estimates for Singular Integral operators

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.Comment: To appear in Mathematische Annale