Resumen de Singular integral operators and rectifiability
The problems that we study in this thesis lie in the area of Harmonic Analysis and Geometric Measure Theory. Namely, we consider the connection between the analytic properties of singular integral operators defined in $L^2(\mu)$ and associated with some Calderón-Zygmund kernels and the geometric properties of the measure $\mu$. Let us be more precise.
Let $E$ be a Borel set in the complex plane with non-vanishing and finite linear Hausdorff measure $H^1$, i.e. such that $0 < H^1(E) < \infty$. G. David and J.C. Léger (1999) proved that the Cauchy kernel $1/z$ (and even its real part $(\Re z)/|z|^2$) has the following property: the $L^2(H^1_E)$-boundedness of the corresponding singular integral operators implies that $E$ is rectifiable. Later on, V. Chousionis, J. Mateu, L. Prat and X. Tolsa (2012) proved the same property for the kernel $(\Re z)^3/|z|^4$. Moreover, there are examples of kernels due to P. Huovinen (2001) and B. Jaye and F. Nazarov (2013) such that the corresponding singular integral operators are $L^2(H^1_E)$-bounded for some purely unrectifiable sets $E$, i.e. the above-mentioned property does not hold.
In the thesis, we present our results related to the behaviour of singular integral operators associated with the class of Calderón-Zygmund kernels $(\Re z)^3/|z|^4+t\cdot (\Re z)/|z|^2$, where $t$ is a real parameter. It is shown that this class of kernels generalizes all above-mentioned ones considered by different authors. Furthermore, we prove that the property “$L^2$-boundedness implies rectifiability” holds for the operators with $t\in (-\infty,-\sqrt{2}) \cup (-t_0,+\infty]$, where $t_0>0$ is a small absolute constant. It is important that for some of the $t$ just mentioned the so called curvature method commonly used to relate $L^2$-boundedness and rectifiability is not available but it is still possible to establish the above-mentioned property. To the best of our knowledge, it is the first example of this type in the plane.
It is also worth mentioning that we extend our results to even more general class of kernels and additionally consider analogous problems for Ahlfors-David regular sets $E$.
