arXiv subjects
Publications and source records attributed to Xavier Tolsa.
In this paper we consider the question of whether the doubling character of a measure supported on a subset of $\RR^m$ determines the regularity of its support (in a classical sense). This problem was studied by David, Kenig and Toro for codimension 1 sets under the assumption that the support be flat. Here we study the higher codimension case and remove the flatness hypothesis.
Let $γ(E)$ be the analytic capacity of a compact set $E$ and let $γ_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $γ(E)\approxγ_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $γ$ is semiadditive, which solves a long standing question of Vitushkin.
Let $μ$ be a Borel measure on $R^d$ which may be non doubling. The only condition that $μ$ must satisfy is $μ(B(x,r))\leq C r^n$ for all $x\in R^d$, $r>0$, and for some fixed $0<n\leq d$. In this paper, we develop Littlewood-Paley theory for functions in $L^p(μ)$. One of the main difficulties is the construction of reasonable approximations of the identity for obtaining a Calderon type reproducing formula. Moreover, it is shown that the T(1) theorem for n-dimensional Calderon-Zygmund operators, without doubling assumptions, can be proved using the Littlewood-Paley decomposition that is obtained for $L^2(μ)$ functions, as in the classical case of homogeneous spaces.
Let $μ$ be a Radon measure on $R^d$, which may be non doubling. The only condition that $μ$ must satisfy is $μ(B(x,r))\leq C r^n$, for all $x,r$ and for some fixed $0<n\leq d$. Recently we introduced spaces of type $BMO(μ)$ and $H^1(μ)$ which proved to be useful to study the $L^p(μ)$ boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic space $H^1(μ)$ in terms of a grand maximal operator $M_Φ$ is given. It is shown that $f$ belongs to $H^1(μ)$ iff $f\in L^1(μ)$, $\int f dμ=0$ and $M_Φ(f)\in L^1(μ)$, as in the usual doubling situation. The lack of any regularity condition on $μ$, apart from the size condition stated above, is one of the main difficulties that appears when one tries to extend the classical arguments to the present situation.
Given a doubling measure $μ$ on $R^d$, it is a classical result of harmonic analysis that Calderon-Zygmund operators which are bounded in $L^2(μ)$ are also of weak type (1,1). Recently it has been shown that the same result holds if one substitutes the doubling condition on $μ$ by a mild growth condition on $μ$. In this paper another proof of this result is given. The proof is very close in spirit to the classical argument for doubling measures and it is based on a new Calderon-Zygmund decomposition adapted to the non doubling situation.
Given a Radon measure $μ$ on $R^d$, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties that hold when $μ$ is doubling remain valid for the space BMO introduced in this paper, without assuming $μ$ doubling. For instance, Calderon-Zygmund operators which are bounded in $L^2$ are bounded from $L^\infty$ into the new BMO space. Moreover, a John-Nirenberg inequality is satisfied, and the predual of BMO is an atomic space $H^1$. Using a sharp maximal function it is proved that operators bounded from $L^\infty$ into BMO and from $H^1$ into $L^1$ are also bounded on $L^p$, $1<p<\infty$. This result gives a new proof of the T(1) theorem for the Cauchy transform with non doubling measures. Finally, a result about commutators is obtained.