Small Furstenberg sets

For $\alpha$ in $(0,1]$, a subset $E$ of \RR is called Furstenberg set of type $\alpha$ or $F_\alpha$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is greater or equal than $\alpha$. In this paper we show that if $\alpha > 0$, there exists a set $E\in F_\alpha$ such that \HH{g}(E)=0 for $g(x)=x^{1/2+3/2\alpha}\log^{-\theta}(\frac{1}{x})$, $\theta>\frac{1+3\alpha}{2}$, which improves on the the previously known bound, that $H^{\beta}(E) = 0$ for $\beta>1/2+3/2\alpha$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for \h_\gamma(x)=\log^{-\gamma}(\frac{1}{x}), $\gamma>0$, we construct a set E_\gamma\in F_{\h_\gamma} of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any E\in F_{\h_\gamma}, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions \h_\gamma.Comment: Final versio

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

Improving dimension estimates for Furstenberg-type sets

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets.Comment: 16 pages, 3 figure