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

Public Capital and Productive Efficiency in the Spanish Regions (1964 1989)

The article analyses the evolution of the differences in economic conditions among Spanish regions from the perspective provided by the recent advances made in economic growth empirics. Although convergence is usually established in terms of Gross Value Added (GVA) per capita, in the case of Spain it is of special interest to break it down into three separate elements: activity rate, employment rate, and productivity of labour. Regional differences in unemployment rates, which persist for long periods of time, are identified as a force against convergence. After describing the distinction between conditional and non conditional convergence, the paper considers the role played by the productive structure in establishing regional differences, with special reference to the weight of the agricultural sector and the role of public capital in conditional convergence. Finally, conditional and non conditional convergence equations are estimated for the 17 Spanish regions for the 1955 1991 period. The paper concludes that the convergence process is concentrated in the first half of the period (1955 1979) and that both, the productive structure and public capital, had a significant role in the convergence process.Regional Convergence, Public Capital

Sharp reverse Hölder property for A∞ weights on spaces of homogeneous type

In this article we present a new proof of a sharp Reverse Hölder Inequality for A∞ weights. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy-Littlewood maximal function involving A∞ constants: kMkLp(w) ≤ c 1 p − 1 [w]Ap [σ]A∞ 1/p , where 1 < p < ∞, σ = w 1 1−p and c is a dimensional constant. Our approach allows us to extend the result to the context of spaces of homogeneous type and prove a weak Reverse H¨older Inequality which is still sufficient to prove the open property for Ap classes and the Lp boundedness of the maximal function. In this latter case, the constant c appearing in the norm inequality for the maximal function depends only on the doubling constant of the measure µ and the geometric constant κ of the quasimetric.European Research CouncilMinisterio de Ciencia e InnovaciónJunta de Andalucí