Controlant la integral singular maximal

Resumen

The main objects of study of this dissertation are the singular integrals. We find an special motivation in three papers originated from the idea of bounding the norm of maximal operator of a singular integral by the norm of the singular integral itself. In the first one, of J. Mateu and J. Verdera from 2006, [MV], they prove pointwise inequalities for the particular cases of the j-th Riesz transform and the Beurling transform. For the first time, one notice that we obtain different bounds depending on the parity of the kernel of each operator. A posteriori, in the papers of J. Mateu, J. Orobitg and J. Verdera from 2011, [MOV], and in [MOPV] from 2010 from same authors plus C. Pérez, they prove pointwise inequalities as the aforementioned for higher order Riesz transforms. In the first work they treat the case of operators with even kernel, and in the second one, they do the same but for odd kernels. Here is when Cotlar inequality takes shows of, because we can notice that the inequality for the even case is an improvement of this one In [MOV] they prove that, for even Calderón-Zygmund singular integrals with smooth kernel, the pointwise inequality of the maximal operator bounded by the operator itself is equivalent to the L^2 estimate and also to an algebraic condition on the kernel of the singular integral. For the odd operators, in [MOPV], it's proved the same result, but in the pointwise inequality we need the second iteration of the Hardy-Littlewood maximal operator. It was proved before, in [MV], that one cannot bound without this iteration in the case of the Riesz trasnform. From here on, in this dissertation we have been working on this kind of estimates. In the first chapter we give a positive answer to one open question in [MOV]. We prove that the L^p estimate (and the weighted L^p) is also equivalent to the pointwise inequality, not only with p=2. This results are reflected in [BMO1]. In the second chapter we work on another open question from the same paper. We deal with the same estimates but relaxing the regularity of the kernel. When we are in the plane, we give a good answer, setting an initial differenciability for the kernel. For higher dimensions, with n bigger than 2, we have a partial answer, in the sense that the initial regularity depends on the degree of a polynomial depending on the kernel. This means that we may should ask for a very big differentiability, but a finite one. In the third chapter, we give an example for which we can't bound de weak L^1 norm of the maximal function in terms of the L^1 norm of the operator. We give the case of a harmonic polynomial of degree 3 in the plane and we explain how we can generalize to all polynomials with odd degree in the plane. However, because of the difficult caracterization of the harmonic polynomials en higher dimensions, the problem in R^n, for n>2, is open. In the last chapter, we consider the same problem of pointwise estimating the maximal operator of a singular integral by the same operator, but in this case we define a new maximal where we truncate by cubes instead of balls. We work with the Beurling transform and we prove that we need the second iteration of the Hardy-Littlewood maximal operator, and that we can't replace it for the first iteration. This results are reflected in [BMO2]. Bibliography [BMO1] A. Bosch-Camós, J. Mateu, J. Orobitg, «L^p estimates for the maximal singular integral in terms of the singular integral», J. Analyse Math. 126 (2015), 287-306. [BMO2] A. Bosch-Camós, J. Mateu, J. Orobitg, «The maximal Beurling transform associated with squares», Ann. Acad. Sci. Fenn. 40 (2015), 215-226. [MOPV] J. Mateu, J. Orobitg, C. Perez, J. Verdera, «New estimates for the maximal singular integral», Int. Math. Res. Not. 19 (2010), 3658-3722. [MOV] J. Mateu, J. Orobitg, J. Verdera, «Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels», Ann. of Math. 174 (2011), 1429-1483. [MV] J. Mateu, J. Verdera, «L^p and weak L^1 estimates for the maximal Riesz transform and the maximal Beurling transform, Math. Res. Lett. 13 (2006), 957-966.