Abstract. The purpose of this paper is twofold. We introduce a general maximal function on the Gaussian setting which dominates the Ornstein-Uhlenbeck maximal operator and prove its weak type (1, 1) by using a covering lemma which is halfway between Besicovitch and Wiener. On the other hand, by taking as a starting point the generalized Cauchy-Riemann equations, we introduce a new class of Gaussian Riesz Transforms. We prove, using the maximal function defined in the first part of the paper, that unlike the ones already studied, these new Riesz Transforms are weak type (1, 1) independently of their orders.

Abstract. In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in (0, ∞) then the one-sided Ap condition, A − p , is a sufficient condition for the singular integral to be bounded in L p (w), 1 < p < ∞, or from L 1 (wdx) into weak-L 1 (wdx) if p = 1. This one-sided Ap condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in (0, ∞). The two-sided version of this result is also obtained: Muckenhoupt's Ap condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in (−∞, 0) or in (0, ∞).

We obtain parabolic Besov smoothness improvement for temperatures on cylindrical regions based on Lipschitz domains. The results extend those for harmonic functions obtained by S. Dahlke and R. DeVore using the wavelet description of Besov regularity.

We prove that having a quasi-metric on a given set X is essentially equivalent to have a family of subsets S(x, r) of X for which y ~ S(x, r) implies both ,..q(y, r) C S(x, Kr) and S(x, r) C S(y, Kr) for some constant K. As an application, starting from the Monge-Ampbre setting introduced in [3], we get a space of homogeneous type modeling the real analysis for such an equation.