Pesos para integrales singulares con núcleo no estándar

M.S. RIVEROS

FaMAF, Cordoba

Congreso; Jornadas de Análisis Armónico y Grupos de Lie; 2006

FaMAF

This is a joint work with M. Lorente, J.M. Martell and C. Pérez.  In [1] and [ 2] it was considered a general framework to deal with singular integral operators and commutators of singular integral operators with BMO functions  where Hörmander type conditions associated with Young functions are assumed on the kernels. More precisely, the kernels  are those that satisfy the following definition Definition:Let Ke L1 loc( Rn-{ 0}), and let A be a Young function. We say that the kernel K satisfies the L A$-Hörmander condition  if there are positive numbers cA \geq1 and C A such that for any x and R>cA |x|,  (H A)   \sum m=1 (2mR)n||K(x-.)-K(-.)cS m,R(.)|| A,B m,R   \leq CA We say that a  kernel K satisfies the L\infty-Hörmander condition if there exist  C and c\geq 1, such that, for any x and R>c|x|, (H\infty) \sum m=1 (2mR)nsup{2mR\leq |y|<2m+1R} |K(x-y)-K(-y)|\leq  C. In these cited articles  are obtained Coifman type estimates, weighted norm inequalities and two-weight estimates. In the present work we give  weighted weak-type  estimates for pairs of weights (w, Sw) where w is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. One-sided singular integrals, as the differential transform operator, are under study. We also provide applications to Fourier multipliers and homogeneous singular integrals. Bibliography[1] M. Lorente, M.S. Riveros and A. de la Torre, Weighted estimatesfor singular integral operators satisfying Hörmander's conditionsof Young type,  J.  Fourier Anal. Apl. 11 (2005), no. 5, 497--509. [2] M. Lorente, J.M. Martell, M.S. Riveros and A. de laTorre,  Weighted norm inequalities for commutators of singularintegral operators satisfying Hömander's condition of Young type, preprint.