Weak-type Inequalities for higher order Riesz-Laguerre Transforms

FORZANI, L; SASSO, E.; SCOTTO, R.

JOURNAL OF FUNCTIONAL ANALYSIS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2009 p. 1 - 12

0022-1236

Let Rm denote the mth Riesz-Laguerre transform with m 2 Zd0, associated with the multidimensionalLaguerre operator L, where  = (1, . . . , d) is amulti-index with i  0, i = 1, 2, . . . , d.In a previous paper [Bull. Austral. Math. Soc. 75 (2007), no. 3, 397–408; MR2331017(2008i:42032)], the second author of the paper under review proves that the first-order Riesz-Laguerre transforms are of weak type (1, 1). Also, a counterexample is given which shows thatRiesz-Laguerre transforms of order at least three are not of weak type (1, 1) with respect to theLaguerre measure μ.In the present paper, the authors investigate the weak type (1, 1) boundedness of higher-orderRm . In particular, by splitting the modified Riesz-Laguerre transforms of second order into a localoperator and a global one, the authors prove that the second-order Riesz-Laguerre transforms mapL1(dμ) continuously into L1,1(dμ). In addition, the authors find the sharp polynomial weight! that makes the Riesz-Laguerre transforms of order greater than two continuous from L1(!dμ)into L1,1(dμ).