RIESZ TRANSFORMS AND MULTIPLIERS FOR THE BESSEL-GRUSHIN OPERATOR

Abstract. We establish that the spectral multiplier M(Gα) associated to the differential oper-ator Gα = −∆x + m∑ j=1 α2j − 1/4 x2j − |x|2∆y on (0,∞)m × Rn, which we denominate Bessel-Grushin operator, is of weak type (1, 1) provided that M is in a suitable local Sobolev space. In order to do this we prove a suitable weighted Plancherel estimate. Also, we study Lp-boundedness properties of Riesz transforms associated to Gα, in the case n = 1