We establish that the spectral multiplier M(Gα) associated
to the differential operator Gα=−Δx+j=1∑mxj2αj2−1/4−∣x∣2Δyon(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.Comment: 33 page