Summary: We study the semigroups \((P^{(\lambda)}_t)_{t\geq 0}\) generated by the operators \[ L^{(\lambda)}= \sum^n_{i,j=1} (\delta_{ij}- x_i x_j){\partial^2\over\partial x_i \partial x_j}- \lambda\sum^n_{i=1} x_i{\partial\over\partial x_i} \] on the open unit ball of \(\mathbb{R}^n\) with the measure \((1-\| x\|^2)^{(\lambda- n-1)/2}dx_1\cdots dx_n\). For \(\lambda\geq n\) we prove, via elementary method using essentially a commutation property between the generator and the gradient, a logarithmic Sobolev inequality. We establish by the same method, in the circle case, a family of inequalities recovering the Sobolev and Onofri inequalities.