1. Introduction We are going to investigate the equation ut(t, x ... - ICMS

18Theorem 3.4. For any γ,ν,p,θ,b, there exist a constant c 0 > 0 suchthat, for any c ≥ c 0 , the operator(−L b,c ) γ : H 2γ+νp,θ→ H ν p,θis bounded and, for any u ∈ H γ+νp,θ,wehave||u|| 2γ+ν,p,θ ≤ N||(−L b,c ) γ u|| ν,p,θ ≤ N||u|| 2γ+ν,p,θ . (3.2)The main difficulty of the theory is that ∆ and L b,c do not commute.Theorem 3.5. For any b, γ, p, and θ, the operator L b,0 is the generatorof an analytic semigroup T bt acting in H γ p,θ .Notice that, if d =1, the theorem is quite simple. Indeed,(D 2 +(b − 1)D)(v(e x )) = (L b,0 v)(e x ),so that the properties of the semigroup related to L b,0 can be easily obtained from thewell-known properties of the semigroup related to the operator D 2 +(b − 1)D withconstant coefficients. However, for d ≥ 2, we do not know any easy way to deal withL b,0 .