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

22in our range of θ.theorem forTherefore, instead of consideringĀ = M∆ ˜RM −1 f.Ã we may and will prove theNotice that, if f ∈ C ∞ 0 (R × Rd ) and t is not in the support of f(s) as a function of s,then ˜T s,t M −1 f(s) is infinitely differentiable in x andĀf(t) =M∆ ˜RM −1 f(t) =∫ t−∞M∆ ˜T s,t M −1 f(s) ds. (5.3)Moreover, the boundedness of Ā as an operator from L p (R,H γ p,θ ) to L p(R,H γ p,θ ) andpoinwise estimates of the operator M∆ ˜T s,t M −1 show that (5.3) holds for almost all toutside the support of f if f is a bounded H γ p,θ-valued function with compact support.In other words, for those t,∫Āf(t) =RK(t, s)f(s) ds,where the operator K(t, s) is defined by the formulaK(t, s)h = I t>s M∆ ˜T s,t M −1 h.Observe that by a general theorem, K(t, s) is a bounded operator from H γ p,θinto itselfwith norm less than N|t − s| −1 .