A maximal L_p -regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes

Let Z=(Zt)t≥0 be an additive process with a bounded triplet (0,0,Λt)t≥0. Then the infinitesimal generators of Z is given by time dependent nonlocal operators as follows: AZ(t)u(t,x)=limh↓0E[u(t,x+Zt+h-Zt)-u(t,x)]h=∫Rd(u(t,x+y)-u(t,x)-y·∇xu(t,x)1|y|≤1)Λt(dy).Suppose that for any Schwartz function φ on R^d whose Fourier transform is in Cc∞(Bcs\Bcs-1), there exist positive constants N, N₁, and N₂ such that ∫Rd|E[φ(x+r-1Zt)]|dx≤N0e-N1ts(r),∀(r,t)∈(0,1)×[0,T],and ‖ψμ(r-1D)φ‖L1(Rd)≤N2s(r),∀r∈(0,1).where s is a scaling function (Definition 2.4), c_s is a positive constant related to s, μ is a symmetric Lévy measure on R^d, ψ^μ(r^{- 1}D) φ(x) = F^{- 1}[ψ^μ(r^{- 1}ξ) F[φ]] (x) and ψμ(ξ):=∫Rd(eiy·ξ-1-iy·ξ1|y|≤1)μ(dy). In particular, above assumptions hold for Lévy measures Λ_t having a nice lower bound and μ satisfying a weak-scaling property (Propositions 3.3, 3.5, and 3.6). We emphasize that there is no regularity condition on Lévy measures Λ_t and they do not have to be symmetric. In this paper, we establish the L_p-solvability to the initial value problem ∂u∂t(t,x)=AZ(t)u(t,x),u(0,·)=u0,(t,x)∈(0,T)×Rd,where u is contained in a scaled Besov space Bp,qs;γ-2q(Rd) (see Definition 2.8) with a scaling function s, exponent p∈ (1 , ∞) , q∈ [1 , ∞) , and order γ∈ [0 , ∞). We show that equation (0.2) is uniquely solvable and the solution u obtains full-regularity gain from the diffusion generated by a stochastic process Z. In other words, there exists a unique solution u to equation (0.2) in Lq((0,T);Hpμ;γ(Rd)), where Hpμ;γ(Rd) is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution u satisfies ‖u‖Lq((0,T);Hpμ;γ(Rd))≤N‖u0‖Bp,qs;γ-2q(Rd),where N is independent of u and u. We finally remark that our operators A_Z(t) include logarithmic operators such as - a(t) log (1 - Δ) (Corollary 3.2) and operators whose symbols are non-smooth such as -∑j=1dcj(t)(-Δ)xjα/2 (Corollary 3.9).