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: (Formula presented.) Suppose that for any Schwartz function φ on R^d whose Fourier transform is in C_c^∞(Bc_s\Bc_s-1), there exist positive constants N₀, N₁, and N₂ such that (Formula presented.) and (Formula presented.) 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^-1D)φ(x)=F^-1ψ^μ(r^-1ξ)F[φ](x) and ψ^μ(ξ):=∫R_d(e^iy·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 (Formula presented.) where u₀ is contained in a scaled Besov space B_p,q^s;γ-2q(R^d) (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 L_q((0,T);H_p^μ;γ(R^d)), where H_p^μ;γ(R^d) is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution u satisfies (Formula presented.) where N is independent of u and u₀. 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=1^dc_j(t)(-Δ)x_j^α/2 (Corollary 3.9).