Fe b 20 15 A NEW APPROACH TO THE L p - THEORY OF − ∆ + b · ∇ , AND ITS APPLICATIONS TO FELLER PROCESSES WITH GENERAL DRIFTS

We develop a detailed regularity theory of −∆+ b · ∇ in L(R), for a wide class of vector fields. The L-theory allows us to construct associated strong Feller process in C∞(R ). 1. Let Ld be the Lebesgue measure on Rd, Lp = Lp(Rd,Ld) and W 1,p = W 1,p(Rd,Ld) the standard Lebesgue and Sobolev spaces, C0,γ = C0,γ(Rd) the space of Hölder continuous functions (0 < γ < 1), Cb = Cb(R d) the space of bounded continuous functions endowed with the sup-norm, C∞ ⊂ Cb the closed subspace of functions vanishing at infinity, Wα,p, α > 0, the Bessel space endowed with norm ‖u‖p,α := ‖g‖p, u = (1 − ∆)− α 2 g, g ∈ Lp, and W−α,p the dual of Wα,p. We denote by B(X,Y ) the space of bounded linear operators between complex Banach spaces X → Y , endowed with operator norm ‖ · ‖X→Y ; B(X) := B(X,X). Set ‖ · ‖p→q := ‖ · ‖Lp→Lq . For each p > 1 and p′ = p/(p − 1), by 〈u, v〉 we denote the (Lp, Lp) pairing, so that 〈u, v〉 = 〈uv̄〉 := ∫ Rd uv̄dL (u ∈ L, v ∈ Lp). 2. Let us define the following classes of vector fields. (1) We say that a b : Rd → Cd belongs to Fδ, the class of form-bounded vector fields, and write b ∈ Fδ, if b is Ld-measurable, and there exists λ = λδ > 0 such that ‖b(λ−∆)− 1 2‖2→2 6 √ δ. (2) We say that a b : Rd → Cd belongs to the Kato class K δ , and write b ∈ K δ , if b is Ld-measurable, and there exists λ = λδ > 0 such that ‖b(λ−∆)− 12 ‖1→1 6 δ. (3) We say that a b : Rd → Cd belongs to F 1 2 δ , the class of weakly form bounded vector fields, and write b ∈ F 1 2 δ , if b is Ld-measurable, and there exists λ = λδ > 0 such that ‖|b| 12 (λ−∆)− 14 ‖2→2 6 √ δ. Simple examples show: Fδ1 −K δ 6= ∅ and K δ1 − Fδ 6= ∅ for any δ, δ1 > 0; K δ ( F 1 2 δ , Fδ1 ( F 1 2 δ for δ1 > √ δ by Heinz inequality [He]; 2010 Mathematics Subject Classification. 35J15, 47D07 (primary), 35J75 (secondary).