Does an elliptic operator with unbounded coefficients generate an L1 semigroup?

Why p = 1 is the hard endpoint

The same operator had already been studied in Lp(R^d) for 1 < p < ∞, where it was shown to generate positive analytic semigroups with characterized domains, consistency and compactness for positive times. The open case was the endpoint p = 1, and the paper explains why it does not follow by taking a limit.

Two tools used for 1 < p < ∞ break down at p = 1. The Calderón–Zygmund estimate used to control all second derivatives is not valid in L1. In the analyticity proof, the equation is tested against u|u|^{p−2} and the real and imaginary parts are compared; after a shift and integration by parts, the negative real part controls a nonnegative gradient term whose coefficient is p − 1. At p = 1 that coefficient vanishes, so the argument no longer produces the needed estimate for a mixed gradient term in the imaginary part.

The paper's route around this is to keep the resolvents built for the Lp case as a tool, and to let the potential term −c|x|^β do the controlling. The condition β > α − 2 is what allows that term to dominate the order |x|^{α−2} contributions coming from the diffusion and the drift.

What the generator domain actually contains

The domain D(Λ) is defined as the set of u in W^{1,1}(R^d) such that Δu lies in L1_loc(R^d) and the three weighted quantities |x|^β u, (1+|x|^{α−1})∇u and (1+|x|^α)Δu all lie in L1(R^d), with derivatives understood in the distributional sense. This is not a generic Sobolev space: the weights are tied to the growth exponents α and β of the operator itself.

The paper proves that the norm built from these weighted pieces, namely ‖u‖1 + ‖|x|^β u‖1 + ‖(1+|x|^{α−1})∇u‖1 + ‖(1+|x|^α)Δu‖1, is equivalent to the graph norm ‖u‖1 + ‖Λu‖1 on D(Λ). It also shows that C∞_c(R^d) is a core for the closure, so the realization is determined by how Λ acts on smooth compactly supported functions.

A weighted interpolation estimate is the technical hinge here. For every ε > 0 there is a constant C_ε with ‖(1+|x|^{α−1})∇u‖1 ≤ ε‖(1+|x|^α)Δu‖1 + C_ε‖(1+|x|^{α−2})u‖1. Combined with the Kato-type inequality and the fact that β > α − 2, this yields L1 control of u, ∇u and Δu and produces the explicit domain description.

Positivity comes from the resolvent

Generation is proved through the Lumer–Phillips theorem. A Kato-type inequality shows that, for every u in C∞_c(R^d;C), the real part of ∫ (u/|u|)Λu dx is bounded above by ∫ ((α−b)(d+α−2)|x|^{α−2} − c|x|^β)|u| dx. With ω1 defined as the supremum over x in R^d of (α−b)(d+α−2)|x|^{α−2} − c|x|^β, this gives (λ−ω1)‖u‖1 ≤ ‖(λ−Λ)u‖1 for real λ > ω1, so Λ − ω1 is dissipative.

The resolvent set contains (ω1, ∞), with ‖R(λ,Λ)‖ ≤ (λ−ω1)^{−1}, and the resolvent is positive for every real λ > ω1. Positivity of the semigroup then follows: for nonnegative f one can approximate by nonnegative compactly supported functions, the resolvents stay nonnegative, and the exponential formula passes that to T1(t)f ≥ 0.

This matters because positivity is not an extra decoration here. It is what makes the L1 realization the natural object for integrable solutions of the associated evolution problem, and it is established through the resolvent rather than assumed.

The optimal exponential bound and contractivity threshold

The growth rate ω1 is not left as an abstract supremum. Setting γ := (α−b)(d+α−2), the paper computes ω1 = 0 when b ≥ α, and otherwise ω1 = ((β−α+2)/β) γ ((α−2)γ/(cβ))^{(α−2)/(β−α+2)}. The maximum of γr^{α−2} − cr^β over r ≥ 0 is attained at r* = ((α−2)γ/(cβ))^{1/(β−α+2)}, and at that point cr*^β = ((α−2)/β) γ r*^{α−2}.

The bound is optimal, not merely sufficient. If ‖T1(t)‖ ≤ e^{ωt} holds for some real ω, then testing against nonnegative compactly supported f with ‖f‖1 = 1 and letting t decrease to 0 forces γ|x|^{α−2} − c|x|^β ≤ ω for every x, hence ω1 ≤ ω.

A concrete threshold follows: ω1 ≥ 0 always, with equality exactly when b ≥ α. So the semigroup is contractive on L1(R^d) if and only if b ≥ α. The drift coefficient b, not the potential strength c alone, decides whether the semigroup grows or contracts.

Compactness for every positive time

The domain D(Λ) with the graph norm embeds compactly into L1(R^d). The proof splits the integral into a large ball and its complement: on each ball the Rellich–Kondrachov theorem gives compactness of W^{1,1}(B_j) into L1(B_j), and a diagonal subsequence converges locally; outside a ball of radius R the weighted bound gives sup_n ∫_{|x|>R} |u_n| dx ≤ C R^{−β}, which can be made arbitrarily small because β > 0.

Compactness of the embedding makes the resolvent R(λ,Λ) compact on L1(R^d) for every λ in the resolvent set. Analyticity is proved separately by a localization and weighted interpolation argument: for some θ in (π/2, π), some ω > ω1 and some M > 0, the sector ω + Σ_θ lies in the resolvent set with ‖R(λ,Λ)‖ ≤ M/|λ−ω|.

Combining the two gives compactness of the semigroup itself. For t > 0, analyticity places T1(t)L1(R^d) inside D(Λ) with ΛT1(t) bounded, so T1(t) = R(ω,Λ)(ωI − Λ)T1(t) is a compact operator composed with a bounded one. Hence T1(t) is compact for every positive time.

Consistency with the Lp semigroups

The L1 realization is not a separate object from the earlier Lp theory. For every 1 < p < ∞ the paper proves T1(t)f = Tp(t)f for f in L1(R^d) ∩ Lp(R^d) and all t ≥ 0, where Tp is the semigroup generated by Λp in the earlier work.

The argument runs through resolvents. For f in L1 ∩ Lp and λ ≥ λ*, one has R(λ,Λ)f = R(λ,Λp)f. This is first checked on C∞_c(R^d) using injectivity of λ − Λ, then extended by approximating f in both the L1 and Lp norms; the two resolvent sequences agree term by term and both converge in L1_loc, so their limits coincide. The same identification is then carried through the exponential formula to the semigroups.

The practical consequence is that the endpoint case does not contradict or replace the Lp results. On functions that belong to both spaces, the two semigroups agree pointwise in time, so the L1 theory extends the existing picture rather than competing with it.

Where the result applies and where it stops

The theorem is stated for d ≥ 3, α > 2, β > α − 2, b ∈ R and c > 0. These are not cosmetic hypotheses. The condition β > α − 2 is what lets the potential −c|x|^β control the order |x|^{α−2} terms from diffusion and drift, and it is also what makes ω1 finite. The requirement c > 0 keeps the potential present; the paper does not treat c = 0 in this L1 setting.

The paper also notes that earlier L1 generation and domain results for operators with constant diffusion and unbounded lower-order coefficients do not apply directly here. In that framework the assumptions imply a(x) ≤ C(1+|x|^2) and require Δa to be bounded; both fail for a(x) = 1 + |x|^α when α > 2, since a grows faster than |x|^2 and Δa = α(d+α−2)|x|^{α−2} is unbounded.

One further boundary is worth stating plainly: the compactness and analyticity conclusions are about the L1 realization on R^d under the stated parameter ranges. The paper does not claim the same endpoint behavior for other growth regimes, such as 0 ≤ α < 2, which are handled elsewhere in the literature under different assumptions.

What decides growth versus contraction

The most decision-relevant quantity in this paper is the pair (b, α). The contractivity criterion is exactly b ≥ α, and in that regime ω1 = 0. When b < α, the exponential bound is strictly positive and is given by the explicit formula in terms of γ = (α−b)(d+α−2), c, α and β, attained at r* = ((α−2)γ/(cβ))^{1/(β−α+2)}.

This is a sharper statement than saying the semigroup is bounded or unbounded. The paper identifies the smallest real ω for which ‖T1(t)‖ ≤ e^{ωt} holds for all t ≥ 0, so the comparison between drift strength b and diffusion exponent α is the threshold that determines whether the L1 semigroup contracts or grows, and by how much when it grows.

For a reader deciding whether this L1 theory applies to a given operator, the checklist is therefore concrete: check d ≥ 3, α > 2, β > α − 2 and c > 0; then compare b with α to know whether the semigroup is contractive; and use the weighted domain description to test whether a specific function lies in D(Λ).

Sources used in this answer