THE NON–AUTONOMOUS KATO CLASS

ROLAND SCHNAUBELT* AND JÜRGEN VOIGT^{†}

Abstract. We discuss singular time dependent absorption–excitation rates for second order parabolic equations. A suitable class, the non–autonomous Kato class, is defined for the heat equation in connection with the non–autonomous Miyadera perturbation theorem. We give sufficient conditions in terms of integrability for a potential to belong to the non–autonomous Kato class.

1. Introduction

The Kato class K of potentials V_{0} : ℝ^{n} → ℝ was introduced in 1982 by M. Aizenman and B. Simon, [1], [10], to study Schrödinger operators Δ + V_{0} for singular V_{0}. It is a special case of a Stummel class defined in terms of the fundamental solution of the Laplacian Δ whose importance for second order elliptic operators was realized by T. Kato in [5]. Aizenman and Simon proved that K coincides with the set of potentials having relative bound 0 with respect to Δ on L^{1}(ℝ^{n}), [1, Thm. 4.5]. Moreover, K is closely related to the set of potentials which are Miyadera perturbations of the semigroup (U(t))_{t≥0} on L^{1}(ℝ^{n}) generated by Δ, see [6, 11, 12, 13]. More precisely, the enlarged Kato class is defined by

K̂ := {V_{0} : ℝ^{n} → ℂ : V_{0} measurable, ∥V_{0}∥_{U} < ∞}, where
∥V_{0}∥_{U} := sup_{f∈D(Δ), ∥f∥_{1}≤1} ∫_{0}^{1} ∥V_{0}U(t)f∥_{1} dt.

(To be precise, in [6, 12, 13] the class K̂ was introduced for real potentials V_{0}. However, the results of these papers which are used below carry over to complex-valued V_{0} by considering |V_{0}|.) We remark that (K̂, ∥ · ∥_{U}) is a Banach space, see e.g. [13]. For V_{0} ∈ K̂ and α > 0, let

c′_{α}(V_{0}) := sup_{f∈D(Δ), ∥f∥_{1}≤1} ∫_{0}^{α} ∥V_{0}U(t)f∥_{1} dt and
c(V_{0}) := lim_{α→0} c′_{α}(V_{0}) = inf_{α>0} c′_{α}(V_{0}),

cf. [6, 13]. Then the Kato class is given by

K = {V_{0} ∈ K̂ : c(V_{0}) = 0},

[12, Prop. 4.7, 5.1]. In particular, if c(V_{0}) < 1 then Δ + V_{0} with domain D(Δ) generates a strongly continuous semigroup on X = L^{1}(ℝ^{n}) by virtue of the Miyadera perturbation theorem, see e.g. [11].

∗Support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

†Partly supported by the Deutsche Forschungsgemeinschaft.

1991 Mathematics Subject Classification. 35K10, 47A55, 47D06.

In the present note, we investigate perturbations of the Laplacian by time dependent potentials

V : [0, ∞) × ℝ^{n} → ℂ.

By V (t) we denote the multiplication operator with maximal domain induced by V (t, ·) on L^{1}(ℝ^{n}). As in the autonomous case, we set

∥V (·)∥_{U} := sup_{s≥0} sup_{f∈D(Δ), ∥f∥_{1}≤1} ∫_{0}^{1} ∥V (s + t)U(t)f∥_{1} dt,

and define the non–autonomous enlarged Kato class by

N̂K = {V : [0, ∞) × ℝ^{n} → ℂ : V measurable, ∥V (·)∥_{U} < ∞}.

Observe that ∥V (·)∥_{U} = ∥V_{0}∥_{U} for a time independent potential V (·) = V_{0}. We further introduce

c′_{α}(V (·)) := sup_{s≥0} sup_{f∈D(Δ), ∥f∥_{1}≤1} ∫_{0}^{α} ∥V (s + t)U(t)f∥_{1} dt and
c(V (·)) := lim_{α→0} c′_{α}(V (·)) = inf_{α>0} c′_{α}(V (·)),

for V ∈ N̂K and α > 0. We remark that, for V ∈ N̂K, the quantity c′_{α}(V (·)) does not change if we replace “f ∈ D(Δ)” by “f ∈ L^{1}(ℝ^{n})” in the second supremum, see [8, Thm. 3.4].

Recently, the Miyadera perturbation theorem has been extended to the non–autonomous situation in [7] and [8]. This result is applied in Theorem 1 to obtain mild solutions of the parabolic problem

\begin{aligned}\frac{∂}{∂t}u(t, x) & = ∑_{k,l=1}^{n} \frac{∂}{∂x_{k}} a_{kl}(t, x) \frac{∂}{∂x_{l}} u(t, x) + V (t, x)u(t, x), \\ u(s, x) & = f(x), t ≥ s, x ∈ ℝ^{n}, \end{aligned}  (1)

provided that V satisfies an estimate similar to c(V (·)) < 1. In the remainder of the paper we investigate the class N̂K. In Theorem 2 we see that the estimate c(V (·)) < q is equivalent to an integrability condition on V which involves the heat kernel

K_{t}(x) = (4πt)^{− \frac{n}{2}} e^{− \frac{|x|^{2}}{4t}} (t > 0, x ∈ ℝ^{n}).

Recall that U(t)f = K_{t} ∗ f. Further, it is shown in Theorem 6 that Hölder continuity and boundedness of t ↦ V (t) ∈ K̂ imply V ∈ N̂K. By means of examples, we first see that continuity is not sufficient for this conclusion and that, second, boundedness in K̂ is not necessary for V ∈ N̂K.

2. Results

First, we study the problem (1) in the space L^{1}(ℝ^{n}), where we assume that (A) a_{kl} ∈ L^{∞}([0, ∞) × ℝ^{n}, ℝ) and ∑_{k,l=1}^{n} a_{kl}(t, x)η_{k}η_{l} ≥ μ |η|^{2} for x, η ∈ ℝ^{n}, t ≥ 0,

k, l = 1, ⋯, n, and a constant μ > 0.

It is known that there exists a unique weak fundamental solution

Γ : {(t, x, s, y) ∈ [0, ∞) × ℝ^{n} × [0, ∞) × ℝ^{n} : t ≥ s} → [0, ∞)

of (1) with V = 0 which satisfies a Gaussian estimate

0 ≤ Γ(t, x, s, y) ≤ M K_{a(t−s)}(x − y) (x, y ∈ ℝ^{n}, t > s ≥ 0)  (2)

for some constants M, a > 0, see [2, Thm. 10], [3, Thm. 5.3], and [4, Thm. 1.6]. In [9, Lemma 5.1] it is shown that

(U(t, s)f)(x) := ∫_{ℝ^{n}} Γ(t, x, s, y)f(y) dy for t > s ≥ 0 and U(s, s) = Id for s ≥ 0

defines an evolution family on L^{1}(ℝ^{n}). This means that

(t, s) ↦ U(t, s) ∈ L(L^{1}(ℝ^{n})) is strongly continuous for t ≥ s ≥ 0 and
U(t, s) = U(t, r)U(r, s) and U(s, s) = Id for t ≥ r ≥ s ≥ 0.

Notice that the estimate (2) yields 0 ≤ U(t, s) ≤ M U(a(t − s)). We call a function u ∈ C([s, ∞), L^{1}(ℝ^{n})) a mild solution of the problem (1) if V (·)u(·) ∈ L_{loc}^{1}([s, ∞), L^{1}(ℝ^{n})) and

u(t) = U(t, s)f + ∫_{s}^{t} U(t, r)V (r)u(r) dr

for all t ≥ s. Let V_{a}(t) = V (t/a) for a > 0.

Theorem 1. Assume that (A) holds and that V : [0, ∞) × ℝ^{n} → ℂ satisfies c(V_{a}(·)) < a/M, where M, a > 0 are given by (2). Then there is a unique evolution family W (t, s), t ≥ s ≥ 0, on L^{1}(ℝ^{n}) satisfying

W (t, s)f = U(t, s)f + ∫_{s}^{t} W (t, τ)V (τ)U(τ, s)f dτ

for f ∈ L^{1}(ℝ^{n}) and t ≥ s ≥ 0. Further, W (·, s)f is the unique mild solution of (1) for all f ∈ L^{1}(ℝ^{n}).

Proof. Let c(V_{a}(·)) \frac{M}{a} < q < 1. Then we derive

∫_{0}^{α} ∥V (s + t)U(s + t, s)f∥_{1} dt ≤ \frac{M}{a} ∫_{0}^{aα} ∥V_{a}(as + τ)U(τ) |f| ∥_{1} dτ

≤ \frac{M}{a} c′_{aα}(V_{a}(·)) ∥f∥_{1} ≤ q ∥f∥_{1}

for s ≥ 0, f ∈ L^{1}(ℝ^{n}), and sufficiently small α > 0. Except for uniqueness, the assertion now follows from the non–autonomous Miyadera perturbation theorem [8, Thm. 3.4].

In order to show the uniqueness of mild solutions, let u be a mild solution of (1) with u(s) = 0. Take α > 0 such that c′_{α}(V_{a}(·)) \frac{M}{a} ≤ q < 1. The function v = V (·)u(·) ∈ L_{loc}^{1}([s, ∞), L^{1}(ℝ^{n})) satisfies

u(t) = ∫_{s}^{t} U(t, r)v(r) dr and v(t) = V (t) ∫_{s}^{t} U(t, r)v(r) dr  (3)

for t ≥ s. The second identity implies

∫_{s}^{s+α} ∥v(r)∥_{1} dr ≤ ∫_{s}^{s+α} ∫_{τ}^{s+α} ∥V (r)U(r, τ)v(τ)∥_{1} dr dτ ≤ q ∫_{s}^{s+α} ∥v(r)∥_{1} dr.

Therefore, we have v(t) = 0 for a.e. t ∈ [s, s + α], and so u(t) = 0 for s ≤ t ≤ s + α due to (3). Continuing to the right, one obtains u(t) = 0 for t ≥ s.

We refer to [2] or [3] concerning related results for a smaller class of potentials V and to [9] for negative V.

We now investigate the quantity c(V (·)) in more detail. A stronger condition than (4) below was introduced in [14]. Compare also [1, 5, 10, 12] concerning the autonomous case.

Theorem 2. A measurable function V : [0, ∞) × ℝ^{n} → ℂ satisfies c(V (·)) < q if and only if

M(α, β) := sup_{s≥0} ess sup_{x∈ℝ^{n}}∫_{s}^{s+α} ∫_{|x−y|≤β} |V (t, y)| K_{t−s}(x − y) dy dt < q  (4)

for some α, β > 0. In particular, c(V (·)) = 0 if and only if inf_{α,β>0} M(α, β) = 0.

Proof. Notice that we have

∫_{0}^{α} ∥V (s + t)U(t)f∥_{1} dt = ∫_{ℝ^{n}} f(x) ∫_{s}^{s+α} ∫_{ℝ^{n}} |V (t, y)| K_{t−s}(x − y) dy dt dx

for 0 ≤ f ∈ L^{1}(ℝ^{n}), and that therefore

c′_{α}(V (·)) = sup_{s≥0} ess sup_{x∈ℝ^{n}}∫_{s}^{s+α} ∫_{ℝ^{n}} |V (t, y)| K_{t−s}(x − y) dy dt.  (5)

This implies M(α, β) ≤ c′_{α}(V (·)) for all β > 0 and inf_{α,β>0} M(α, β) ≤ c(V (·)).

In order to prove the converse implication, we assume that (4) holds for some α′, β > 0; without restriction α′ ≤ 1. For 0 < α ≤ α′, s ≥ 0, and x ∈ ℝ^{n}, we estimate

q ≥ sup_{s≥0} ess sup_{x∈ℝ^{n}}∫_{s}^{s+α} ∫_{|y−x|≤ \frac{β}{2}} |V (t, y)| (4π(t − s))^{− \frac{n}{2}} e^{− \frac{|y−x|^{2}}{4(t−s)}} dy dt
≥ sup_{s≥0} ess sup_{x∈ℝ^{n}}∫_{0}^{α}(4πt)^{− \frac{n}{2}} e^{− \frac{β^{2}}{16t}} ∫_{|y|≤ \frac{β}{2}} |V (s + t, x + y)| dy dt,  (6)
∫_{s}^{s+α} ∫_{|y−x|≥β} |V (t, y)| K_{t−s}(y − x) dy dt
≤ ∫_{0}^{α}(4πt)^{− \frac{n}{2}} e^{− \frac{β^{2}}{8t}} ∫_{|y|≥β} |V (s + t, x + y)| e^{− \frac{|y|^{2}}{8}} dy dt
= C_{1} ∫_{0}^{α}(4πt)^{− \frac{n}{2}} e^{− \frac{β^{2}}{8t}} ∫_{|y|≥β} ∫_{|z−y|≤ \frac{β}{3}} |V (s + t, x + y)| e^{− \frac{|y|^{2}}{8}} dz dy dt
≤ C_{1} ∫_{0}^{α}(4πt)^{− \frac{n}{2}} e^{− \frac{β^{2}}{8t}} ∫_{|z|≥ \frac{2}{3} β} ∫_{|y−z|≤ \frac{β}{3}} |V (s + t, x + y)| e^{− \frac{|y|^{2}}{8}} dy dz dt
≤ C_{1} ∫_{0}^{α}(4πt)^{− \frac{n}{2}} e^{− \frac{β^{2}}{8t}} ∫_{|z|≥ \frac{2}{3} β} e^{− \frac{|z|^{2}}{32}} ∫_{|y|≤ \frac{β}{3}} |V (s + t, x + z + y)| dy dz dt
≤ C_{1} e^{− \frac{β^{2}}{16α}} ∫_{ℝ^{n}} e^{− \frac{|z|^{2}}{32}} ∫_{0}^{α}(4πt)^{− \frac{n}{2}} e^{− \frac{β^{2}}{16t}} ∫_{|y|≤ \frac{β}{3}} |V (s + t, x + z + y)| dy dt dz
≤ C_{2} q e^{− \frac{β^{2}}{16α}}  (7)

with constants C_{1} and C_{2} (depending on β), where we have used (6) in the last step. Now (5), (4), and (7) imply

c′_{α}(V (·)) ≤ M(α, β) + C_{2} q e^{− \frac{β^{2}}{16α}} < q

for small α > 0. The last assertion is then clear.

As an easy consequence of the above characterization, we obtain a slight generalization of a well-known integrability condition for the existence of weak solutions of (1), see e.g. [2]. We use the space L_{loc,u}^{p}(Ω) of locally integrable functions f having a finite norm

∥f∥_{L_{loc,u}^{p}(Ω)} = sup_{x∈Ω} ∥χ_{Ω∩B(x,1)}f∥_{p} ,

where Ω ⊆ ℝ^{n}, B(x, 1) = {y ∈ ℝ^{n} : |x − y| ≤ 1}, and 1 ≤ p ≤ ∞.

Corollary 3. Let 1 ≤ p, q ≤ ∞, \frac{1}{q} + \frac{n}{2p} < 1, and V ∈ L_{loc,u}^{q}([0, ∞), L_{loc,u}^{p}(ℝ^{n})). Then c(V (·)) = 0.

Proof. Notice that ∥K_{t}∥_{p′} = C t^{− \frac{n}{2p}} for \frac{1}{p} + \frac{1}{p′} = 1 and a constant C depending on n and p. Therefore Hölder’s inequality yields

∫_{0}^{α} ∫_{|y|≤1} |V (s + t, x − y)| K_{t}(y) dy dt ≤ C ∫_{0}^{α} t^{− \frac{n}{2p}} ∥V (s + t)∥_{L_{loc,u}^{p}(ℝ^{n})} dt
≤ C_{1} ∥V ∥ α^{\frac{1}{q′} − \frac{n}{2p}} .  □

Remark 4. For q = ∞, Corollary 3 yields the condition p > \frac{n}{2} which is the usual integrability condition in the t–independent case; cf. [1, Prop. 4.3], [10]. In the other extreme case, p = ∞, we get the condition q > 1. It is also easy to show that

L^{1}([0, ∞), L^{∞}(ℝ^{n})) ⊆ NK := {V ∈ N̂K : c(V (·)) = 0}.

We now discuss the relationship between the classes N̂K and K̂. Our first observation indicates that N̂K is an appropriate extension of K̂ to the non–autonomous situation.

Remark 5. Let E = L^{1}([0, ∞), L^{1}(ℝ^{n})) ≅ L^{1}([0, ∞) × ℝ^{n}). It is easy to see that the operator L = −\frac{d}{dt} + Δ defined on

{g ∈ W^{1,1}([0, ∞), L^{1}(ℝ^{n})) : g(0) = 0, g(t) ∈ D(Δ) for a.e. t ≥ 0, Δg ∈ E}

has a closure G in E which generates a positive semigroup T (·); namely the evolution semigroup on E corresponding to the semigroup U(·), compare [7, pp. 525]. Let V : [0, ∞) × ℝ^{n} → ℂ be measurable. By the proof of [8, Thm. 3.4] the potential V belongs to N̂K if and only if

sup_{g∈E, ∥g∥≤1} ∫_{0}^{1} ∥V T (t)g∥_{E} dt < ∞.

The latter condition is equivalent to the G–boundedness of V due to [12, Prop. 4.7]. In other words, N̂K is the class of potentials which are relatively bounded with respect to the closure of the parabolic operator L in L^{1}([0, ∞) × ℝ^{n}), whereas K̂ is the space of Δ–bounded potentials V_{0} : ℝ^{n} → ℂ by [12, Prop. 5.1].

Theorem 6. Assume that [0, ∞) ∋ s ↦ V (s) ∈ K̂ is Hölder continuous and that c′_{α}(V (s)) ≤ q for all s ≥ 0 and some α > 0. Then c(V (·)) ≤ q.

Proof. Due to [12, Prop. 4.7, 5.1], the assumption yields

∥(V (s + t) − V (s))(1 − Δ)^{−1}∥ ≤ C t^{β}

for t, s ≥ 0 and some constants C ≥ 0 and 0 < β ≤ 1. Recall that ∥ΔU(t)∥ ≤ \frac{c}{t} for a suitable constant c. Thus,

∫_{0}^{α} ∥V (s + t)U(t)f∥_{1} dt ≤ ∫_{0}^{α} t^{β} ∥(1 − Δ)U(t)f∥_{1} dt + ∫_{0}^{α} ∥V (s)U(t)f∥_{1} dt
≤ (C_{1} (α^{β} + α^{β+1}) + q) ∥f∥_{1}

for a constant C_{1}. This implies the assertion.

The next two examples show that neither of the spaces N̂K and C_{b}([0, ∞), K̂) is included in the other.

Example 7. Set V (t, x) = φ(t)V_{0}(x) for φ ∈ L^{1}([0, ∞)) and V_{0} ∈ L^{∞}(ℝ^{n}). Then c(V (·)) = 0, but V ∈ C_{b}([0, ∞), K̂) if and only if φ ∈ C_{b}([0, ∞)).

Example 8. Let n ≥ 3. The potential

V (t, x) = \begin{cases}∥K_{t}∥_{U}^{−1} K_{t}(x), & t ≥ 1/e, x ∈ ℝ^{n}, \\ (|ln t| ∥K_{t}∥_{U})^{−1} K_{t}(x), & 0 < t ≤ 1/e, x ∈ ℝ^{n}, \\ 0, & t = 0, x ∈ ℝ^{n}, \end{cases}

belongs to C_{b}([0, ∞), K̂), but not to N̂K.

Proof. The first assertion is clear. From [12, Prop. 4.7] we derive by a change of variables that

∥K_{t}∥_{U} = ∥K_{t} ∫_{0}^{1} U(s) ds∥_{L(L^{1}(ℝ^{n}))}

= ess sup_{x∈ℝ^{n}} \frac{1}{4} π^{− \frac{n}{2}} ∫_{ℝ^{n}} K_{t}(x − y) |y|^{2−n} ∫_{|y|^{2}/4}^{∞} τ^{\frac{n}{2} −2} e^{−τ} dτ dy

≤ C_{1} ∫_{ℝ^{n}} K_{t}(y) |y|^{2−n} dy

= C_{2} t^{− \frac{n}{2}} ∫_{0}^{∞} r^{n−1} r^{2−n} e^{− \frac{r^{2}}{4t}} dr

= 2C_{2} t^{1− \frac{n}{2}} ∫_{0}^{∞} e^{−r} dr = C_{3} t^{1− \frac{n}{2}} .

As a consequence,

∫_{0}^{1/e} ∫_{ℝ^{n}} V (t, y) K_{t}(y) dy dt ≥ ∫_{0}^{1/e} \frac{t^{\frac{n}{2}}}{C_{3} t |ln t|} ∫_{ℝ^{n}} K_{t}(y)^{2} dy dt

= C_{4} ∫_{0}^{1/e} \frac{dt}{t |ln t|} = ∞.

Using (5), this easily implies V ∉ N̂K.

References

[1] M. Aizenman, B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209–273.

[2] D.G. Aronson, Non–negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1968), 607–694.

[3] S.D. Eidelman, F.O. Porper, Two–sided estimates of fundamental solutions of second–order parabolic equations, and some applications, Russian Math. Surveys 39 (1984), 119–178.

[4] E.B. Fabes, D.W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327–338.

[5] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135–148.

[6] E.-M. Ouhabaz, P. Stollmann, K.-T. Sturm, J. Voigt, The Feller property for absorption semigroups, J. Funct. Anal. 138 (1996), 351–378.

[7] F. Räbiger, A. Rhandi, R. Schnaubelt, Perturbation and an abstract characterization of evolution semigroups, J. Math. Anal. Appl. 198 (1996), 516–533.

[8] F. Räbiger, A. Rhandi, R. Schnaubelt, J. Voigt, Non-autonomous Miyadera perturbations, submitted.

[9] F. Räbiger, R. Schnaubelt, Absorption evolution families and exponential stability of non– autonomous diffusion equations, to appear in Differential Integral Equations.

[10] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526.

[11] J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann. 229 (1977), 163–171.

[12] J. Voigt, Absorption semigroups, their generators, and Schrödinger semigroups, J. Funct. Anal. 67 (1986), 167–205.

[13] J. Voigt, Absorption semigroups, Feller property, and Kato class, Oper. Theory Adv. Appl. 78, Birkhäuser, Basel, 1995, 389–396.

[14] Qi S. Zhang, Global existence and local continuity of solutions for semilinear parabolic equations, Comm. Partial Differential Equations 22 (1997), 1529–1558.

R.S., Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.

E-mail address: rosc@michelangelo.mathematik.uni-tuebingen.de

J.V., Technische Universität Dresden, Fachrichtung Mathematik, 01062 Dresden, Germany.

E-mail address: voigt@math.tu-dresden.de