GAUSSIAN ESTIMATES FOR SECOND ORDER OPERATORS WITH UNBOUNDED COEFFICIENTS

STEFAN KARRMANN

Abstract. We study second order differential operators A with lower order coefficients in some L_{q} + L_{∞}. We prove the generation of positive, quasi-contractive C_{0}-semigroups on L_{p} for all p ∈ (1, ∞). If the second order coefficients are in some L_{q} + L_{∞} we get upper pseudo Gaussian bounds of the heat kernel.

Maximal regularity, spectral independence on L_{p}, and analyticity of the generated semigroup on L_{1} are studied for these operators.

Introduction

There is now an extensive literature on Gaussian bounds for heat kernels associated with strongly elliptic operators (see Davies [8], Robinson [19] and references therein). The best upper bounds have been derived by a technique introduced by Davies in [7]. We use this technique and ideas developed in Stampacchia [20], Daners [6], Arendt and ter Elst [2] to analyse the Cauchy Problem

\begin{matrix}d_{t}u(t) + Au(t) & = 0  on Ω, for all t ∈ (0, ∞) \\ u(0) & = u_{0}  on Ω \\ u(t) & = 0  on ∂Ω, for all t ∈ (0, ∞) \end{matrix} \}  (CP)

where Ω is an open subset of R^{n} and the operator A is associated with the closure of the second order form E given by

E(u, v) = ∑_{i,j=1}^{n} ∫_{Ω} a_{i,j}(x)∂_{i}u(x)∂_{j}v(x) dx + ∑_{j=1}^{n} ∫_{Ω} b_{1,j}(x)∂_{i}u(x)v(x) dx +

+ ∑_{j=1}^{n} ∫_{Ω} b_{2,j}(x)u(x)∂_{j}v(x) dx + ∫_{Ω} c(x)u(x)v(x) dx. (1)

This form is well defined for all u, v ∈ W_{∞,c}^{1} if the coefficients a_{i,j}, b_{l,j} and c are in L_{1,loc} and real valued.

Suppose c_{−} ∈ Ł_{q} := L_{q} + L_{∞} and b_{l,j} ∈ Ł_{2q} for some q ≥ 1. Then we show that the operator −A generates a semigroup T on L_{2}. We prove that in this case T can be extended to consistent, quasi-contractive, positive C_{0}-semigroups on L_{p} for all p ∈ (1, ∞). If b_{l,j} ∈ Ł_{p} and c_{−} ∈ Ł_{q} for some p > 2, q > 1 then T (t) is in L(L_{1}, L_{∞}) and is quasi ultra-contractive, in particular ∥T (t)∥_{L(L_{1},L_{∞})} ≤ Ct^{−n/2}e^{ωt}. Of course, in this situation T is associated with a kernel k such that

Date: 29th December 1999.

1991 Mathematics Subject Classification. Primary 35K15, 35G10; Secondary 35J20, 47A10.

Key words and phrases. Gaussian estimates; unbounded coefficients; second order; elliptic; Dirichlet boundary condition; non-symmetric.

The author is supported by the Studienstiftung des Deutschen Volkes.

T (t)f(x) = ∫_{Ω} k(t, x, y)f(y) dy for all f ∈ L_{1}.  (2)

Assume, in addition, that a_{i,j} ∈ Ł_{p} for some p > 2. Then we prove a pseudo Gaussian bound of order m ≥ 2, explicitly

|k(t, x, y)| ≤ Ct^{−n/2} exp(ω_{1}t − ω_{2}(|x − y|^{m} /t)^{\frac{1}{m−1}} )  (3)

for almost all x, y ∈ Ω, t ∈ (0, ∞) and some C, ω_{i} > 0. If a_{i,j} ∈ L_{∞} we show m = 2 and obtain a classical Gaussian kernel as an upper bound. This extends the result of Aronson [4], Arendt and ter Elst [2] and Daners [6]. Using estimate (2) we obtain some results about spectral independence, maximal regularity and analyticity.

The plan of this article is as follows. In section 1 we state the exact assumptions and the main results. Then we do some estimates for the coefficients in section 2. In section 3 we derive the key estimate (6) and use it to show that the form E generates a C_{0}-semigroup on L_{2}. That E generates quasi-dissipative C_{0}-semigroups on L_{p} (1 < p < ∞) is proved in section 4. There we give an L_{1}-L_{∞} estimate, too. In section 5 we derive the upper Gaussian estimates and apply these results in section 6.

In this article we fix the open set Ω ⊂ R^{n} and use the following notation. By N and R we denote the set of all natural numbers (without zero) and the set of all real numbers, respectively. We use R^{+} = [0, ∞) for short. We write L_{p} to denote the real Banach space of all functions from Ω into R with norm ∥f∥_{p} = (∫_{Ω} |f(x)|^{p} dx)^{1/p}, if 1 ≤ p < ∞ and ∥f∥_{∞} = ess sup_{x∈Ω} |f(x)|. We use Ł_{p} = L_{p} + L_{∞} for short. If p = 2 then L_{2} is a Hilbert space with scalar product ⟨u, v⟩ = ∫_{Ω} u(x)v(x) dx. By L_{1,loc} we denote the set of all real, measurable functions, that are integrable on every compact subset of Ω. The space W_{∞,c}^{1} is the subset of L_{∞}, whose functions have compact support and (weak) derivatives in L_{∞}, while D is the set of all infinitely differentiable functions on Ω with compact support. By H^{1} we denote the Hilbert space of all functions in L_{2}, whose first (weak) derivatives are in L_{2}, too, with scalar product ⟨u, v⟩_{H_{1}} = ⟨u, v⟩ + ∑_{k=1}^{n} ⟨∂_{k}u, ∂_{k}v⟩. Finally, H_{0}^{1} denotes the closure of D with respect to H^{1}.

For functions with two arguments like f(t, x) we write f(t) for the mapping x ↦ f(t, x) and f(x) for the mapping t ↦ f(t, x) if there is no ambiguity. By f_{±} we denote max {0, ±f} for short, while sgn f = 1_{{x;f(x)≠0}}f/ |f| where 1_{A} is the characteristic function of the set A. Given a form E on a Hilbert space H, we write E_{α}(u, v) = E(u, v) + α ⟨u, v⟩_{H} where α ∈ R.

A family (A_{p})_{p∈Π} with Π ⊂ [1, ∞] and A_{p} : L_{p} → L_{p} for all p ∈ Π is called consistent, if for all p, q ∈ Π, f ∈ L_{p} ∩ L_{q} holds A_{p}f = A_{q}f. A continuous form E on some Hilbert space H is called coercive, if there exists a constant c > 0 such that E(u, u) ≥ c ∥u∥_{H}^{2} for all u ∈ D(E).

1. Conditions and Main Results

We distinguish three different conditions; each one enables us to state interesting theorems.

Condition 1.1. The coefficients of the form E satisfy (a) the growth condition: We have a_{i,j}, c ∈ L_{1,loc}, b_{l,j} ∈ Ł_{2q} for i, j = 1, …, n, l = 1, 2 and

c_{−} ∈ Ł_{q} where

q ∈ \begin{cases}\{1\} & if n = 1 \\ (1, ∞) & if n = 2 \\ [n/2, ∞) & if n > 2 \end{cases} .

(b) the strong ellipticity condition: There exists ν ∈ (0, ∞) such that

ν |ξ|^{2} ≤ ∑_{i,j=1}^{n} a_{i,j}(x)ξ_{i}ξ_{j}  (4)

for all ξ ∈ R^{n} and almost all x ∈ Ω.

(c) the antisymmetric boundedness condition: For i, j = 1, …, n we have a_{i,j} − a_{j,i} ∈ L_{∞}.

These requirements are quite weak. The second order coefficients are as arbitrary as possible and the lower order ones may still be unbounded. The boundedness of a_{i,j} −a_{j,i} is only needed in Lemma 3.5 to prove the continuity of E. The closure of E is associated with an generator A and its generated C_{0}-semigroup T. These weak assumptions are strong enough to prove

Theorem 1.2. If Condition 1.1 is satisfied then there is a consistent family (T_{p})_{p∈(1,∞)} of quasi-contractive, positive C_{0}-semigroups T_{p} on L_{p} with generator −A_{p} and A_{2} = A.

We cannot expect to get always a quasi-contractive semigroup on L_{1}. There is a counter example by Ouhabaz [18, Remark 4.3.a].

Condition 1.3. In addition to Condition 1.1 we assume that the coefficients of the form E satisfy b_{l,j} ∈ Ł_{p_{1}} for all j = 1, …, n, l = 1, 2 and c_{−} ∈ Ł_{p_{0}} where p_{1} > 2q and p_{0} > q; i.e., any choice of

p_{1} ∈ \begin{cases}(2, ∞] & if n = 1 \\ (n, ∞] & if n ≥ 2 \end{cases}  p_{0} ∈ \begin{cases}(1, ∞] & if n = 1 \\ (n/2, ∞] & if n ≥ 2 \end{cases}

is possible, if the coefficients are appropriate.

This condition is strong enough to prove the following kernel bound.

Theorem 1.4. If Condition 1.3 is satisfied then T is ultra-contractive. Moreover

∥T (t)∥_{L(L_{1},L_{∞})} ≤ Ct^{−n/2}e^{(β)t} for all t ∈ (0, ∞)

and some C, β > 0 and δ(2) > 0 (see Theorem 3.3 and Remark 4.7).

Condition 1.5. The coefficients of the form E satisfy condition 1.3 and a_{i,j} ∈ Ł_{p_{2}} for all i, j = 1, …, n and some p_{2} > 2q.

Theorem 1.6. If Condition 1.5 is satisfied then the associated semigroup T is given by (T (t)f)(x) = ∫_{Ω} k(t, x, y)f(y) dy and the kernel k : (0, ∞) × Ω × Ω → R satisfies 0 ≤ k(t, x, y) and a pseudo Gaussian estimate of order m; i.e.,

|k(t, x, y)| ≤ Ct^{−n/2} exp(ω_{1}t − ω_{2}(|x − y|^{m} /t)^{\frac{1}{m−1}} )  (5)

for all x, y ∈ Ω, t ∈ (0, ∞), some C, d, ω_{i} > 0 and m = 2 + \frac{4q}{p_{2}−2q}. If m = 2 then we have the classical Gaussian estimate (of order 2).

Theorem 1.7. If Condition 1.5 is satisfied then σ(A_{p}) = σ(A_{2}) for all p ∈ (1, ∞). If the semigroup extends to a C_{0}-semigroup on L_{1} with generator A_{1} (see Remark 6.4) then σ(A_{1}) = σ(A_{2}), too.

Theorem 1.8. If Condition 1.5 is satisfied and a_{i,j} ∈ L_{∞} for all i, j = 1, …, n then A has maximal L_{p}-L_{q} regularity for all 1 < p, q < ∞; i.e., for every f ∈ L_{p}(R^{+}, L_{q}(Ω)) exists an unique solution u ∈ W _{p}^{1}(R^{+}, L_{q}(Ω)) ∩ L_{p}(R^{+}, D(A_{q})) of u′(t) = Au(t) + f(t) for t ∈ R^{+} and u(0) = 0.

Theorem 1.9. If Condition 1.5 is satisfied and the associated semigroup T : Σ_{θ} → L(L_{2}) is analytic with Σ_{θ} = {re^{iφ}; r > 0, |φ| < θ} where θ ∈ (0, π/2], then for all 0 < θ_{1} < θ there is an analytic mapping T_{1} : Σ_{θ_{1}} → L(L_{1}), such that T (z) and T_{1}(z) are consistent for all z ∈ Σ_{θ_{1}}.

If we have a classical Gaussian estimate of the kernel then we obtain an analytic semigroup on L_{1}.

2. Coefficients

As we are discussing quite general forms E, we introduce Ł_{p} a larger space than L_{p} as the space where the coefficients are taken from.

Definition 2.1. For 1 ≤ p ≤ ∞ let Ł_{p} = L_{p} + L_{∞}. Moreover, for 0 ≤ ϵ < ∞ and f ∈ Ł_{p} let

[f]_{ϵ,p} = inf {k ∈ [0, ∞]; ∃f_{i} ∈ Ł_{p} : f = f_{1} + f_{2}, ∥f_{1}∥_{p} ≤ ϵ, ∥f_{2}∥_{∞} ≤ k}

and ∥f∥_{Ł_{p}} = inf {ϵ + [f]_{ϵ,p}; 0 ≤ ϵ < ∞}.

Remark 2.2. Clearly (Ł_{p}, ∥·∥_{Ł_{p}}) is a Banach space and [f]_{ϵ,p} is non-increasing in ϵ. For all f, g ∈ Ł_{p} with 0 ≤ f ≤ g we have [f]_{ϵ,p} ≤ [g]_{ϵ,p}.

Some other properties of the spaces Ł_{p} are collected in

Lemma 2.3. Let 1 ≤ p ≤ ∞, f_{i} ∈ Ł_{p}, g ∈ L_{∞} and 0 ≤ ϵ < ∞. Then

(i) [g]_{ϵ,p} ≤ ∥g∥_{∞}, (ii) for all θ ∈ [0, 1] we have [f_{1} + f_{2}]_{ϵ,p} ≤ [f_{1}]_{θϵ,p} + [f_{2}]_{(1−θ)ϵ,p}, (iii) for all ρ ∈ R we have [ρf_{1}]_{ϵ|ρ|,p} = |ρ| [f_{1}]_{ϵ,p},

(iv) If p < q < ∞, ϵ ≠ 0 and f ∈ L_{q} then f ∈ Ł_{p} and [f]_{ϵ,p} ≤ ∥f∥q^{\frac{q}{q−p}} ϵ^{− \frac{p}{q−p}}.

Proof. (i) and (ii) are clear by the definition of [f]_{ϵ}. We show (iii).

[ρf_{1}]_{ϵ|ρ|,p} = inf {k ∈ [0, ∞]; ∃g_{i} ∈ Ł_{p} : f = g_{1} + g_{2}, ∥ρg_{1}∥_{p} ≤ ϵ |ρ| , ∥ρg_{2}∥_{∞} ≤ k}
= |ρ| inf {k ∈ [0, ∞]; ∃g_{i} ∈ Ł_{p} : f = g_{1} + g_{2}, ∥g_{1}∥_{p} ≤ ϵ, ∥g_{2}∥_{∞} ≤ k}
= |ρ| [f_{1}]_{ϵ,p}

GAUSSIAN ESTIMATES FOR SECOND ORDER OPERATORS 345

To proof (iv) let k = ∥f∥q^{\frac{q}{q−p}} ϵ^{− \frac{p}{q−p}} and set f_{k} = min \{k, max \{−k, f\}\}, g_{k} = f − f_{k}. Then we have ∥f_{k}∥_{∞} ≤ k and

∥g_{k}∥_{p} ≤ ∥f1_{\{|f|>k\}}∥_{p} ≤ ∥f∥_{q} ∥1_{\{|f|>k\}}∥_{1}^{1/p−1/q} ≤ ∥f∥_{q} ∥\frac{|f|}{k}∥_{q}^{q(1/p−1/q)} ≤ ∥f∥_{q}^{q/p} k^{−(q/p−1)} = ϵ.

Thus, [f]_{ϵ,p} ≤ k.

To estimate terms in which the coefficients appear, we need the well known Lemma 2.4 (Sobolev embedding). Let n ∈ N and

q ∈ \begin{cases}\{∞\} & if n = 1 \\ [2, ∞) & if n = 2 \\ [2, \frac{2n}{n−2}] & if n > 2 \end{cases} .

Then exists a constant C ∈ R^{+} such that

∥u∥_{q} ≤ C(∥∇u∥_{2} + ∥u∥_{2})

for all functions u ∈ D. We call the infimum of such constants C the Sobolev embedding constant for n and q.

See for example Adams [1, Theorem 5.4].

Lemma 2.5. Let q ∈ \begin{cases}\{2\} & if n = 1 \\ (2, ∞] & if n = 2 \\ [n, ∞] & if n > 2 \end{cases}. For all a_{i} ∈ Ł_{q}, v, u ∈ W _{∞,c}^{1}, ϵ ∈ (0, ∞) we

have

∑_{i=1}^{n} ∥a_{i}u∂_{i}v∥_{1} ≤ ϵ (∥∇u∥_{2} + ∥u∥_{2}) ∥∇v∥_{2} + (∑_{i=1}^{n} [a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q}^{2})^{1/2} ∥u∥_{2} ∥∇v∥_{2}

≤ ϵ ∥∇u∥_{2} ∥∇v∥_{2} + ϵ ∥∇v∥_{2}^{2} + \frac{1}{2ϵ} (ϵ^{2} + ∑_{i=1}^{n} [a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q}) ∥u∥_{2}^{2}

with C_{1} the Sobolev embedding constant for n and q. Proof. For all η > 0 there are b_{i}, c_{i} ∈ Ł_{q} such that a_{i} = b_{i} + c_{i}, ∥b_{i}∥_{q} ≤ \frac{ϵ}{\sqrt{n}C_{1}} and ∥c_{i}∥_{∞} ≤ [a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q} + η. Let 1/2 = 1/p + 1/q. Then, by Lemma 2.4,

∥a_{i}u∂_{i}v∥_{1} ≤ ∥b_{i}u∂_{i}v∥_{1} + ∥c_{i}u∂_{i}v∥_{1} ≤ ∥b_{i}u∥_{2} ∥∂_{i}v∥_{2} + ∥c_{i}∥_{∞} ∥u∂_{i}v∥_{1}

≤ ∥b_{i}∥_{q} ∥u∥_{p} ∥∂_{i}v∥_{2} + ([a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q} + η) ∥u∂_{i}v∥_{1}

≤ \frac{ϵ}{\sqrt{n}} (∥∇u∥_{2} + ∥u∥_{2}) ∥∂_{i}v∥_{2} + ([a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q} + η) ∥u∂_{i}v∥_{1} .

Letting η ↘ 0 we obtain

∥a_{i}u∂_{i}v∥_{1} ≤ \frac{ϵ}{\sqrt{n}} (∥∇u∥_{2} + ∥u∥_{2}) ∥∂_{i}v∥_{2} + [a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q} ∥u∂_{i}v∥_{1} .

Summing up and using that ∑_{i=1}^{n} ∥∂_{i}f∥_{2} ≤ \sqrt{n} ∥∇f∥_{2} we obtain

∑_{i=1}^{n} ∥a_{i}u∂_{i}v∥_{1} ≤ \frac{ϵ}{\sqrt{n}} (∥∇u∥_{2} + ∥u∥_{2}) ∑_{i=1}^{n} ∥∂_{i}v∥_{2} + ∑_{i=1}^{n} [a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q} ∥u |∂_{i}v|∥_{1}

≤ ϵ (∥∇u∥_{2} + ∥u∥_{2}) ∥∇v∥_{2} + (∑_{i=1}^{n} [a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q}^{2})^{1/2} ∥u∥_{2} ∥∇v∥_{2}

≤ ϵ ∥∇u∥_{2} ∥∇v∥_{2} + ϵ ∥∇v∥_{2}^{2} + (2ϵ)^{−1} (ϵ^{2} + ∑_{i=1}^{n} [a_{i}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,q}^{2}) ∥u∥_{2}^{2}

by xy ≤ ϵx^{2} + (4ϵ)^{−1}y^{2} for all ϵ > 0.

Lemma 2.6. Let q ∈ \begin{cases}\{1\} & if n = 1 \\ (1, ∞] & if n = 2 \\ [n/2, ∞] & if n > 2 \end{cases}. For all f ∈ Ł_{q}, v, u ∈ W _{∞,c}^{1}, ϵ ∈ R^{+} we

have

∥fuv∥_{1} ≤ ϵ (∥∇u∥_{2} ∥∇v∥_{2} + ∥∇u∥_{2}^{2} + ∥∇v∥_{2}^{2} + ∥u∥_{2} ∥v∥_{2}) +

+ \frac{ϵ}{4} (∥u∥_{2}^{2} + ∥v∥_{2}^{2}) + [f]_{_{\frac{ϵ}{C^{2}_{2}}} ,q} ∥uv∥_{1}

with C_{2} the Sobolev embedding constant for n and q.

Proof. For all η > 0 there are g, h ∈ Ł_{q} such that f = g + h, ∥g∥_{q} ≤ ϵ/C_{2}^{2} and ∥h∥_{∞} ≤ [f]_{ϵ/C_{2}^{2},q} + η. Let 1 = 2/p + 1/q. Then, by Lemma 2.4,

∥fuv∥_{1} ≤ ∥guv∥_{1} + ∥huv∥_{1} ≤ ∥gu∥_{\frac{p}{p−1}} ∥v∥_{p} + ∥h∥_{∞} ∥uv∥_{1}
≤ ∥g∥_{q} ∥u∥_{p} ∥v∥_{p} + ([f]_{_{\frac{ϵ}{C^{2}_{2}}} ,q} + η) ∥uv∥_{1}
≤ ϵ (∥∇u∥_{2} + ∥u∥_{2}) (∥∇v∥_{2} + ∥v∥_{2}) + ([f]_{_{\frac{ϵ}{C^{2}_{2}}} ,q} + η) ∥uv∥_{1} .

Letting η ↘ 0 we obtain

∥fuv∥_{1} ≤ ϵ (∥∇u∥_{2} + ∥u∥_{2}) (∥∇v∥_{2} + ∥v∥_{2}) + [f]_{_{\frac{ϵ}{C^{2}_{2}}} ,q} ∥uv∥_{1}
≤ ϵ (∥∇u∥_{2} ∥∇v∥_{2} + ∥∇u∥_{2}^{2} + ∥∇v∥_{2}^{2} + ∥u∥_{2} ∥v∥_{2}) +

+ \frac{ϵ}{4} (∥u∥_{2}^{2} + ∥v∥_{2}^{2}) + [f]_{_{\frac{ϵ}{C^{2}_{2}}} ,q} ∥uv∥_{1}

by xy ≤ x^{2} + y^{2}/4.

3. Approximation

In this section we prove the key to this article, Theorem 3.3. We use this to prove, that E is a closable, coercive, continuous form. Since we need estimates about L_{p} in the next section, we use a substitution and the L_{2}-norm. This enables us to approximate the L_{p}-norm using a technique of Daners [6].

Definition 3.1. For any x ∈ Ω, u : Ω → R, k ∈ [1, ∞), p ∈ [2, ∞) let

f_{k,p}(x) = \begin{cases}|x|^{p/2} & if |x| ≤ k \\ k^{p/2} + \frac{p}{2}k^{\frac{p}{2} −1} (|x| − k) & if |x| > k \end{cases} ,
g_{k,p}(x) = f_{k,p}(x)∂f_{k,p}(x),
w_{k,p,u}(x) = f_{k,p}(u(x)),
v_{k,p,u}(x) = g_{k,p}(u(x)).

We put together some properties of these functions.

Lemma 3.2. Let u ∈ W _{loc}^{1}, k ∈ [1, ∞), p ∈ [2, ∞), then

f_{k,p}(x) = \begin{cases}|x|^{p/2} & if |x| ≤ k \\ \frac{p}{2}k^{\frac{p}{2} −1} (|x| − \frac{p−2}{p}k) & if |x| > k \end{cases} ,

f′_{k,p}(x) = \begin{cases}\frac{p}{2} |x|^{\frac{p}{2} −2} x & if |x| ≤ k \\ \frac{p}{2}k^{\frac{p}{2} −1} |x|^{−1} x & if |x| > k \end{cases} ,

f′′_{k,p}(x) = \begin{cases}\frac{p}{2} \frac{p−2}{2} |x|^{\frac{p}{2} −2} & if |x| ≤ k \\ 0 & if |x| > k \end{cases} ,

f′_{k,p}, g′_{k,p} ∈ L_{∞}(R),

w_{k,p,u}^{2} ≤ uv_{k,p,u} ≤ \frac{p}{2}w_{k,p,u}^{2},

∂_{i}uv_{k,p,u} = w_{k,p,u}∂_{i}w_{k,p,u} a.e.,

|u∂_{i}v_{k,p,u}| ≤ (p − 1)w_{k,p,u} |∂_{i}w_{k,p,u}| a.e.

Proof. We obtain the derivatives of f_{k,p} by simple differentiation and see that f′_{k,p}, g′_{k,p} ∈ L_{∞}(R). Now we fix k, p, u and write w = w_{k,p,u}, v = v_{k,p,u} for short. Then we have w^{2} = |u|^{p} and uv = \frac{p}{2} |u|^{2} if |u(x)| ≤ k, while \frac{w^{2}}{uv} = 1 − \frac{p−2}{p} \frac{k}{|u|} ∈ [2/p, 1] if |u(x)| ≥ k. This leads to

w^{2} ≤ uv ≤ \frac{p}{2}w^{2}.

The next identity is shown using ∂_{i}uv = ∂_{i}uwf′_{k,p}(u) = w∂_{i}w a.e. by Gilbarg and Trudinger [11, Theorem 7.8]. Again by [11], we have for |u(x)| ≤ k

u∂_{i}v = u∂_{i}u\frac{p}{2} |u|^{p−2} (\frac{p}{2} + \frac{p − 2}{2}) = (p − 1)w∂_{i}w a.e.

and

u∂_{i}v = u∂_{i}u (\frac{p}{2})^{2} k^{p−2} = \frac{p}{2}k^{\frac{p}{2} −1} |u| ∂_{i}w a.e.

if |u(x)| ≥ k. As k^{\frac{p}{2} −1} |u| = k^{\frac{p}{2}} + k^{\frac{p}{2} −1}(|u| − k) ≤ w and \frac{p}{2} ≤ p − 1 the last inequality of the Lemma is proved.

In section 5 we need to control δ. Hence, we provide its value in the following theorem.

Theorem 3.3. Let k ≥ 1, η, γ ∈ [0, 1), 2 ≤ p < ∞ and u ∈ W _{∞,c}^{1}. Assume Condition 1.1 is satisfied. Then

E(u, v_{k,p}) + δ(p, η, γ) ∥w_{k,p}∥_{L_{2}}^{2} ≥ γE′(u, v_{k,p}) + νη(1 − γ) ∥∇w_{k,p}∥_{L_{2}}^{2}  (6)

with

δ(p, η, γ) = \frac{3(1 − γ)(1 − η)ν}{8} + \frac{2(p − 1)^{2}}{(1 − γ)(1 − η)ν} ∑_{{j=1 \atop l=1,2}}^{n} [b_{l,j}]_{\frac{ν(1−η)(1−γ)}{4\sqrt{n}C_{1}(p−1)} ,2q}^{2} + \frac{p}{2} [c_{−}]_{_{\frac{ν(1−η)(1−γ)}{3C^{2}_{2}p}} ,q}(7)

E′(u, v) = ∑_{i,j=1}^{n} ∫_{Ω} a_{i,j}(x)∂_{i}u(x)∂_{j}v(x) dx

and C_{1} (respectively C_{2}) the Sobolev constant for n and q (respectively 2q). Let δ(p) = δ(p, 1/2, 0) for short.

Proof. Let E_{k} = {x ∈ Ω; |u(x)| ≤ k}, F_{k} = Ω\E_{k}. Fix the constants k and p, the functions u, w = w_{k,p,u} and v = v_{k,p,u} in the rest of the proof. Note that w, v ∈ W_{∞,c}^{1}, cf. Gilbarg and Trudinger [11, Theorem 7.8]. This is the reason why we don’t use D to define the form E, although the closure would be the same.

On E_{k} we have

∂_{j}w∂_{l}w = \frac{p}{2(p − 1)}∂_{j}u∂_{l}v a.e.

Applying this and Lemma 3.2 to (4) we obtain almost everywhere

(1 − γ)ν |∇w|^{2} ≤ (1 − γ) ∑_{i,j=1}^{n} a_{i,j}∂_{i}w∂_{j}w ≤ \frac{(1 − γ)p}{2(p − 1)} ∑_{i,j=1}^{n} a_{i,j}∂_{i}u∂_{j}v

≤ −γa′(·, u, v) + a(·, u, v) + ∑_{j=1}^{n}(|b_{1,j}∂_{j}uv| + |b_{2,j}v∂_{j}v|) + c_{−} |uv|

≤ −γa′(·, u, v) + a(·, u, v) + ∑_{j=1}^{n}(|b_{1,j}w∂_{j}w| + (p − 1) |b_{2,j}w∂_{j}w|) + \frac{p}{2}c_{−}w^{2}

where a′(x, u, v) = ∑_{i,j=1}^{n} a_{i,j}(x)∂_{i}u(x)∂_{j}v(x) and a(·, u, v) is the integrand of (1).

On F_{k} we have ∂_{j}w∂_{l}w = ∂_{j}u∂_{l}v almost everywhere. Applying this and Lemma 3.2 to (4) we obtain almost everywhere

(1 − γ)ν |∇w|^{2} ≤ (1 − γ) ∑_{i,j=1}^{n} a_{i,j}∂_{i}w∂_{j}w ≤ (1 − γ) ∑_{i,j=1}^{n} a_{i,j}∂_{i}u∂_{j}v

≤ −γa′(·, u, v) + a(·, u, v) + ∑_{j=1}^{n}(|b_{1,j}∂_{j}uv| + |b_{2,j}v∂_{j}v|) + c_{−} |uv|

≤ −γa′(·, u, v) + a(·, u, v) + ∑_{j=1}^{n}(|b_{1,j}w∂_{j}w| + \frac{p}{2} |b_{2,j}w∂_{j}w|) + \frac{p}{2}c_{−}w^{2}

GAUSSIAN ESTIMATES FOR SECOND ORDER OPERATORS 349

As 2 ≤ p and E_{k} ∪ F_{k} = Ω the inequality for E_{k} holds on Ω. Integration over Ω gives:

(1 − γ)ν ∥∇w∥_{L_{2}}^{2} ≤ −γE′(u, v) + E(u, v) + (p − 1) ∑_{{j=1 \atop l=1,2}}^{n} ∥b_{l,j}w∂_{j}w∥_{L_{1}} + \frac{p}{2} ∥c_{−}w^{2}∥_{L_{1}}

≤ −γE′(u, v) + E(u, v) + \frac{p}{2} (3ϵ_{2} ∥∇w∥_{L_{2}}^{2} + \frac{3ϵ_{2}}{2} ∥w∥_{L_{2}}^{2} + [c_{−}]_{\frac{ϵ_{2}}{C^{2}_{2}}} ∥w∥_{2}^{2}) +

+ (p − 1)2ϵ_{1} ∥∇w∥_{L_{2}}^{2} + \frac{p − 1}{2ϵ_{1}} ∥w∥_{L_{2}}^{2} (ϵ_{1}^{2} + ∑_{{j=1 \atop l=1,2}}^{n} [b_{l,j}]2_{\frac{ϵ_{1}}{\sqrt{n}C_{1}}} ),

where we used Lemma 2.5, 2.6 and 2.3. By choosing ϵ_{1} = \frac{ν(1−η)(1−γ)}{4(p−1)}, ϵ_{2} = \frac{ν(1−η)(1−γ)}{3p} the Theorem is proved.

We deduce from this the following corollary. Corollary 3.4. The form E_{δ(2,η,γ)} is positive for all η, γ ∈ [0, 1) if condition 1.1 is satisfied. Proof. Choose p = 2 in (6) and use u = v_{k,2,u} = w_{k,2,u} sgn u to see the positivity. Lemma 3.5. If Condition 1.1 is satisfied then the form E_{δ(2,η,γ)} is continuous with respect to its graph norm ∥u∥_{E_{δ(2,η,γ)}}^{2} = (1+δ(2, η, γ)) ∥u∥_{2}^{2}+E(u, u) for all η ∈ [0, 1) and γ ∈ (0, 1). Proof. By Lemma 3.3 and equation (4) we know that E(u, u)+δ(2, η, γ) ∥u∥_{2}^{2} ≥ γE′(u, u) ≥ γν ∥∇u∥_{2}^{2} ≥ 0. Fix η, γ ∈ [0, 1) and let δ := δ(2, η, γ) in the following.

Let Ě(u, v) = \frac{1}{2}(E(u, v) + E(v, u)) and Ě′(u, v) = \frac{1}{2}(E′(u, v) + E′(v, u)) be the antisymmetric parts of E respective E′. Choose M > 0 such that |a_{i,j} − a_{j,i}| ≤ Mν for all i, j = 1, …, n. Then the Hölder inequality and (4) lead to

4 |Ě′(u, v)|^{2} ≤ (∑_{i,j=1}^{n} ∫_{Ω} Mν |∂_{i}u∂_{i}v|)^{2} ≤ M^{2} (∫_{Ω} ν |∇u| |∇v|)^{2}
≤ M^{2} ∥ν∇u∥_{1} ∥ν∇v∥_{1} ≤ M^{2}E′(u, u)E′(v, v)
≤ M^{2}γ^{−2}E_{δ+1}(u, u)E_{δ+1}(v, v)

for any u, v ∈ W_{∞,c}^{1}. Now we can estimate by Lemma 2.5 2 |Ě(u, v) − Ě′(u, v)|

= |∑_{j=1}^{n}(b_{1,j} − b_{2,l})(v∂_{j}u − u∂_{j}v)| ≤ ∑_{j=1}^{n}(∥(b_{1,j} − b_{2,j})v∂_{j}u∥_{1} + ∥(b_{1,j} − b_{2,j})u∂_{j}v∥_{1})

≤ ϵ(∥∇u∥_{2} + ∥u∥_{2}) ∥∇v∥_{2} + (∑_{j=1}^{n} [b_{1,j} − b_{2,j}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,2q})^{1/2} ∥u∥_{2} ∥∇v∥_{2} +

+ ϵ(∥∇v∥_{2} + ∥v∥_{2}) ∥∇u∥_{2} + (∑_{j=1}^{n} [b_{1,j} − b_{2,j}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,2q})^{1/2} ∥v∥_{2} ∥∇u∥_{2}

As γν ∥∇u∥_{2}^{2} ≤ γE′(u, u) ≤ E_{δ+1}(u, u) and γ ∥u∥_{2}^{2} ≤ γE′_{1}(u, u) ≤ E_{δ+1}(u, u) by (4) and Lemma 3.3, we can continue the estimate by

2 |Ě(u, v) − Ě′(u, v)| ≤ 2E_{δ+1}(u, u)^{1/2}E_{δ+1}(v, v)^{1/2}γ^{−1}

(\frac{ϵ}{ν} + \frac{ϵ}{\sqrt{ν}} + \frac{1}{\sqrt{ν}} (∑_{j=1}^{n} [b_{1,j} − b_{2,j}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,2q})^{1/2})

Using these inequalities we obtain

|Ě(u, v)| ≤ |Ě′(u, v)| + |Ě(u, v) − Ě′(u, v)|

≤ E_{δ+1}(u, u)^{1/2}E_{δ+1}(v, v)^{1/2}γ^{−1} (\frac{ϵ}{ν} + \frac{ϵ}{\sqrt{ν}} + \frac{1}{\sqrt{ν}} (∑_{j=1}^{n} [b_{1,j} − b_{2,j}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,2q})^{1/2} + M/2).

Together with the Cauchy-Schwarz inequality for the symmetric part of E_{δ} this implies

|E_{δ}(u, v)| ≤ KE_{δ+1}(u, u)^{1/2}E_{δ+1}(v, v)^{1/2}

with K = 1+δ +γ^{−1} (\frac{ϵ}{ν} + \frac{ϵ}{\sqrt{ν}} + \frac{1}{\sqrt{ν}} (∑_{j=1}^{n} [b_{1,j} − b_{2,j}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,2q})^{1/2} + \frac{M}{2} ) for all ϵ > 0. This

ends the proof of the Lemma.

Lemma 3.6. If Condition 1.1 is satisfied, then the form E_{δ(2,η,γ)} is closable for all η ∈ [0, 1) and γ ∈ (0, 1). The closure is a coercive form. The forms E_{δ(2,η,γ)} and E′ define equivalent norms.

Proof. Let Ê′(u, v) = \frac{1}{2}(E′(u, v) − E′(v, u)) be the symmetric part of E′. Fix η, γ ∈ [0, 1) and let δ̂ = δ(2, η, γ) in the following. Using Lemma 2.5 and 2.6 we see that

|E(u, u)| ≤ |Ê′(u, u)| + ∑_{j=1}^{n} ∫_{Ω} |b_{1,j} + b_{2,j}| |u∂_{j}u| + ∫_{Ω} |c| u^{2}

≤ Ê′(u, u) + 2ϵ ∥∇u∥_{2}^{2} + \frac{1}{2ϵ} (ϵ^{2} + ∑_{j=1}^{n} [b_{1,j} + b_{2,j}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,2q}) ∥u∥_{2}^{2} +

+ 3ϵ ∥∇u∥_{2}^{2} + (ϵ + \frac{1}{2ϵ} + [c]_{_{\frac{ϵ}{C^{2}_{2}}} ,q}) ∥u∥_{2}^{2}

≤ E′_{1}(u, u) (1 + \frac{5ϵ}{ν} + \frac{3ϵ}{2} + \frac{1}{2ϵ} + \frac{1}{2ϵ} ∑_{j=1}^{n} [b_{1,j} + b_{2,j}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,2q} + [c]_{_{\frac{ϵ}{C^{2}_{2}}} ,q}),

by (4) and the inequality E(u, u)+δ ∥u∥_{2}^{2} ≥ γÊ′(u, u) ≥ γν ∥u∥_{2}^{2} ≥ 0 derived from Theorem 3.3. Using these inequalities we obtain

c^{−1}E′_{1}(u, u) ≤ E_{δ̂+γ}(u, u) ≤ E_{δ̂+1}(u, u) ≤ cE′_{1}(u, u)

with c = γ^{−1} +δ̂+2+ \frac{5ϵ}{ν} + \frac{3ϵ}{2} + \frac{1}{2ϵ} + \frac{1}{2ϵ} ∑_{j=1}^{n} [b_{1,j} + b_{2,j}]_{\frac{ϵ}{\sqrt{n}C_{1}} ,2q} +[c]_{_{\frac{ϵ}{C^{2}_{2}}} ,q}. Thus the associated

norms are equivalent. By Ma and Röckner [15, I.3 Proposition 3.5] the form E_{δ̂+γ} is closable since E′ is closable, see [15, II.2d Case 1].

Since E_{δ} is continuous (Lemma 3.5), we have |E_{δ}(u, v)|^{2} ≤ K^{2}E_{δ+1}(u, u)E_{δ+1}(v, v) for some K > 0. Thus

|E_{δ+1}(u, v)|^{2} ≤ (K + 1)^{2}Ê_{δ̂+1}(u, u)Ê_{δ̂+1}(v, v) ≤ (K + 1)^{2}c^{2}E′_{1}(u, u)E′_{1}(v, v).

Together with Ma and Röckner [15, I.3 Proposition 3.5] this shows that the closure of E_{δ} is a closed, coercive form.

Definition 3.7. If Condition 1.1 is satisfied, then E_{δ(2,η,γ)} has a unique, positive, closed, coercive, continuous extension to the real Hilbert space

V = \overline{W_{∞,c}^{1}} with respect to ⟨u, v⟩_{V} = ⟨u, v⟩ + \frac{1}{2} (E′(u, v) + E′(v, u))

for all η ∈ [0, 1), γ ∈ (0, 1). This extension is the well-known Friedrichs extension and denoted again by E_{δ(2,ν,γ)} and E, respectively.

It is well known that in this situation E is associated with an unique operator A with D(A) = {u ∈ V; ∃w ∈ L_{2}∀v ∈ V : ⟨w, v⟩ = E(u, v)} and Au = w for all u ∈ D(A) where ⟨w, v⟩ = E(u, v) for all v ∈ V. The operator −A is the generator of a C_{0}-semigroup T on L_{2}.

Lemma 3.8. Suppose f ∈ C(R) is piecewise in C^{1} with f′ ∈ L_{∞}(R) and f(0) = 0. The substitution u ↦ f ∘ u is continuous on V and on L_{2}.

Proof. First, we have to show that the range of the substitution is in V. Let u ∈ V and choose u_{n} ∈ W_{∞,c}^{1} such that u_{n} → u in V. Then, by equation (4),

sup_{n∈N} E′(f ∘ u_{n}, f ∘ u_{n}) = sup_{n∈N} ∫_{Ω} ∑_{i,j=1}^{n} a_{i,j}∂_{i}u_{n}(x)∂_{j}u_{n}(x)(f′ ∘ u_{n})^{2}

≤ sup_{n∈N} ∥f′∥_{∞}^{2} E′(u_{n}, u_{n}) < ∞

since E′(u_{n}, u_{n}) → E′(u, u) for n → ∞ and ∂_{i}f ∘u_{n} = f′∘u_{n}∂_{i}u_{n}, see Gilbarg and Trudinger [11, Theorem 7.8]. By the Banach-Alaoglu Theorem f ∘ u_{n_{k}} → v weakly in V for some subsequence u_{n_{k}} of u_{n} and some v ∈ V. Since f ∘u_{n} → f ∘u in L_{2} we obtain f ∘u = v ∈ V.

Second, let u_{n} ∈ V such that u_{n} → u ∈ V.Then, again by equation (4) and Gilbarg and Trudinger [11, Theorem 7.8], we have

E′(f ∘ u_{n}, f ∘ u_{n}) = ∫_{Ω} ∑_{i,j=1}^{n} a_{i,j}∂_{i}u_{n}(x)∂_{j}u_{n}(x)(f′ ∘ u_{n})^{2}
≤ ∥f′∥_{∞}^{2} E′(u_{n}, u_{n}) → ∥f′∥_{∞}^{2} E′(u, u)

for n → ∞. Thus, there exists a subsequence u_{n_{k}} and a function g ∈ L_{1} such that

0 ≤ (f′ ∘ u_{n_{k}})^{2} ∑_{i,j=1}^{n} a_{i,j}∂_{i}u_{n_{k}}(x)∂_{j}u_{n_{k}}(x) ≤ g

almost everywhere. By the dominated convergence Theorem E′(f ∘ u_{n_{k}}, f ∘ u_{n_{k}}) converges to E′(f ∘ u, f ∘ u) for n → ∞. This implies continuity, since there is such a dominated subsequence for all subsequences of u_{n}.

Third, f ∘ u ∈ L_{2} whenever u ∈ L_{2} as |f ∘ u| ≤ ∥f′∥_{∞} |u| almost everywhere. Finally, |f ∘ u − f ∘ v| ≤ ∥f′∥_{∞} |u − v| almost everywhere for all u, v ∈ L_{2}. Thus, we have ∥f ∘ u − f ∘ v∥_{2} ≤ ∥f′∥_{∞} ∥u − v∥_{2}.

Corollary 3.9. If Condition 1.1 is satisfied, then the semigroup T associated with E is positive. The mapping u ↦ |u| is continuous on V.

Proof. Fix η, γ ∈ [0, 1) and δ = δ(2, η, γ) in the proof. Lemma 3.8 implies that V is a lattice and u ↦ |u| is continuous on V, since x ↦ |x| is smooth on R \ {0} with essentially bounded derivative. Thus, we can apply the Beurling-Deny criteria for nonsymmetric forms, see Ouhabaz [18, Theorem 2.4]. Since ∂_{i}u_{+} = ∂_{i}u1_{{x∈Ω;u(x)>0}} we have E_{δ}(u_{+}, u_{−}) = 0 for any u ∈ V, we obtain the positivity of T immediately.

4. Upper Bounds

From the previous section we know that there are unique solutions of (CP). In this section we derive upper bounds of these solutions. To do this, we need a simple generalisation of Theorem 3.3.

Remark 4.1. Since f_{k,p} and g_{k,p} are piecewise smooth with bounded derivative, see Lemma 3.2, w_{k,p,u}, v_{k,p,u} ∈ V for all u ∈ V, by Lemma 3.8.

Obviously Theorem 3.3 holds for all u ∈ V. Since all terms of (6) are defined and the first part of the proof is a pointwise calculation, while the second part only uses that w_{k,p,u} ∈ H^{1} ⊃ V, the proof carries over to this more general situation.

Lemma 4.2. If Condition 1.1 is satisfied and u is an orbit with initial value u(0) ∈ D(A) then

d_{t} ∥w_{k,p,u}(t)∥_{2}^{2} ≤ −ν ∥∇w_{k,p,u}(t)∥_{2}^{2} + 2δ(p, η, γ) ∥w_{k,p,u}(t)∥_{2}^{2}

for all t > 0, η, γ ∈ [0, 1).

Proof. Since u(0) ∈ D(A) we have u ∈ C(R^{+}, D(A)) ∩ C^{1}(R^{+}, L_{2}). Thus, the mapping t ↦ f_{k,p} ∘ u(t) : R → L_{2} has the derivative f′_{k,p} ∘ u(t)∂_{t}u(t) almost everywhere, by Gilbarg and Trudinger [11, Theorem 7.8]. Now, we see that

d_{t} ∥w_{k,p,u}(t)∥_{2}^{2} = 2 ⟨f′_{k,p} ∘ u(t)∂_{t}u(t), w_{k,p,u}(t)⟩ = 2 ⟨∂_{t}u(t), v_{k,p,u}(t)⟩ = −2 ⟨Au(t), v_{k,p,u}(t)⟩
= −2E(u(t), v_{k,p,u}(t)) ≤ −ν ∥∇w_{k,p,u}(t)∥_{2}^{2} + 2δ(p, η, γ) ∥w_{k,p,u}(t)∥_{2}^{2}

as of Remark 4.1.

Proof of Theorem 1.2. By Corollary 3.9 we have the positivity of the semigroup. Thus, we have only to show the quasi-contractivity on L_{p}.

If u is an orbit with initial value u(0) ∈ D(A) then by Lemma 4.2 we have

d_{t} ∥w_{k,p,u}(t)∥_{2}^{2} ≤ −ν ∥∇w_{k,p,u}(t)∥_{2}^{2} + 2δ(p) ∥w_{k,p,u}(t)∥_{2}^{2} ≤ 2δ(p) ∥w_{k,p,u}(t)∥_{2}^{2} .

From this differential inequality it follows that

∥w_{k,p,u}(t)∥_{2}^{2} ≤ ∥w_{k,p,u}(0)∥_{2}^{2} e^{2δ(p)t}.

Since D(A) is dense in L_{2}, we can choose any initial value in L_{2}(Ω) ∩ L_{p}(Ω) by approximation, see Lemma 3.8. Since w_{k,p,u}(t, x) ↗ |u(t, x)|^{p/2} as k → ∞ we have

∥|u(t)|^{p/2}∥_{2}^{2} ≤ ∥|u(0)|^{p/2}∥_{2}^{2} e^{2δ(p)t} and ∥u(t)∥_{p} ≤ ∥u(0)∥_{p} e^{2δ(p)t/p}

The proposition is finished by a limiting argument.

The subsequent L_{p}-estimates of semigroup kernels are based upon an inequality first derived by Nash, see [17] to discuss parabolic partial differential equations on L_{2}(R^{n}).

Proposition 4.3 (Nash-inequality). There exists a constant M_{1} > 0 such that

∥u∥_{2}^{2+4/n} ≤ M_{1} ∥∇u∥_{2}^{2} ∥u∥_{1}^{4/n}

for all u ∈ H _{0}^{1} ∩ L_{1}. We call M_{1} = \frac{4πn}{n+2} (\frac{Γ(1+n/2)}{1+n/2})^{2/n} the Nash constant.

Proof. Extend the function u by 0 to R^{n} and use Nash’s result [17, page 936], or see Davies [9, Theorem 2.4.6].

Lemma 4.4. If Condition 1.1 is satisfied and u is an orbit with initial value u(0) ∈ D(A) then we have

(∥w_{k,2p,u}(t)∥_{2}^{1/p} e^{− \frac{δ(2p)}{p} t})^{−4p/n} ≥ \frac{2ν}{M_{1}n} ∫_{0}^{t} (∥w_{k,p,u}(s)∥_{2}^{2/p} e^{− \frac{δ(2p)}{p} s})^{−4p/n} ds

for all t ∈ J := {t ∈ R^{+}; inf_{0≤s≤t} ∥u(s)∥_{2} > 0}, p ≥ 2, k > 1 and M_{1} the Nash constant.

Proof. By Lemma 4.2 and Proposition 4.3 we have

d_{t} ∥w_{k,2p,u}(t)∥_{2}^{2} ≤ −ν ∥∇w_{k,2p,u}(t)∥_{2}^{2} + 2δ(2p) ∥w_{k,2p,u}(t)∥_{2}^{2}

≤ −\frac{ν}{M_{1}} ∥w_{k,2p,u}(t)∥_{2}^{2+4/n} ∥w_{k,2p,u}(t)∥_{1}^{−4/n} + 2δ(2p) ∥w_{k,2p,u}(t)∥_{2}^{2}

≤ −\frac{ν}{M_{1}} ∥w_{k,2p,u}(t)∥_{2}^{2+4/n} ∥w_{k,p,u}(t)∥_{2}^{−8/n} + 2δ(2p) ∥w_{k,2p,u}(t)∥_{2}^{2} .

The last inequality holds by the definition of w_{k,p,u}.

For positive functions f with f′ ≤ g + cf on R^{+} we have

f^{−M}(t)e^{cMt} ≥ −M ∫_{0}^{t} f^{−M−1}(s)g(s)e^{cMs} ds

for all M > 0. Choosing M = 2/n we obtain

∥w_{k,2p,u}(t)∥_{2}^{−4/n} e^{4 \frac{δ(2p)}{n} t} ≥ \frac{2ν}{M_{1}n} ∫_{0}^{t} ∥w_{k,p,u}(s)∥_{2}^{−8/n} e^{4 \frac{δ(2p)}{n} s} ds

which implies the claim.

Lemma 4.5. If Condition 1.3 is satisfied and r ≥ max \{2 + \frac{4q}{p_{1}−2q}, 1 + \frac{q}{p_{0}−q}\} then there exists δ_{0} > 0 such that δ(p) ≤ f(p)δ_{0} for all p ≥ 2 with f(p) = (p − 1)^{r}. Further we have ∑_{j=1}^{∞} 2^{−j} log f(2^{j+1}) ≤ 3r log 2.

Proof. Choose b̂_{l,j}, ĉ ∈ L_{∞}, b̌_{l,j} ∈ L_{p_{1}}, č ∈ L_{p_{0}} such that c_{−} = č + ĉ, and b_{l,j} = b̌_{l,j} + b̂_{l,j}. We obtain by a direct estimate using Theorem 3.3 and Lemma 2.3 (iv)

δ_{0} = \frac{3}{16}ν + \frac{4}{ν} ∑_{{j=1 \atop l=1,2}}^{n} (∥b̂_{l,j}∥_{∞} + ∥b̌_{l,j}∥p_{1}^{\frac{p_{1}}{p_{1}−2q}} (\frac{8\sqrt{n}C_{1}}{ν} )^{\frac{2q}{p_{1}−2q}} )^{2} +

+ \frac{6}{ν} + \left∥{ \atop ĉ}\right∥_{∞} + \left∥{ \atop č}\right∥p_{0}^{\frac{p_{0}}{p_{0}−q}} (\frac{6C_{2}^{2}}{ν} )^{\frac{q}{p_{0}−q}}

which satisfies the inequality.

The last inequality of this Lemma is quite easy, since

∑_{j=1}^{∞} 2^{−j} log f(2^{j+1}) = ∑_{j=1}^{∞} 2^{−j} log(2^{j+1})^{r} ≤ r ∑_{j=1}^{∞} 2^{−j}(j + 1) log 2 = r log 2 ∑_{j=0}^{∞} 2^{−j}(j + 1)

= 2r log 2

using the derivative of the geometric series at 1/2.

Lemma 4.6. Let k > 1, l ∈ N, p = 2^{l} and u be an orbit. Then there exists C, γ > 0 such that

∥w_{k,p,u}(t)∥_{2}^{2/p} ≤ 2^{−n/4}\sqrt{C}t^{− \frac{n}{2}(\frac{1}{2} − \frac{1}{p})}e^{βt} ∥u(0)∥_{2}  (8)

holds for all t ∈ (0, ∞) if condition 1.3 is satisfied.

Remark 4.7. Taking δ_{0}, f, r like in Lemma 4.5 and choosing ϵ ∈ (0, 1), the constants in Lemma 4.6 are given by

β = ϵδ_{0}δ(2) and C = (\frac{M_{1}n12^{r}}{νϵ(3^{r} − ϵ)})^{n/2}

Proof. Let u be an orbit with initial value u(0) ∈ D(A), ϵ ∈ (0, 1), the function h_{2}(t) = e^{δ(2)t} and

h_{2p}(t) = h_{p}(t)e^{\frac{δ(2p)ϵ}{pf(2p)} t} (\frac{M_{1}n}{2ν3^{r}ϵ(3^{r} − ϵ)})^{\frac{n}{4p}} (f(2p) − ϵ)^{\frac{n}{4p}},

then h_{p}(t) ≤ 2^{−n/4}\sqrt{C}e^{βt} holds, by Lemma 4.5, and h_{p} is non-decreasing. To proof the claim we do an induction over l = log_{2} p. We see that ∥w_{k,2}(s)∥_{2} = ∥u(t)∥_{2} ≤ h_{2}(t) ∥u(0)∥_{2}. By Lemma 4.4 and the uniqueness of the zero solution we obtain

∥w_{k,2p,u}(t)∥_{2}^{1/p} ≤ e^{\frac{δ(2p)}{p} t} (\frac{M_{1}n}{2ν})^{\frac{n}{4p}} (∫_{0}^{t} (∥w_{k,p,u}(s)∥_{2}^{2/p} e^{− \frac{δ(2p)}{p} s})^{− \frac{4p}{d}} ds)^{− \frac{n}{4p}}

≤ e^{\frac{δ(2p)}{p} t} (\frac{M_{1}n}{2ν})^{\frac{n}{4p}} (∫_{0}^{t} (s^{− \frac{n}{2} ( \frac{1}{2} − \frac{1}{p})}h_{p}(s)e^{− \frac{δ(2p)}{p} s})^{− \frac{4p}{d}} ds)^{− \frac{n}{4p}} ∥u(0)∥_{2}

≤ e^{\frac{δ(2p)}{p} t} (\frac{M_{1}n}{2ν})^{\frac{n}{4p}} h_{p}(t) (∫_{t(1−ϵ/f(2p))}^{t} s^{p−2} ds)^{− \frac{n}{4p}} e^{− \frac{δ(2p)}{p} t(1−ϵ/f(2p))} ∥u(0)∥_{2}

≤ e^{\frac{δ(2p)ϵ}{pf(2p)} t} (\frac{M_{1}n}{2ν})^{\frac{n}{4p}} h_{p}(t) (\frac{1}{p − 1}t^{p−1} (1 − (1 − \frac{ϵ}{f(2p)})^{p−1}))^{− \frac{n}{4p}} ∥u(0)∥_{2}

≤ h_{2p}(t)t^{− \frac{n}{2} ( \frac{1}{2} − \frac{1}{2p})} ∥u(0)∥_{2}

since 1 − (1 − ϵ/f(2p))^{p−1} ≥ \frac{(f(4)−ϵ)(p−1)}{f(p)−ϵ}(ϵ/f(4)) which can be seen by the following elementary calculus. Firstly, g(x) = \frac{e^{x}−1}{x} is increasing on R. Secondly, the function γ(p) = −(p − 1) log(1 − ϵ/f(2p)) = ϵ\frac{p−1}{f(2p)} ∑_{j=0}^{∞} \frac{ϵ^{j}}{(j+1)(f(2p))^{j}} is decreasing, since f(p) = (p − 1)^{r} and r > 2. Thirdly, γ̃(x) = −(x − ϵ) log(1 − ϵ/x) is increasing for x > ϵ, as its derivative is γ̃′(x) = − log(1−ϵ/x)−ϵ/x ≥ 0. Finally, γ̃(f(2p))g(−γ(p)) = \frac{f(2p)−ϵ}{p−1}(1−(1−ϵ/f(2p))^{p−1}) is non-decreasing.

Lemma 3.8 with regard to f_{k,p} shows that we can approximate any initial value in L_{2}.

Corollary 4.8. If Condition 1.3 is satisfied then there exists C, β > 0 such that

∥T (t)∥_{L(L_{2},L_{∞})} ≤ 2^{−n/4}\sqrt{C}t^{− \frac{n}{4}} e^{βt} for all t > 0.

The constants are given in Remark 4.7.

Proof. Let h(t) = 2^{−n/4}\sqrt{C}e^{βt}. By definition we have

∥T (t)∥_{L(L_{2},L_{∞})} = sup_{∥u_{0}∥_{2}=1} ∥T (t)u_{0}∥_{∞} ≤ sup_{∥u_{0}∥_{2}=1} lim sup_{j→∞} ∥T (t)u_{0}∥_{2^{j}}

= sup_{∥u_{0}∥_{2}=1} lim sup_{j→∞} lim_{k→∞} ∥f_{k,2^{j}} ∘ T (t)u_{0}∥_{2}^{2/2^{j}}

≤ sup_{∥u_{0}∥_{2}=1} lim sup_{j→∞} lim_{k→∞} t^{− \frac{n}{2} ( \frac{1}{2} −2^{−j})}h(t) ∥u_{0}∥_{2}

= t^{− \frac{n}{4}} h(t).

Proof of Theorem 1.4. By duality we obtain

∥T (t)∥_{L(L_{1},L_{∞})} ≤ ∥T (t/2)∥_{L(L_{2},L_{∞})} ∥T (t/2)∥_{L(L_{1},L_{2})} ≤ Ct^{−n/2}e^{(δ(2)+γ)t}.

5. Gaussian Estimates

In this section we use a technique developed by Davies [7] to obtain pseudo Gaussian upper bounds of the kernel of T. We assume Condition 1.5 in this section.

Definition 5.1. Let Ψ = {ψ ∈ D(R^{n}); |∇ψ| ≤ 1}. For all real numbers ρ and ψ ∈ Ψ let

(B_{ρ,ψ}f)(x) = e^{ρψ(x)}f(x)
T_{ρ,ψ}(t) = B_{ρ,ψ}T (t)B_{−ρ,ψ}.

Lemma 5.2. For any real ρ and ψ ∈ Ψ the semigroup T_{ρ,ψ} is associated with the form E_{ρ,ψ} given by

E_{ρ,ψ}(u, v) = ∑_{i,j=1}^{n} ∫_{Ω} a_{i,j}∂_{i}u∂_{j}v + ∑_{j=1}^{n} ∫_{Ω}(b_{1,j,ρ,ψ}∂_{j}uv + b_{2,j,ρ,ψ}v∂_{j}v) + ∫_{Ω} c_{ρψ}uv

where b_{1,j,ρ,ψ} = b_{1,j} + ρ ∑_{k=1}^{n} a_{j,k}∂_{k}ψ,  b_{2,j,ρ,ψ} = b_{2,j} − ρ ∑_{k=1}^{n} a_{k,j}∂_{k}ψ,

c_{ρ,ψ} = c − ρ^{2} ∑_{i,j=1}^{n} a_{i,j}∂_{j}ψ∂_{i}ψ − ρ ∑_{j=1}^{n}(b_{1,j} − b_{2,j})∂_{j}ψ

for all u, v ∈ V.

Proof. For every u ∈ D(A), v ∈ W_{∞,c}^{1} we have

∂_{t} ⟨T_{ρ,ψ}(t)u, v⟩ = ∂_{t} ⟨T (t)e^{−ρψ}u, e^{ρψ}v⟩ = ⟨AT (t)e^{−ρψ}u, e^{ρψ}v⟩ = E(T (t)e^{−ρψ}u, e^{ρψ}v)
= E(e^{−ρψ}T_{ρ,ψ}(t)u, e^{ρψ}v) = E_{ρ,ψ}(T_{ρ,ψ}(t)u, v).

By approximation this holds actually for all u ∈ V, from which the Lemma follows by a direct calculation.

Theorem 5.3. If ρ ∈ R, ψ ∈ Ψ, t ∈ (0, ∞)2 then there exists C, d, δ_{i} > 0 such that

∥T_{ρ,ψ}(t)∥_{L(L_{1},L_{∞})} ≤ Ct^{−n/2} exp((δ_{2}ρ^{m} + δ_{1})t)  (9)

where

m = 2 + \frac{4q}{p_{2} − 2q} ∈ [2, ∞).  (10)

Remark 5.4. The exact values of the constants are not needed in the following, but are given here for completeness. Choosing ϵ ∈ (0, 1) and analysing the proof provides

C = (\frac{M_{1}n12^{r}}{ν(3^{r} − ϵ)ϵ})^{n/2} , δ_{i} = (1 + ϵ)δ̂_{i},

f(p) = (p − 1)^{r}, r = max \{2 + \frac{4q}{min \{p_{2}, p_{1}\} − 2q} , 1 + \frac{q}{p_{0} − q} \}

δ̂_{1} = δ̂_{2} + \frac{12}{ν} ∑_{{j=1 \atop l=1,2}}^{n} (∥b̂_{l,j}∥_{∞} + ∥b̌_{l,j}∥p_{1}^{\frac{p_{1}}{p_{1}−2q}} (\frac{16\sqrt{n}C_{1}}{ν} )^{\frac{2q}{p_{1}−2q}} )^{2} +

+ \left∥{ \atop ĉ}\right∥_{∞} + \left∥{ \atop č}\right∥p_{0}^{\frac{p_{0}}{p_{0}−q}} (\frac{18C_{2}^{2}}{ν} )^{\frac{q}{p_{0}−q}} + \frac{3}{16}ν

δ̂_{2} = \frac{12}{ν} ∑_{j=1}^{n} ((∑_{i=1}^{n} \left∥{ \atop â_{i,j}}\right∥_{∞})^{2} + (∑_{i=1}^{n} \left∥{ \atop ǎ_{i,j}}\right∥p_{2}^{\frac{p_{2}}{p_{2}−2q}} (\frac{32n\sqrt{n}C_{1}}{ν} )^{\frac{2q}{p_{2}−2q}} )^{2} +

+ (∑_{i=1}^{n} \left∥{ \atop â_{i,j}}\right∥_{∞})^{2} + (∑_{i=1}^{n} \left∥{ \atop ǎ_{i,j}}\right∥p_{2}^{\frac{p_{2}}{p_{2}−2q}} (\frac{32n\sqrt{n}C_{1}}{ν} )^{\frac{2q}{p_{2}−2q}} )^{2}) +

+ ∑_{i,j=1}^{n} (\left∥{ \atop â_{i,j}}\right∥_{∞} + \left∥{ \atop ǎ_{i,j}}\right∥p_{2}^{\frac{p_{2}}{p_{2}−q}} (\frac{18n^{2}C_{2}^{2}}{ν} )^{\frac{q}{p_{2}−q}} ) +

+ ∑_{j=1}^{n} (∥b̂_{1,j} − b̂_{2,j}∥_{∞} + ∥b̌_{1,l} − b̌_{2,l}∥p_{1}^{\frac{2p_{1}}{p_{1}−q}} (\frac{18nC_{2}^{2}}{ν} )^{\frac{q}{2p_{1}−q}} )

for any choice of â_{i,j}, b̂_{l,j}, ĉ ∈ L_{∞}, ǎ_{i,j} ∈ L_{p_{2}}, b̌_{l,j} ∈ L_{p_{1}} and č ∈ L_{p_{0}} such that p_{2} > 2q or p_{1} = ∞ and a_{i,j} = ǎ_{i,j} + â_{i,j}, b_{l,j} = b̌_{l,j} + b̂_{l,j} for all i, j = 1, …, n, l = 1, 2 and c_{−} = č+ ĉ. If a_{i,j} ∈ L_{∞} for all i, j = 1 …, n then choose ǎ_{i,j} = 0, p_{2} = ∞, m = 2.

If a_{i,j} ∈ L_{∞} for all i, j = 1, …, n then we obtain a classical Gaussian estimate.

Proof. Since the form E_{ρ,ψ} satisfies condition 1.3 Theorem 1.4 holds with δ_{ρ,ψ} corresponding to (7). To estimate δ_{ρ,ψ} we choose functions â_{i,j}, b̂_{l,j}, ĉ ∈ L_{∞}, ǎ_{i,j} ∈ L_{p_{2}}, b̌_{l,j} ∈ L_{p_{1}} and č ∈ L_{p_{0}} such that a_{i,j} = ǎ_{i,j} + â_{i,j}, b_{l,j} = b̌_{l,j} + b̂_{l,j} and c_{−} = č+ ĉ. If a_{i,j} ∈ L_{∞} then choose ǎ_{i,j} = 0, p_{2} = ∞, m = 2. With the help of these decompositions and Lemma 2.3 (ii),(iv) we obtain

[b_{1,j,ρ,ψ}]_{θ} ≤ [b_{1,j}]_{θ/2} + |ρ| ∑_{k=1}^{n} \left∥{ \atop â_{j,k}}\right∥_{∞} + |ρ| ∑_{k=1}^{n} \left∥{ \atop ǎ_{j,k}}\right∥_{p_{2}} ^{\frac{p_{2}}{p_{2}−2q}} (\frac{2n |ρ|}{θ})^{\frac{2q}{p_{2}−2q}}

≤ [b_{1,j}]_{θ/2} + |ρ| ∑_{k=1}^{n} \left∥{ \atop â_{j,k}}\right∥_{∞} + |ρ| ∑_{k=1}^{n} \left∥{ \atop ǎ_{j,k}}\right∥_{p_{2}} ^{\frac{p_{2}}{p_{2}−2q}} (\frac{2n |ρ|}{θ})^{\frac{2q}{p_{2}−2q}}

[b_{1,j,ρ,ψ}]_{θ}^{2} ≤ 3 [b_{1,j}]_{θ}^{2} + 3(1 + |ρ|^{m}) (∑_{k=1}^{n} \left∥{ \atop â_{j,k}}\right∥_{∞})^{2} +

+ 3(1 + |ρ|^{m}) (∑_{k=1}^{n} \left∥{ \atop ǎ_{j,k}}\right∥_{p_{2}} ^{\frac{p_{2}}{p_{2}−2q}} (\frac{2n}{θ})^{\frac{2q}{p_{2}−2q}} )^{2}

[b_{2,j,ρ,ψ}]_{θ}^{2} ≤ 3 [b_{2,j}]_{θ}^{2} + 3(1 + |ρ|^{m}) (∑_{k=1}^{n} \left∥{ \atop â_{k,j}}\right∥_{∞})^{2} +

+ 3(1 + |ρ|^{m}) (∑_{k=1}^{n} \left∥{ \atop ǎ_{k,j}}\right∥_{p_{2}} ^{\frac{p_{2}}{p_{2}−2q}} (\frac{2n}{θ})^{\frac{2q}{p_{2}−2q}} )^{2}

[c_{ρ,ψ −}]_{θ} ≤ [c_{−}]_{θ/3} + ρ^{2} ∑_{i,j=1}^{n} \left∥{ \atop â_{i,j}}\right∥_{∞} + ρ^{2} ∑_{i,j=1}^{n} \left∥{ \atop ǎ_{i,j}}\right∥_{p_{2}} ^{\frac{p_{2}}{p_{2}−q}} (\frac{3n^{2}ρ^{2}}{θ} )^{\frac{q}{p_{2}−q}} +

+ |ρ| ∑_{j=1}^{n} ∥b̂_{1,j} − b̂_{2,j}∥_{∞} + |ρ| ∑_{j=1}^{n} ∥b̌_{1,j} − b̌_{2,j}∥p1^{\frac{p_{1}}{p_{1}−q}} (\frac{3n |ρ|}{θ})^{\frac{q}{p_{1}−q}}

≤ [c_{−}]_{θ/3} + (1 + |ρ|^{m}) ∑_{i,j=1}^{n} \left∥{ \atop â_{i,j}}\right∥_{∞} + (1 + |ρ|^{m}) ∑_{i,j=1}^{n} \left∥{ \atop ǎ_{i,j}}\right∥_{p_{2}} ^{\frac{p_{2}}{p_{2}−q}} (\frac{3n^{2}}{θ} )^{\frac{q}{p_{2}−q}} +

+ (1 + |ρ|^{m}) ∑_{j=1}^{n} ∥b̂_{1,j} − b̂_{2,j}∥_{∞} + (1 + |ρ|^{m}) ∑_{j=1}^{n} ∥b̌_{1,j} − b̌_{2,j}∥p1^{\frac{p_{1}}{p_{1}−q}} (\frac{3n}{θ})^{\frac{q}{p_{1}−q}} .

Together with (7) we obtain the estimate δ_{ρ,ψ}(p) ≤ (δ̂_{1} + |ρ|^{m} δ̂_{2})f(p). Then, by Theorem 1.4 and the values of the constants given in Remark 4.7,

∥T_{ρ,ψ}(t)∥_{L(L_{1},L_{∞})} ≤ Ct^{−n/2} exp((δ_{2}ρ^{m} + δ_{1})t)

Proof of Theorem 1.6. Let ρ ∈ R, ψ ∈ Ψ, k (respectively k_{ρ,ψ}) be the kernel associated with T (respectively T_{ρ,ψ}). Then k(t, x, y) = e^{−ρ(ψ(x)−ψ(y))}k_{ρ,ψ}(x, y, t) ≥ 0 for all t > 0 and almost all x, y ∈ Ω, since the semigroup T is positive (Theorem 1.2). By Theorem 5.3 we obtain

|k(t, x, y)| ≤ Ct^{−n/2} exp((δ_{2} |ρ|^{m} + δ_{1})t − ρ(ψ(x) − ψ(y)))

Replacing ρ by −ρ it follows that

|k(t, x, y)| ≤ Ct^{−n/2} exp((δ_{2} |ρ|^{m} + δ_{1})t − ρ |ψ(x) − ψ(y)|)

Since sup_{ψ∈Ψ} |ψ(x) − ψ(y)| = |x − y|, we have

|k(t, x, y)| ≤ Ct^{−n/2} exp((δ_{2} |ρ|^{m} + δ_{1})t − ρ |x − y|)

Choosing ρ^{m−1} = (δ_{2}mt)^{−1} |x − y| proves the claim with ω_{2} = (δ_{2}m)^{− \frac{m}{m−1}} (1 − 1/m), ω_{1} = δ_{1}.

6. Applications

Here we can use the results in the literature about kernel-estimates. Firstly we get p-independence of the spectrum σ(A) using

Theorem 6.1 (Kunstmann [14, Theorem 1.1]). Let Q be an integral operator on L_{2}(Ω) given by a kernel k which satisfies

|k(x, y)| ≤ g(x − y) ∀x, y ∈ Ω

where g is a function satisfying

x ↦ exp(ρ(|x|))g(x) ∈ L_{1}(R^{n}) ∩ L_{p_{0}}(R^{n})

for some p_{0} ∈ (1, ∞] and some Ω-admissible function ρ (i.e. ρ : R^{+} → R^{+} is nondecreasing, subadditive, lim_{t→0} ρ(t) = 0 and x ↦ exp(−ρ(|x|)) ∈ L_{1}(Ω̃) where Ω̃ = {x − y; x, y ∈ Ω}). Then Q extends to consistent operators Q_{p} on all L_{p}(Ω), p ∈ [1, ∞] and the spectrum σ(Q_{p}) does not depend on p.

Corollary 6.2. Let m > 1 and d ∈ R such that n > m(d − 1). Let R_{2} ∈ L(L_{2}) be given by the formula

R_{2}f(x) = ∫_{Ω} k(x, y)f(y) dx ∀f ∈ L_{2}(Ω)

with some measurable kernel k : Ω × Ω → C. Assume

|k(x, y)| ≤ g_{μ,b}(x − y) = C ∫_{0}^{∞} t^{−d}e^{−μt−b( \frac{|x−y|^{m}}{t})^{\frac{1}{m−1}}} dt

for some μ, b, C > 0. Then, there is a consistent family (R_{p} ∈ L(L_{p}))_{p∈[1,∞]} and the spectrum σ(R_{p}) = σ(R_{2}) is independent of p ∈ [1, ∞].

Proof. Using the continuous Minkowski inequality and the substitution x = yt^{1/m}, we obtain ∥g_{μ,b}∥_{p} < ∞ if p^{−1} > \frac{m}{n}(d − 1). Observe that the range of p does not depend on b. For any c ∈ (0, b) and a > 0 we have

sup_{s≥0} as − c(s^{m}/t)^{1/(m−1)} = t (\frac{a}{m})^{m} (\frac{m − 1}{c})^{m−1} =: tγ(a, m, c).

Thus, choosing a > 0 and c ∈ (0, b) such that γ(a, m, c) < μ, we have

|e^{a|x|}g_{μ}(x)| ≤ g_{μ−γ(a,m,c),b−c}(x)

for all x ∈ R^{n}. Now choose ρ(r) = ar and p_{0} > 1 such that p_{0}^{−1} > \frac{m}{n}(d − 1). Then the assumptions of Theorem 6.1 are given. Therefore R_{2} extends to a consistent family (R_{p})_{p∈[1,∞]} and the spectrum of R_{p} is independent of p ∈ [1, ∞].

Lemma 6.3. Let n ∈ N, M_{i} ⊂ {z ∈ C; Re z < w} for some w ∈ R and N_{i,λ} = {(λ − z)^{n}; z ∈ M_{i}} where i = 1, 2. If N_{1,λ} = N_{2,λ} for all λ > w then M_{1} = M_{2}.

Proof. First let B(x, r) = \{y ∈ C; |x − y| < r\}, S_{i,λ} = M_{i} ∩ B (λ, \frac{λ−w}{cos π/n}) and T_{i,λ} =

\{(λ − z)^{n}; z ∈ S_{i,λ}\}. Then T_{i,λ} = N_{i,λ}∩B (0, (\frac{λ−w}{cos π/n})^{n}), λ−S_{i,λ} = (λ−M_{i})∩B (0, \frac{λ−w}{cos π/n})

is a subset of Σ_{π/n} := {re^{iφ}; r > 0, |φ| < π/n} and T_{1,λ} = T_{2,λ}. Of course the mapping z ↦ z^{n} : Σ_{π/n} → C is injective. Thus, λ − S_{1,λ} = λ − S_{2,λ} and then S_{1,λ} = S_{2,λ}, too, for all λ > w.

Now, we finish the proof by the equations

M_{1} = M_{1} ∩ ⋃_{λ>w} B (λ, \frac{λ − w}{cos π/n}) = ⋃_{λ>w} M_{1} ∩ B (λ, \frac{λ − w}{cos π/n}) = ⋃_{λ>w} S_{1,λ}

= ⋃_{λ>w} S_{2,λ} = ⋃_{λ>w} M_{2} ∩ B (λ, \frac{λ − w}{cos π/n}) = M_{2} ∩ ⋃_{λ>w} B (λ, \frac{λ − w}{cos π/n}) = M_{2}

since C = ⋃_{λ>w} B (λ, \frac{λ−w}{cos π/n}).

Proof of Theorem 1.7. If T generates a C_{0}-semigroup on L_{1} let P = [1, ∞), otherwise P = (1, ∞).

Using the resolvent identity and Theorem 1.6 we find

(R(λ, A_{p}))^{n+1}f(x) = \frac{(−1)^{n}}{n!} \frac{d^{n}}{dλ^{n}} R(λ, A_{p})f(x) = ∫_{Ω} ∫_{0}^{∞} \frac{1}{n!}t^{n}e^{−λt}k(t, x, y) dtf(y) dy

for all f ∈ L_{p} and λ > ω_{1}. Applying Corollary 6.2 we obtain σ((R(λ, A_{2}))^{n+1}) = σ((R(λ, A_{p}))^{n+1}) for all λ > ω_{1} and p ∈ P.

Using the spectral mapping theorem for the resolvent, i.e.

σ(R(λ, A)) \ \{0\} = \frac{1}{λ − σ(A)} = \{\frac{1}{λ − μ}; μ ∈ σ(A)\}

and for bounded operators S and polynomials g (σ(g(S)) = g(σ(S))), we obtain

{1/z; z ∈ σ((R(λ, A_{p}))^{n+1}) \ {0}} = {(λ − μ)^{n+1}; μ ∈ σ(A_{p})}

for all p ∈ P and λ > ω_{1}. Applying Lemma 6.3 to N_{p,λ} := {(λ − z)^{n+1}; z ∈ σ(A_{p})} finishes the proof.

Remark 6.4. If a_{i,j} ∈ L_{∞} then by Theorem 1.6 the kernel associated with the semigroup satisfies a classical Gaussian estimate. Thus the semigroup extends to a C_{0}-semigroup on L_{1}.

Another way to show that we have a C_{0}-semigroup on L_{1} uses L_{1}-contractivity of S(t) = e^{−kt}T (t). The semigroup S is L_{1}-contractive if k + c − ∑_{j=1}^{n} ∂_{j}b_{1,j} ≥ 0 as a distribution. In fact, by Lemma 3.8 min {u_{+}, 1} ∈ V whenever u ∈ V. Now, a direct calculation shows E(u − min {u_{+}, 1} , min {u_{+}, 1}) ≥ 0 for all u ∈ D. This remains true for all u ∈ V as D is dense in V. By Ma and Röckner [15, Proposition I.4.3 and Theorem I.4.4] T is L_{1}-contractive.

The same arguments show that T is quasi L_{∞}-contractive if there is an k ∈ R such that k + c − ∑_{j=1}^{n} ∂_{j}b_{2,j} ≥ 0 as a distribution. In general, E is not quasi L_{∞}-contractive, see Ouhabaz [18, Remark 4.3.a] even though Condition 1.5 is satisfied.

Secondly, we have maximal L_{p}-L_{q} regularity, if Condition 1.5 is satisfied and a_{i,j} ∈ L_{∞} for all i, j = 1, …, n. Since we get classical Gaussian estimates by Theorem 1.6, we can use the result of Hieber and Pruess [12, Theorem 3.1] to prove Theorem 1.8. Moreover, there exists a constant M > 0 such that

∫_{0}^{∞} ∥u(t)∥_{q}^{p} dt + ∫_{0}^{∞} ∥u′(t)∥_{q}^{p} dt + ∫_{0}^{∞} ∥A_{q}u(t)∥_{q}^{p} dt ≤ M ∫_{0}^{∞} ∥f(t)∥_{q}^{p} dt

for all f ∈ L_{p}(R^{+}, L_{q}(Ω)).

Thirdly, we can prove Theorem 1.9 getting an analytic mapping on L_{1} in an open sector.

Proof of Theorem 1.9. This proof uses Theorem 1.6 and a technique of Arendt [3]. Let 0 < (m − 1)θ_{1} < (m − 1)θ_{2} < θ_{3} < θ_{4} < θ. Replacing T by (e^{−wt}T (t))_{t≥0} for some w > ω_{1}, we can assume that

∥T (z)∥_{L(L_{2})} ≤ M_{1} for all z ∈ Σ_{θ_{4}}
0 ≤ k(t, x, y) ≤ Ct^{−n/2}e^{−b(|x−y|^{m}/t)^{\frac{1}{m−1}}} for all t > 0, x, y ∈ Ω

By Corollary 4.8 and duality we know, if w is sufficiently large, that

∥T (t)∥_{L(L_{2},L_{∞})} ≤ M_{2}t^{−n/4} and ∥T (t)∥_{L(L_{1},L_{2})} ≤ M_{3}t^{−n/4} for all t > 0.

Now, choose δ ∈ (0, 1) such that δt+is ∈ Σ_{θ_{4}} whenever t+is ∈ Σ_{θ_{3}}. For any z = t+is ∈ Σ_{θ_{3}} we have

∥T (z)∥_{L(L_{1},L_{∞})} ≤ ∥T ((1 − δ)t/2)∥_{L(L_{1},L_{2})} ∥T (δt + is)∥_{L(L_{2})} ∥T ((1 − δ)t/2)∥_{L(L_{2},L_{∞})}

≤ M_{1}M_{2}M_{3}t^{−n/2} (\frac{1 − δ}{2} )^{−n/2} =: M_{4}(Re z)^{−n/2}.

Applying [3, Theorem 4.2] obtain the analyticity of T : Σ_{θ_{3}} → L(L_{1}, L_{∞}) ∼ L_{∞}(Ω × Ω). By [3, Lemma 4.1] there exists K : Σ_{θ_{3}} × Ω × Ω → C such that K(·, x, y) is analytic for all x, y ∈ Ω, K(z, ·, ·) ∈ L_{∞}(Ω × Ω) for all z ∈ Σ_{θ_{3}} and T (z)f(x) = ∫_{Ω} K(z, x, y)f(y) dy for all f ∈ L_{1} ∩ L_{2}. As L_{∞}(Ω × Ω) is isometric isomorph to L(L_{1}, L_{∞}) by k ↦ (f ↦ ∫_{Ω} k(·, y)f(y) dy), we obtain |K(z, x, y)| ≤ M_{4}(Re z)^{−n/2} for all z ∈ Σ_{θ_{3}} and almost all x, y ∈ Ω and K(t, x, y) = k(t, x, y) for all t > 0 and almost all x, y ∈ Ω.

Applying [3, Lemma 4.4] to H(z, x, y) = K(z^{m−1}, x, y)z^{(m−1)n/2} we obtain the estimate

|K(z, x, y)| ≤ M_{0} |z|^{−n/2} e^{−β(|x−y|^{m}/|z|)^{\frac{1}{m−1}}} for all z ∈ Σ_{θ_{1}} and almost all x, y ∈ Ω where M_{0} = sup \{M_{4}, C\} and β = \frac{sin(γ−θ_{1})}{sin γ}b.

Thus, sup_{z∈Σ_{θ_{1}},|z|>r} ∥T_{p}(z)∥_{p} < ∞ for all r > 0, p ∈ [1, ∞). Since the operators T_{p} are consistent, we can finish the proof by applying [3, Theorem 4.2] again.

We cannot prove the boundedness of the semigroup T on L_{1} near 0 using only a pseudo Gaussian estimate of order m ≠ 2.

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University of Ulm, Abteilung Angewandte Analysis, D-89069 Ulm, Germany

E-mail address: S.Karrmann@gmx.net
URL: http://www.mathematik.uni-ulm.de/m5/sk