DESINGULARIZING WEIGHTS AND THE HEAT KERNEL BOUNDS

Pierre D. Milman and Yu.A. Semenov

Abstract. We obtain sharp upper weighted bounds on the heat kernel for the parabolic operator ∂_{t} = Δ_{x} + V (t, x), x ∈ ℝ^{d}, d ≥ 3, with potential V having singularities of the type c|x|^{−2}, with a suitably small constant c. These bounds differ from the standard ones by a multiplicative factor of the form ψ(t − s) · ϕ(x) · ϕ(y) with appropriate weight functions ϕ(x) and the diagonal term ψ(τ) that differs from the standard case (τ^{−d/2}) for the positive sign singularity. The growth rate of the weights ϕ(x) depends (explicitly) on the constants c. This is the first result dealing with a situation where the lower and upper bounds behave differently at t = 0 and/or x = 0. Standard bounds (ϕ ≡ 1) and methods (estimates in L^{p}-spaces without desingularizing weights) fail for singular potentials. Even in the standard case of time dependent potentials from the parabolic Kato class our result is new.

I. INTRODUCTION. We obtain sharp upper weighted bounds on the heat kernel Z_{V} (t, x; s, y) for the parabolic operator ∂_{t} − Δ_{x} + V (t, x), x ∈ R^{d}, d ≥ 3, with potentials V having singularities of the type |x|^{−2}. These bounds are of the form Z_{V} (t, x; s, y) ≤ Cψ(t−s)ϕ(x)ϕ(y)e^{−c|x−y|^{2}/(t−s)} for appropriate weight functions ϕ(x), ”the diagonal term” functions ψ(τ) and constants C, c. Standard bounds (ϕ ≡ 1) and methods (estimates in L^{p}-spaces without desingularizing weights) fail for singular potentials. For example, in the case of the time independent potential satisfying the inequalities W +γV_{0} ≤ V ≤ W +βV_{0}, where W belongs to the Kato class, V_{0}(x) ≡ (d−2/2)^{2}|x|^{−2} and γ, β are arbitrary constants such that 0 ≤ γ ≤ β < 1, we establish two sided bounds on Z_{−V} (t, x; s, y) with weights ϕ_{i}(x) = |x|^{−σ_{i}} for |x| ≤ 1 and ϕ_{i}(x) = 1 for |x| ≥ 1, where (d − 2/2)(−1 + \sqrt{1 + γ}) < σ_{1} < σ_{2} < (d − 2/2)(1 − \sqrt{1 − β}), and ψ_{2}(τ) for the upper bound being the usual τ^{−d/2} while ψ_{1}(τ) for the lower bound equal [ψ_{2}(τ)]^{(q−1)/q}, 2 < q < d/σ_{1}. Even in the standard case of time dependent potentials from the parabolic Kato class our result is new.

Kato parabolic class. The following question is important. What is the “maximal” class of potentials for which a (unique) weak heat kernel exists and satisfies the upper and lower Gaussian bounds? Our answer to this question is as follows. Given a w ∈ L_{loc}^{1}(R_{+}^{1} × R^{d}), set

k(w; r) := ess sup_{u≥0, x∈R^{r}} ∫_{u}^{u+r} ∫_{R^{d}} Γ_{t−u}(x − y)|w|(t, y) dy dt

where Γ_{t}(z) := (4πt)^{−d/2} exp ( − \frac{|z|^{2}}{4t} ).

We say that w belongs to the Kato parabolic class and we write w ∈ K_{d}^{P} if and only if

k(w) := lim_{r→0} k(w; r)

exists and is finite.

Theorem 1. Let d ≥ 2 and V = V^{+}−V^{−}, V^{±} ≥ 0. Assume that V ∈ K_{d}^{P} and k(V^{−}) < 1. Then the unique weak heat kernel, Z_{V} (t, x; s, y), for parabolic operator ∂_{t} − Δ_{x} + V (t, x) in R^{d} exists and, for any ε ∈]0, 1 − k(V^{−})[ and δ > 0, there are constants c_{ε}^{(2)}, c_{ε,δ}^{(1)} > 0 and ω_{i} such that the estimates

c_{ε,δ}^{(1)}e^{(t−s)ω_{1}}Γ_{\frac{t−s}{1+k(V ^{+})+δ}} (x − y) ≤ Z_{V} (t, x; s, y) ≤ c_{ε}^{(2)}e^{(t−s)ω_{2}}Γ_{\frac{t−s}{1−k(V ^{−})−ε}} (x − y)

are satisfied for all t > s ≥ 0 and a.e. (x, y) ∈ R^{d} × R^{d}.

Its proof rests upon some consequences of DuHamel’s principle and the Lie product formula for propagators. This theorem generalizes and strengthens (some) results in [Se1] and [Zh1-3]. For its applications see [Se2].

Singular potentials. We now consider the case of d ≥ 3 and assume that potential V is time independent and satisfies the following constraints.

There are constants λ_{i}, γ, β; 0 ≤ γ ≤ β < 1, such that λ_{1} + γV_{0}(x) ≤ V (x) ≤ λ_{2} + βV_{0}(x) (pointwise a.e.).

Due to the Hardy inequality, we know that V is a −Δ-form bounded operator in the Hilbert space L^{2} = L^{2}(R^{d}, dx) with relative bound β so that the form difference H^{−} := −Δ ˙−V is a well defined (self-adjoint bounded from below) operator in L^{2}. Due to [Se2, Th.III.5.1], exp(−tH^{−}) extends by continuity to a holomorphic semigroup on L^{r} for all r ∈ I_{m} :=]p(β), p′(β)[ where

p′(β) = \frac{d}{d − 2} \frac{2}{1 − \sqrt{1 − β}} , p(β) = p′(β)/(p′(β) − 1) .

Moreover, for all t > 0, ℓ = 0, 1, …, p(β) < r ≤ s < p′(β), there are constants M = M(r, s, ℓ, λ_{2}) such that

∥∂_{t}^{ℓ}e^{−tH^{−}}∥_{r→s} ≤ Me^{λ_{2}t}t^{−ℓ−d(s−r)/2rs} .

Here and below we denote by ∥T ∥_{r→s} the operator norm of the map T: L^{r} → L^{s}.

It is well known that for such a singular potential the heat kernel of exp(−tH^{−}) does not admit the standard upper Gaussian bound and the semigroup is not ultracontractive. However, we show that for a suitable weight ϕ the weighted semigroup exp(−tH_{ϕ}^{−}): L_{ϕ}^{2} → L_{ϕ}^{2} is ultracontractive and derive an upper heat kernel bound of the form e^{−tH^{−}}(x, y) ≤ ct^{−d/2}ϕ(x)ϕ(y). We also consider operator H^{+} := −Δ ∔ V, the form sum of −Δ and V, and establish an upper heat kernel bound of the form e^{−tH^{+}}(x, y) ≤ C_{t}φ(x)φ(y) for appropriate functions φ and C_{t}. As a byproduct of these estimates we obtain two-sided weighted Gaussian bounds on e^{−tH^{−}}(x, y). These bounds become sharp when γ = β. To formulate precisely the results let us define weight functions

ϕ_{i}(x) := |x|^{−σ_{i}} if |x| ≤ 1 and ϕ_{i}(x) = 1 if |x| ≥ 1, i = 1, 2, 3 ,

where

σ_{1} = \frac{d − 2}{2}( − 1 + \sqrt{1 + γ}), σ_{2} = \frac{d − 2}{2}(1 − \sqrt{1 − β}) and σ_{3} = −σ_{1} .

Let q be any number from the interval ]2, d/σ_{1}[.

Theorem 2. There are constants c_{i} = c_{i}(q) > 0 and ω_{i} such that the inequalities

\begin{aligned}c_{1}t^{−d(q−1)/(2q)}e^{−ω_{1}t− \frac{|x−y|^{2}}{2t}} ϕ_{1}(x)ϕ_{1}(y) & ≤ e^{−tH^{−}}(x, y) ≤ c_{2}t^{−d/2}e^{ω_{2}t}ϕ_{2}(x)ϕ_{2}(y) , \\ e^{−tH^{+}}(x, y) & ≤ c_{3}t^{−d(1+q)/(2q)}e^{ω_{1}t}ϕ_{3}(x)ϕ_{3}(y) , \end{aligned}

hold for all t > 0 and x, y ∈ R^{d}.

Theorem 2 seems to be the first result dealing with a situation where the lower and upper bounds on e^{−tH^{−}}(x, y) behave differently at t = 0 and/or at x = 0.

Combined Kato-Singular potentials.

Theorem 2′. Assume that 1) W ∈ K_{d}^{P} with k(W) = 0; 2) For some constants 0 ≤ γ ≤ β < 1, the inequality γV_{0}(x) ≤ V (t, x) ≤ βV_{0}(x) is valid pointwise a.e. t > 0 and x ∈ R^{d}. Then there exist weak heat kernels Z_{W −V} , Z_{W +V} for parabolic operators ∂_{t} − Δ + W − V and ∂_{t} − Δ + W + V respectively and they are unique. There are constants c_{i} > 0, ω_{i} such that the estimates

c_{1}(t − s)^{−d(q−1)/(2q)}e^{−(t−s)ω_{1}−c_{0} \frac{|x−y|^{2}}{t−s}} ϕ̃_{1}(x)ϕ̃_{1}(y) ≤ Z_{W −V} (t, x; s, y)
≤ c_{2}(t − s)^{−d/2}e^{(t−s)ω_{2}} ϕ̃_{2}(x)ϕ̃_{2}(y) ,
Z_{W +V} (t, x; s, y) ≤ c_{3}t^{−d(1+q)/(2q)}e^{(t−s)ω_{1}} ϕ̃_{3}(x)ϕ̃_{3}(y)

hold for all t > 0, x, y ∈ R^{d}, where ϕ̃_{i} are defined in the same way as ϕ_{i} but with σ̃_{1} > σ_{1}, σ̃_{2} > σ_{2}.

To formulate an analogous result with the Gaussian factor in the upper bounds let us define, for any ε ∈ ]0, 1 ∧ (\frac{1}{β} − 1)[,

ϕ_{1,ε}(x) := |x|^{−σ_{i}(ε)} if |x| ≤ 1 and ϕ_{i,ε}(x) = 1 if |x| ≥ 1, i = 1, 2, 3,

where σ_{3}(ε) = −σ_{1}(ε) and

σ_{1}(ε) = \frac{d − 2}{2(1 − ε)}( − 1 + \sqrt{1 + γ − εγ}), σ_{2}(ε) = \frac{d − 2}{2(1 + ε)}(1 − \sqrt{1 − β − εβ}) .

Let q be any number from the interval ]2, d/σ_{1}(ε)[.

Theorem 3. There are constants c_{i} = c_{i}(ε, q) > 0 and ω_{i} = ω_{i}(ε) such that the inequalities

e^{−tH^{−}}(x, y) ≤ c_{1}t^{−d/2}ϕ_{2,ε}(x)ϕ_{2,ε}(y) exp ⌋ω_{1}t − c_{2} \frac{|x − y|^{2}}{4t} ⌊,

e^{−tH^{+}}(x, y) ≤ c_{3}t^{−d(1+q)/(2q)}ϕ_{3,ε}(x)ϕ_{3,ε}(y) exp ⌋ω_{2}t − c_{4} \frac{|x − y|^{2}}{4t} ⌊

are satisfied for all t > 0 and x, y ∈ R^{d}.

Acknowledgement. We would like to thank Yu.A. Ovchinnikov for fruitful discussions of Theorem 2 .

II. AUXILIARY RESULTS. Let X be a Banach space and assume that for each t the operator −A(t) generates a holomorphic semigroup in X. Assume that there is a family U of strongly continuous functions U^{t,s} (propagators) defined on 0 ≤ s ≤ t < ∞, which takes values in L(X), the space of all bounded linear operators from X into X; U^{s,s} = I for each s ≥ 0; U^{t,r}U^{r,s} = U^{t,s} for 0 ≤ s ≤ r ≤ t < ∞; there is a dense subspace D_{U} such that ∂_{t}U^{t,s}f + A(t)U^{t,s}f = 0 and ∂_{t}U^{t,s}f − U^{t,s}A(s)f = 0 for all f ∈ D_{U}.

Let P be another such family generated by −B(t) with D_{P} = D = D_{U}. Let us assume that P^{t,s}D ⊆ D for 0 ≤ s ≤ t < ∞ and P^{t,s}f (resp. U^{t,s}f) are strongly differentiable for each f ∈ D with respect to t (resp. s ∈ [0, t]). Then the following DuHamel formula

U^{t,s}f − P^{t,s}f = ∫_{s}^{t} U^{t,τ} (B(τ) − A(τ))P^{τ,s}f dτ, ∀f ∈ D

is valid. Set W (t) := B(t) − A(t) and assume that W (t) ∈ L(X). Let ∥P^{t,s}∥ ≤ Me^{ω(t−s)}. Then the following Miyadera condition holds: There are numbers η > 0 and 0 < a < 1 such that

(M)  sup_{u≥0} ∫_{u}^{u+η} ∥W (τ)P^{τ,u}f∥e^{−ωτ} dτ ≤ a∥f∥, ∀f ∈ D .

A simple iteration process based on the DuHamel formula yields

(∗)  ∥U^{t,s}∥ ≤ \frac{M}{1 − a} exp ((ω + \frac{1}{η} log \frac{M}{1 − a})(t − s)) .

We will also use a product formula for quasi contractions R, U, T generated by −C(t), −A(t), −V (t) respectively. If D(C(t)) = D(A(t)) ∩ D(V (t)) and C(t) = A(t) + V (t) for all t, and ∥C(τ)R^{τ,r}C^{−1}(r)∥ ≤ δ(t_{0}) < ∞ for all 0 ≤ τ − r ≤ t_{0} < ∞, then the Lie product formula

(∗∗)  R^{t,s} = str. lim_{n→∞} F_{n}^{t,s}

holds, where F_{n}^{t,s} := U^{t,t_{n−1}}T^{t,t_{n−1}}U^{t_{n−1},t_{n−2}}T^{t_{n−1},t_{n−2}} · . . . · U^{t_{1},s}T^{t_{1},s}, t_{i} := s + \frac{i}{n}(t − s). One can easily prove this by mimicking the corresponding proof for semigroups in [Ne].

Let now (M, μ) be a measure space with σ-finite measures μ ≥ 0, X = L^{r} = L^{r}(M, μ) for some r ∈ [1, ∞[. Let U, T from (∗∗) be positivity preserving, and U ultracontractive. The latter means that ∥U^{t,s}∥_{1→∞} ≤ S(t−s) for all t−s > 0 where function S(τ) is bounded on any compact interval from ]0, ∞[. Suppose further that for some p > 1 operators −pV (t) and −A(t) − pV (t) generate T_{p} and R_{p} respectively, and that (∗∗) is also valid for R_{p}, U, T_{p}. If V (t) is an operator of multiplication by a real valued function V (t, x), then the following (pointwise a.e.) inequality

(∗ ∗ ∗)  R^{t,s}h ≤ (R_{p}^{t,s}h^{αp})^{1/p}(U^{t,s}h^{(1−α)p′})^{1/p′}

is valid for all 0 ≤ h ∈ L^{1}∩L^{∞}, 0 ≤ α ≤ 1, where p′ = p/(p−1). This inequality is standard in the functional integration theory. In our setting it follows via the Lie product formula (cf. [HS]). Indeed, by ultracontractivity, U^{t,s} is an integral operator. Let N(t, x; s, y) denote its integral kernel, which is non-negative due to positivity preserving of U. Then

F_{n}^{t,s}(V)h(x) ≡ F_{n}^{t,s}h(x)
= ∫_{M} ⋯ ∫_{M} N(t, x; t_{n−1}, x_{n−1})e^{− \frac{t−s}{n} V (t_{n−1},x_{n−1})} · . . .
· N(t_{1}, x_{1}; s, y)e^{− \frac{t−s}{n} V (s,y)}h(y)dx_{n−1} · . . . · dx_{1}dy

(here we are using Hölder’s inequality and the reproductive property of U)

≤ (F_{n}^{t,s}(pV)h^{αp}(x))^{1/p}(U^{t,s}h^{(1−α)p′}(x))^{1/p′} .

We now pass to the limit n → ∞ in this inequality, using (∗∗), which proves (∗ ∗ ∗).

Proposition 1. Let p = m/(m − 1) for some integer m ≥ 2. In addition to the above conditions assume that ∥R_{p}^{t,s}1∥_{∞} ≤ θ_{∞}(t_{0}) and ∥R_{p}^{t,s}∥_{1→1} ≤ θ_{1}(t_{0}) for all 0 ≤ t − s ≤ t_{0} < ∞. Then there is a constant c = c(θ_{1}(t_{0}), θ_{∞}(t_{0})) such that, for all 0 ≤ t − s ≤ t_{0},

∥R^{t,s}∥_{1→∞} ≤ c^{m+1}S(\frac{t − s}{m}) .

Proof. Applying (∗ ∗ ∗) with α = 0 yields

\begin{aligned}∥R^{t,s}f∥_{∞} & ≤ ∥R_{p}^{t,s}1∥_{∞}^{1/p}∥U^{t,s}|f|^{p′}∥_{∞}^{1/p′} \\ & ≤ ∥R_{p}^{t,s}1∥_{∞}^{1/p}∥U^{t,s}∥_{1→∞}^{1/p′}∥f∥_{m} . \end{aligned}

Applying (∗ ∗ ∗) with α = n/(n + 1) where 1 ≤ n ≤ m − 1 yields

∥R^{t,s}f∥_{m/n} ≤ ∥R_{p}^{t,s}|f|^{αp}∥_{m/np}^{1/p}∥U^{t,s}|f|^{(1−α)m}∥_{∞}^{1/m}
≤ ∥R_{p}^{t,s}∥_{(m−1)/n→(m−1)/n}^{1/p}∥f∥_{αm/n}^{α}∥U^{t,s}∥_{1→∞}^{1/m}∥f∥_{(1−α)m}^{1−α}
≤ ∥R_{p}^{t,s}∥_{(m−1)/n→(m−1)/n}^{1/p}∥U^{t,s}∥_{1→∞}^{1/m}∥f∥_{m/(n+1)} .

By interpolation, ∥R_{p}^{t,s}∥_{q→q} ≤ c(θ_{1}(t_{0}), θ_{∞}(t_{0})) for all 1 ≤ q ≤ ∞ and 0 ≤ t − s ≤ t_{0}. By the ultracontractivity of U, ∥U^{t,s}∥_{1→∞} ≤ S(t − s). The desired assertion follows now from the reproductive property of R^{t,s}.

Proposition 1 is particularly useful if one can effectively estimate θ_{i}(t_{0}). The latter can be carried out by means of (∗).

III. PROOF OF THEOREM 1. We are going now to apply the techniques developed in section II to the parabolic operator ∂_{t}−Δ+V in R^{d}, d ≥ 2. Let V fulfill the assumptions of Theorem 1. Choose a sequence {V_{n}, n ≥ 1} of bounded smooth potentials such that V_{n}^{±} ≤ V^{±} and k(V^{±} − V_{n}^{±}) → 0 as n → ∞. Set P^{t,s} := e^{(t−s)Δ}. Then it is easy to see that the Miyadera condition (M) in L^{1} = L^{1}(R^{d}, dx) is satisfied for V_{n}^{−} and (p_{0} − 1)V_{n}^{+} uniformly in n, where constant p_{0} is determined by the condition: 0 < k(V^{+})(p_{0} − 1) < 1. Set A(t) := −Δ + V_{n}(t) with domain D(A(t)) := (1 − Δ)^{−1}L^{1}. Then, due to the standard theory (see e.g. [Fr]), U, P fulfill all of the assumptions which led us to (∗). As a result we have for 0 ≤ s ≤ t < ∞

(∗′) ∥U^{t,s}∥_{1→1} = ∥U^{t,s} 1∥_{∞} ≤ \frac{1}{1 − α} exp ((ω + \frac{1}{η} log \frac{1}{1 − α})(t − s))

with constants α, ω and η, dependent only on k(V^{−}).

The same estimate holds for U_{p} generated by A(t) := −Δ + pV_{n}(t) for any p ∈ ]1, 1/k(V^{−})[. We will denote it by (∗′_{p}). Of course, constants α, ω and η from (∗′_{p}) depend on pk(V^{−}).

At this stage it is easy to see that Proposition 1 is applicable in this setting. As a consequence it follows

∥U^{t,s}∥_{1→∞} ≤ c^{m+1}m^{d/2}(4π(t − s))^{−d/2} (0 ≤ s < t < ∞),

where m ≥ 2 is the maximal integer satisfying the inequality k(V^{−}) < (m − 1)/m, and c denotes the R.H.S. of (∗′_{p}).

Setting p_{12} = p_{1}p_{2}, m_{12} = \frac{m_{1}m_{2}}{(m_{1}−1_{(}m_{2}−1)}, p_{i} = \frac{m_{i}}{m_{i}−1} where m_{i} > 2, m_{1}m_{2}k(V^{−}) < (m_{1} − 1)(m_{2} − 1), we immediately conclude that

∥U_{p_{12}}^{t,s}∥_{1→∞} ≤ c^{m_{12}+1}m_{12}^{d/2}(4π(t − s))^{−d/2} (0 ≤ s < t < ∞) .

Using this bound and (∗ ∗ ∗), we have U^{t,s}(x, y) ≤ (U_{p_{12}}^{t,s}(x, y))^{1/p_{12}}(P^{t,s}(x, y))^{1/p′_{12}} or

(1_{−})  Z_{−V_{n}^{−}}(t, x; s, y) ≤ c_{2}e^{(t−s)ω_{2}}Γ_{(t−s)p′_{12}}(x − y)

for all x, y ∈ R^{d} and all 0 ≤ s < t < ∞.

The same argument gives

(1_{−(p_{0}−1)})  Z_{−(p_{0}−1)V_{n}^{+}}(t, x; s, y) ≤ c_{3}e^{(t−s)ω_{3}}Γ_{(t−s)c_{4}}(x − y)

for all x, y ∈ R^{d} and all 0 ≤ s < t < ∞.

Finally, put W ≡ \frac{p_{0}−1}{p_{0}}V_{n}^{+} and let H_{W} (t) := −Δ + W (t) denote the generator of H_{W}^{t,s}. Since −Δ = H_{W} (t) − W (t), we obtain, using (∗ ∗ ∗), (e^{(t−s)Δ}h)^{p_{0}} ≤ (H_{p_{0}W}^{t,s} h)^{p_{0}−1}H_{−p_{0}W}^{t,s} h so that

(Γ_{t−s}(x − y))^{p_{0}} ≤ Z_{−(p_{0}−1)V_{n}^{+}}(t, x; s, y)(Z_{V_{n}^{+}}(t, x; s, y))^{p_{0}−1}

(here we are using (1_{−(p_{0}−1)}))

≤ c_{3}e^{(t−s)ω_{3}}Γ_{(t−s)c_{4}}(x − y)(Z_{V_{n}^{+}}(t, x; s, y))^{p_{0}−1}

or

(1_{+})  c_{4}e^{(t−s)ω_{1}}Γ_{(t−s)c_{5}}(x − y) ≤ Z_{V_{n}^{+}}(t, x; s, y) .

As soon as (a priori) estimates (1_{−}) and (1_{+}) are established the proof of Theorem 1 can be completed using standard approximation arguments (see [Se2], [Zh1-3]).

IV. PROOF OF THEOREM 2. THE UPPER BOUND ON e^{−tH^{−}}(x, y). Let 0 ≤ ϕ, 1/ϕ ∈ L_{loc}^{2}(R^{d}, dx). We define a unitary map Φ from L_{ϕ}^{2} = L^{2}(R^{d}, ϕ^{2}(x)dx) onto L^{2} = L^{2}(R^{d}, dx) by Φf = ϕf and set H_{ϕ}^{−} = Φ^{−1}H^{−}Φ. We denote L^{r}(R^{d}, ϕ^{2}(x)dx) by L_{ϕ}^{r} , ∥ · ∥_{r,ϕ} denotes the norm in L_{ϕ}^{r} and ⟨f, g⟩, ⟨f, g⟩_{ϕ} denote pairings between dual spaces. We write V ∈ P K(β) if there are constants β and C(β) such that

(∗)  |⟨V h, h⟩| ≤ β⟨∇h, ∇h⟩ + C(β)⟨h, h⟩

for all h ∈ H^{1}(R^{d}) := the standard Sobolev space.

Let us recall the Nash inequality

⟨∇h, ∇h⟩ ≥ C_{N}⟨h, h⟩^{1+2/d}∥h∥_{1}^{−4/d} , ∀h ∈ L^{1} ∩ H^{1}(R^{d}) .

We will usually make no difference in writing an operator and its extension by continuity.

The following conditions imply ultracontractivity of e^{−tH_{ϕ}^{−}}.

Proposition 2. Let V ∈ P K(β) for some β < 1. Set H^{−} := −Δ ˙−V. If ϕ ≥ 1 and e^{−tH_{ϕ}^{−}} ∈ L(L_{ϕ}^{1}) with

(∗∗)  ∥e^{−tH_{ϕ}^{−}}∥_{1→1,ϕ} ≤ ce^{ωt} for all t ≥ 0 and some constants c, ω,

then e^{−tH_{ϕ}^{−}} is ultracontractive and, for all t > 0,

∥e^{−tH_{ϕ}^{−}}∥_{1→∞,ϕ} ≤ (\frac{d}{2C_{N}(d)(1 − β)})^{d/2}c^{2}e^{λt}t^{−d/2} ,

where λ = max{ω, C(β)} and C_{N}(d) = C_{N} is the (sharp) constant from the Nash inequality.

Proof. Let U ∈ L_{ϕ}^{2} ∩ L_{ϕ}^{1}. Set u_{t} := e^{−t(λ+H_{ϕ}^{−})}u. Since e^{−tH_{ϕ}^{−}} = Φ^{−1}e^{−tH^{−}}Φ and e^{−tH^{−}}L^{2} ⊂ D(H^{−}) ∀t > 0, we have ϕu_{t} ∈ D(H^{−}). By the definition of H^{−}, D(H^{−}) ⊂ H^{1}(R^{d}) and ⟨H^{−}g, h⟩ = ⟨∇g, ∇h⟩ − ⟨V g, h⟩, g ∈ D(H^{−}), h ∈ H^{1}(R^{d}). Thus, using the equation ∂_{t}u_{t} + (λ + H_{ϕ}^{−})u_{t} = 0, we have

−\frac{1}{2} \frac{d}{dt}⟨u_{t}, u_{t}⟩_{ϕ} = ⟨(λ + H_{ϕ}^{−})u_{t}, u_{t}⟩_{ϕ}
= λ⟨u_{t}, u_{t}⟩_{ϕ} + ⟨H^{−}(ϕu_{t}), ϕu_{t}⟩
≥ (1 − β)⟨∇(ϕu_{t})∇(ϕu_{t})⟩
≥ (1 − β)C_{N}⟨u_{t}, u_{t}⟩_{ϕ}^{1+2/d}∥ϕu_{t}∥_{1}^{−4/d}

or, setting v = ⟨u_{t}, u_{t}⟩_{ϕ},

\frac{d}{dt}(v^{−2/d}) ≥ \frac{4(1 − β)}{d}C_{N}∥ϕu_{t}∥_{1}^{−4/d} .

Noting that ϕ ≥ 1 implies ∥ϕu_{t}∥_{1} ≤ ∥u_{t}∥_{1,ϕ}, one has, using (∗∗),

\frac{d}{dt}(v^{−2/d}) ≥ \frac{4(1 − β)}{d}C_{N}c^{−4/d}∥u∥_{1,ϕ}^{−4/d} .

Integrating this inequality over the interval [0, t] yields

∥e^{−tH_{ϕ}^{−}}∥_{1→2,ϕ} ≤ (\frac{d}{4(1 − β)C_{N}} )^{d/4}ct^{−d/4}e^{tλ} .

By duality,

∥e^{−tH_{ϕ}^{−}}∥_{1→∞,ϕ} ≤ (\frac{d}{2(1 − β)C_{N}} )^{d/2}c^{2}t^{−d/2}e^{tλ} ,

which completes the proof.

Remark. There is an alternative proof of Proposition 2, based on the ideas from preceding sections.

Proposition 3. Let V ∈ P K(β) for some β < 1. Suppose that

ϕ, |∇ϕ|, |\frac{∇ϕ}{ϕ}|, \frac{Δϕ}{ϕ} ∈ L_{loc}^{1}(R^{d}, dx) and − \frac{Δϕ}{ϕ} − V ≥ −ω for some ω < ∞ .

Then

∥e^{−tH_{ϕ}^{−}}∥_{1→1,ϕ} ≤ e^{ωt} .

Proof. Since C_{0}^{∞}(R^{d}) is a form core of H^{−}, Φ^{−1}C_{0}^{∞}(R^{d}) is a form core of H_{ϕ}^{−}. The quadratic form h_{ϕ} associated with H_{ϕ}^{−} on Φ^{−1}C_{0}^{∞}(R^{d}) has the form

h_{ϕ}[u] = ⟨∇u, ∇u⟩_{ϕ} − ⟨(\frac{Δϕ}{ϕ} + V )u, u⟩_{ϕ}

which clearly shows that the quadratic form h_{ϕ} + ω is Markovian. Proposition 3 is proved.

Now we construct ϕ satisfying the conditions of Propositions 2 and 3 for V (x) ≤ λ_{2} + βV_{0}(x) (see the Introduction). Let ϕ(x) = |x|^{−σ_{2}} if |x| ≤ \frac{1}{2}, ϕ(x) = 1 if |x| ≥ 1 and 1 ≤ ϕ ∈ C^{2}(R^{d}\B_{1/2}(0)). We have −\frac{Δϕ}{ϕ} − V ≥ −λ_{2} on the ball B_{1/2}(0) so ϕ does fulfill all the assumptions. The required part of Theorem 2 is proved.

V. PROOF OF THEOREM 2. THE UPPER BOUND ON e^{−tH^{+}}(x, y). We use the same idea of passing to a weighted space. Denoting a new weight function by φ we construct H_{φ}^{+} in L_{φ}^{2} by means of the unitary transform Φ_{1}: L_{φ}^{2} ↦ L^{2} defined by Φ_{1}f = φf so that H_{φ}^{+} = Φ_{1}^{−1}H^{+}Φ_{1}.

First let us formulate a proposition analogous to Proposition 2. Proposition 4. Let V ∈ P K(β) for some β < 1. Let H := −Δ ∔ V. Suppose that

(∗)  ∥e^{−tH}φ^{−1}g∥_{1} ≤ ĉe^{ω̂t}t^{−d/(2q)}∥g∥_{1}

for all g from a dense subspace of L^{1}. Then there are constants C = C(d, β, ĉ) and λ such that the inequality

∥e^{−tH_{φ}}∥_{1→∞,φ} ≤ Ce^{λt}t^{−d(1+q)/(2q)}

is valid for all t > 0.

We skip a proof of Proposition 4. In contrast to ϕ from Proposition 2, we now take a weight φ which has the form |x|^{ℓ}, ℓ > 0, in B_{1/2}(0), and this means that (∗) is fairly non-trivial.

Let us also suppose that 1 ≥ φ(x), if |x| ≥ 1. Using evident inequalities φ^{−1} ≤ 1 + φ^{−1}1_{|x|≤1}, ∥e^{−tH}φ^{−1}g∥_{1} ≤ ∥φe^{−tH}φ^{−1}g∥_{1} + ∥φ^{−1}1_{|x|≤1}∥_{q}∥φe^{−tH}φ^{−1}g∥_{q′}, we reduce the problem to (L^{1}, L^{1}) and (L^{1}, L^{q′}) estimates for φe^{−tH}φ^{−1} acting in ordinary L^{r}-spaces. Now equality φe^{−tH}φ^{−1} = e^{−tφHφ^{−1}} (in (L(L_{φ^{−1}}^{2})) is no use. Following [Se1] we will construct −H(φ), the generator of C_{0}-semigroup in L^{2}, such that φe^{−tH}φ^{−1}g = e^{−tH(φ)}g for all g from a dense subset of L^{2}. We stress that H(φ) only formally coincides with φHφ^{−1} of domain {φu; u ∈ D(H)}. We then show that e^{−tH(φ)} for V ≥ λ_{1} + γV_{0} is indeed (quasi) contractive and (L^{1}, L^{q′}) smooth (with ∥e^{−tH(φ)}∥_{1→q′} ≤ ct^{−d/(2q)}e^{tω}).

Let ρ: R^{d} ↦ R_{+} = [0, ∞[ and let H := −Δ ∔ V ≥ 0 be such that H ≥ −c_{2}Δ − c_{3} (in the sense of the quadratic forms in L^{2}) for some constants c_{3}, 0 < c_{2}. We say that a weight ρ and the operator H are compatible if

1) ρ, \frac{1}{ρ}, |∇ρ|, |\frac{∇ρ}{ρ} | ∈ L_{loc}^{2}(R^{d});

2) (\frac{∇ρ}{ρ})^{2} ≤ c_{0}H + c_{1} (in the sense of the quadratic forms in L^{2})

for some constants c_{0} < 1, c_{1} < ∞. Proposition 5. Let a weight ρ and the operator H be compatible. In the complex Hilbert space L^{2} with the inner product ⟨u, v⟩, linear with respect to u, define the quadratic forms t, ℓ, ℓ_{i}, i = 1, 2, 3, by

t[u, v] = ⟨H^{1/2}u, H^{1/2}v⟩ − ℓ[u, v], ℓ = ℓ_{1} + ℓ_{2} + ℓ_{3}, D(t) = D(ℓ_{i}) = Q(H) × Q(H), ℓ[u, v] := ℓ(ρ)[u, v], ℓ_{i}[u, v] := ℓ_{i}(ρ)[u, v], ℓ_{1}(ρ)[u, v] := ⟨u, ρ^{−1}∇ρ · ∇v⟩, ℓ_{2}(ρ)[u, v] := −\overline{ℓ_{1}(ρ)[v, u]}, ℓ_{3}(ρ)[u, v] := ⟨uρ^{−2}(∇ρ)^{2}v⟩.

Then i) the form t is m-sectorial,

t[u, v] = ⟨H(ρ)u, v⟩, u ∈ D(H(ρ)) ⊂ Q(H), v ∈ Q(H),

where H(ρ) is the quasi m-accretive operator associated with t.

ii) For all λ ≥ λ_{0}, where λ_{0} > 0 ∨ \frac{c_{1}}{c_{0}} ∨ \frac{c_{3}}{c_{2}} , and all g ∈ L_{com}^{∞}(R^{d}),

\begin{aligned}ρ(λ + H)^{−1}ρ^{−1}g & = (λ + H(ρ))^{−1}g , \\ ρe^{−tH}ρ^{−1}g & = e^{−tH(ρ)}g . \end{aligned}

Proof. Define the (complex) Hilbert spaces X_{+} ⊂ L^{2} ⊂ X_{−} setting X_{+} = (Q(H), ∥u∥_{+}) with ∥u∥_{+} = ∥(λ_{0} + H)^{1/2}u∥_{2}, X_{−} = (X_{+})^{∗}, where, given u, v ∈ X_{+}, one has

0 ≤ ℓ_{3}[u] ≤ c_{0}∥u∥_{+}^{2} ,

|ℓ_{1}[u, v]| ≤ ⟨|uρ^{−1}∇ρ|, |∇v|⟩ ≤ \sqrt{\frac{c_{0}}{c_{2}}} ∥u∥_{+}∥v∥_{+} ,

|ℓ_{2}[u, v]| = |ℓ_{1}[v, u]| ≤ \sqrt{\frac{c_{0}}{c_{2}}} ∥u∥_{+}∥v∥_{+} ,

Re t[u] = ∥u∥_{+}^{2} − ℓ_{3}[u] − λ_{0}∥u∥_{2}^{2} , Im t[u] = \sqrt{−1} (ℓ_{1}[u] − ℓ_{2}[u]) , (1 − c_{0})∥u∥_{+}^{2} ≤ Re t[u] + λ_{0}∥u∥_{2}^{2} ≤ ∥u∥_{+}^{2} .

The above shows that t is m-sectorial and therefore (see [K1, Ch. VI]) i) is proved. It is also clear that

⟨H^{1/2}u, H^{1/2}v⟩ = ⟨Ĥu, v⟩ where Ĥ : X_{+} ↦ X_{−} , and D(H) = {f ∈ X_{+} : Ĥf ∈ L^{2}} , Ĥ | D(H) = H, ℓ[u, v] = ⟨Êu, v⟩, ℓ_{i}[u, v] = ⟨Ê_{i}u, v⟩, Ê, Ê_{i} : X_{+} ↦ X_{−} , H(ρ) = Ĥ − Ê | D(H(ρ)), D(H(ρ)) = {f ∈ X_{+} : Ĥf − Êf ∈ L^{2}} .

Set B̂ := λ + Ĥ − Ê_{3}, λ ≥ λ_{0}, and define B := B̂ | {f ∈ X_{+} : B̂f ∈ L^{2}} – the form sum of λ + H and −(ρ^{−1}∇ρ)^{2}. Then B^{−1/2}: X_{−} ↦ L^{2}, B^{−1/2}: L^{2} ↦ X_{+}. Define G := −\sqrt{−1} B^{−1/2}(Ê_{1} + Ê_{2})B^{−1/2} so that G = G^{∗}: L^{2} ↦ L^{2} and

(∗∗)  (λ + H(ρ))^{−1} = B^{−1/2}(1 − \sqrt{−1} G)^{−1}B^{−1/2} .

Define ρ_{n}, n ∈ N, by

ρ_{n}(x) ; = \{ \begin{matrix}ρ(x) & if \frac{1}{n} ≤ ρ(x) ≤ n, \\ n & if ρ(x) ≥ n, \\ 1/n & if \frac{1}{n} ≥ ρ(x), \end{matrix}

Then ρ_{n} and H are compatible (with the same constants c_{i}) and hence (∗∗) also holds for ρ_{n}. Let us show that

(λ + H(ρ_{n}))^{−1} → (λ + H(ρ))^{−1} weakly in L^{2}.

Indeed, since B^{−1/2}(ρ_{n}) → B^{−1/2}(ρ) strongly in L^{2} by the monotone convergence theorem for a non-increasing sequence of symmetric quadratic forms [K1, Ch. VIII, sec. 3], we only have to prove that lim_{n} ⟨(Ê_{i}(ρ) − Ê_{i}(ρ_{n}))f, g⟩ = 0 for i = 1, 2 and all f, g ∈ X_{+}. But, for instance, |⟨(Ê_{1}(ρ) − Ê_{1}(ρ_{n}))f, g⟩| ≤ ∥∇f∥_{2}⟨(1 − 1_{n})(ρ^{−1}∇ρ)^{2}g, g⟩^{1/2} where 1_{n} is the indicator of the support of ρ_{n}, and hence the required convergence follows e.g., from the Fatou lemma.

Next it is easily seen that u ∈ Q(H) implies that ρ_{n}^{±1}u ∈ Q(H) and

(∗ ∗ ∗∗)  ⟨H^{1/2}ρ_{n}^{−1}u, H^{1/2}ρ_{n}v⟩ = t(ρ_{n})[u, v],  u, v ∈ Q(H) .

Note that T^{t} := ρ_{n}e^{−tH}ρ_{n}^{−1}, t ≥ 0, is a C_{0}-semigroup on L^{2} with ∥T^{t}∥_{2→2} ≤ n^{2}. Let −A denote its generator. We claim that A = H(ρ_{n}). Indeed, given f ∈ L^{2} and λ > 0, define u = (λ + A)^{−1}f. Since (λ + A)^{−1} = θ(λ + H)^{−1}θ^{−1}, θ ≡ ρ_{n}, we see that θ^{−1}u ∈ D(H) and (λ + H)θ^{−1}u = θ^{−1}f. Therefore, for all v ∈ X_{+}, ⟨(λ + H)θ^{−1}u, θv⟩ = ⟨f, v⟩ or ⟨f − λu, v⟩ = ⟨H^{1/2}θ^{−1}u, H^{1/2}θv⟩. The latter means, due to (∗ ∗ ∗∗), that u ∈ D(H(ρ_{n})) and ⟨H(ρ_{n})u, v⟩ = ⟨f − λu, v⟩ or H(ρ_{n})u = Au. Thus −A ⊂ −H(ρ_{n}), and since both of them are generators, the claim is proved. In particular, we have exp [ − tH(ρ_{n})] = ρ_{n} exp[−tH]ρ_{n}^{−1}. Assertion ii) of Proposition 5 now follows from (∗ ∗ ∗∗).

Now we construct a weight φ satisfying the conditions of Propositions 4 and 5 for H^{+} = −Δ ∔ V where V (x) ≥ λ_{1} + γV_{0}(x) (see the Introduction). Set φ(x) = |x|^{σ_{1}} if |x| ≤ \frac{1}{2}, φ(x) = 1 if |x| ≥ 1 and 2^{−σ_{1}} ≤ φ ∈ C^{2}(R^{d}\B_{1/2}(0)).

Proposition 6. Let ρ = φ. Then i) ρ and H^{+} are compatible with c_{0} ≤ (\sqrt{1 + γ}−1)^{2} < \frac{1}{6}; ii) e^{−tH^{+}(ρ)} is L^{1}-quasi contraction; iii) ∥e^{−tH^{+}(ρ)}∥_{1→q′} ≤ ce^{tω}t^{−d/(2q)}.

Proof. i) follows directly from the definitions of φ and H^{+}. Since −ρ^{−1}(Δρ)+V ≥ −ω for some finite constant ω and Δ ˙−2∇ · (ρ^{−1}∇ρ) generates a L^{1}-contraction semigroup [LS1], we conclude, e.g., by the Trotter-Kato semigroup formula [K2], that −H^{+}(ρ) = Δ ˙−2∇·(ρ^{−1}∇ρ) ˙−(V −ρ^{−1}(Δρ)) generates a quasi L^{1}-contraction semigroup, which proves ii). iii) follows from [LS1].

Proposition 6 shows that our weight φ does satisfy the conditions of Propositions 4 and 5 and therefore the upper bound on e^{−tH^{+}}(x, y) is proved.

The lower bound on e^{−tH^{−}}(x, y) follows now from the upper bounds on e^{−tH^{±}}(x, y) via the inequality Λ_{t}(x − y) ≤ (e^{−tH^{+}}(x, y))^{1/2}(e^{−tH^{−}}(x, y))^{1/2}.

Concluding Remarks. 1. The proof of Theorem 2′ can be obtained by a simple combination of Theorem 1 and the proof of Theorem 2. Slightly modifying the proof of Theorem 2 one can derive Theorem 3 either by applying non-linear inequality (∗ ∗ ∗) from section II or using the method of E.B. Davies [Da] as it was presented in [LS2]. We skip the details.

2. The ideas of this paper might be applied to the operator −Δ + b · ∇ + V with a singular drift and potential. We hope to investigate it in a forthcoming paper.

References

[Ar] D.G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup.Pisa 22 (1968), 607-694.

[Da] E.B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics 92 (Cambridge University Press, 1989).

[HS] I.W. Herbst and A.D. Sloan, Perturbation of translation invariant positivity preserving semigroups on L^{2}(ℝ^{N}), Trans. Amer. Math. Soc. 233 (1978), 325–360.

[Ka1] T. Kato, Perturbation theory for linear operators, 2nd edition, Springer-Verlag, Berlin-Heidelberg-New York, 1980.

[Ka2] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in Advance in Math. Suppl. Studies 3 Academic Press, New York 1978, 185–195.

[LS1] V. Liskevich and Yu.A. Semenov, Some problems on Markov semigroups, in Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top. II, Akademie Verlag, Berlin, 1996, 163-217.

[LS2] V. Liskevich and Yu.A. Semenov, Two-sided Estimates of the Heat Kernel of the Schrödinger operator, Bull. London Math. Soc. 30 (1998), 596-602.

[Ne] E. Nelson, Feynman integrals and the Schrödinger equations, J. Math. Phys. 5 (1964), 332-343.

[Se1] Yu.A. Semenov, Stability of L^{p}-spectrum of generalized Schrödinger operators and equivalence of Green’s functions, IMRN, No. 12 (1997), 573-593.

[Se2] Yu.A. Semenov, Heat kernel bounds. L^{1}-iteration technique. The Nash algorithm, Bull. AMS, to appear.

[SV] R. Schnaubelt and J. Voigt, The non-autonomous Kato class, Preprint,1998.

[Zh1] Qi S. Zhang, On a parabolic equation with a singular lower order term, Transactions of AMS, 348 7 (1996), 2811-2844.

[Zh2] Qi S. Zhang, On a parabolic equation with a singular lower order term, Part II: The Gaussian bounds, Indiana Univ. Math. J., 43 3 (1997), 989-1020.

[Zh3] Qi S. Zhang, Global existance and local continuity of semilinear parabolic equations, Comm. Partial D.E. 22 (1997), 1529-1558.

P.D. Milman and Yu.A. Semenov Department of Mathematics University of Toronto Toronto, Ontario M5S 3G3, Canada milman@math.toronto.edu semenov@math.toronto.edu