Some Analytic Results for Kimura Diffusion Operators

Abstract

In this note, we prove several analytical results about generalized Kimura diffusion operators, L, defined on compact manifolds with corners, P. It is shown that the $\mathcal C^{0}(P)$-graph closure of L acting on $\mathcal C^{2}(P)$ always has a compact resolvent. In the 1d-case, where P = [0,1], we also establish a gradient estimate $\|\partial _{x} f\|_{\mathcal C^{0}([0,1])}\leq C\| L f\|_{\mathcal C^{0}([0,1])},$ provided that L has strictly positive weights at ∂[0,1] = {0,1}. This in turn leads to a precise characterization of the domain of the $\mathcal C^{0}$-graph closure in this case.

1 Introduction

The Kimura diffusion operator is used to define the standard “forward time model” in population genetics and other fields in evolutionary biology. The classical Kimura diffusion operator is defined on the simplex, ${\Sigma }_{N}\subset \mathbb R^{N},$ by the second-order differential operator

$$ L_{\operatorname{Kim}}u=\sum\limits_{1\leq i,j\leq N}x_{i}(\delta_{ij}-x_{j})\partial_{x_{i}}\partial_{x_{j}}+\sum\limits_{1\leq i\leq N} b_{i}(x)\partial_{x_{i}}, $$

(1.1)

where ${\Sigma }_{N}=\{(x_{1},\dots ,x_{N}) : 0\leq x_{i}, i=1,\dots ,N\text { and }x_{1}+{\cdots } + x_{N}\leq 1\}.$ The first-order part

$$ \sum\limits_{1\leq i\leq N} b_{i}(x)\partial_{x_{i}} $$

(1.2)

is required to be inward pointing along the boundary of Σ_N. In most applications to biology, the coefficients b_i(x) are polynomials. The linear terms in {b_i(x)} model mutation and the higher order terms model effects like selection and migration. In a recent monograph [2], this class of operators on simplices has been generalized to a broad class of operators, called “generalized Kimura diffusions”, which are defined on manifolds with corners. These are briefly defined below; we refer the reader to the cited monograph for complete definitions.

A topological space, P, is an N-dimensional manifold with corners if every point, p, has a coordinate chart isomorphic to [0,1)ⁿ × (− 1,1)^m, where where n + m = N, n ≥ 0 and m ≥ 0. Clearly ∂P is not usually a smooth manifold, but is a stratified space. We denote these coordinates by (x;y). If p corresponds to (0;y), then we say that p lies on a boundary stratum of codimension n.

A second-order partial differential operator, L, defined on P, is a generalized Kimura diffusion operator, if, in such a coordinate chart, L is given by

$$ \begin{array}{@{}rcl@{}} Lu&=&\sum\limits_{i=1}^{n}[x_{i}a_{i}(x;y)\partial_{x_{i}}^{2}+b_{i}(x;y)\partial_{x_{i}}]u+\sum\limits_{i,j=1}^{n}+ x_{i}x_{j}a_{ij}(x;y)\partial_{x_{i}}\partial_{x_{j}}u\\ &&+\sum\limits_{i=1}^{n}\sum\limits_{l=1}^{m}x_{i}c_{il}\partial_{x_{i}}\partial_{y_{l}}u+ \sum\limits_{k,l=1}^{m}d_{kl}(x;y)\partial_{y_{k}}\partial_{y_{l}}u+\sum\limits_{l=1}^{m}d_{l}(x;y)\partial_{y_{l}}u, \end{array} $$

(1.3)

where the second-order part is strictly elliptic in the interior, and the coefficients satisfy

$$ \begin{array}{@{}rcl@{}} a_{i}(0;y)>0,\quad\sum\limits_{l,m=1}^{m}d_{kl}(0;y)\xi_{k}\xi_{l}\geq C|\xi|^{2} \end{array} $$

(1.4)

for some C > 0, and b_i(x;y) ≥ 0 where x_i = 0. More detailed definitions can be found in [2]. The principal symbol, σ₂(L), defines an incomplete Riemannian metric on P. The WF-Hölder spaces are defined with respect to the distance function defined by this metric.

If P is a compact manifold with corners and L is a generalized Kimura diffusion on P, then it is shown that the initial value problem,

$$ \partial_{t}u-Lu=0 \text{ with } u(x,0)=f(x), $$

(1.5)

has a unique solution that is in $\mathcal C^{\infty }(P\times (0,\infty ))\cap \mathcal C^{0}(P\times [0,\infty ))$ provided that f belongs to a WF-Hölder space. This is called the regular solution. Optimal results in these spaces are also obtained for the inhomogeneous problem ∂_tu − Lu = g. These results have been extended to data in $\mathcal C^{0}(P)$ in [5] and [3]. At least as regards initial data in $\mathcal C^{0}(P)$ there are several points that have not yet been satisfactorily understood: (1). Estimates on ∇u, for the solution to (1.5), in terms of the initial data do not seem to be optimal. (2) It has not been shown that the $\mathcal C^{0}(P)$-graph closure of L acting on $\mathcal C^{2}(P)$ has a compact resolvent. This short note is directed towards improving our state of knowledge on these points.

We show below that the $\mathcal C^{0}(P)$-graph closure of L acting on $\mathcal C^{2}(P)$ always has a compact resolvent. We also consider the gradient estimate in 1-dimension for the elliptic problem Lu = h on [0,1]. Under the additional assumption that the first-order part of L is strictly inward pointing at 0 and 1, we show that the $\mathcal C^{0}$-norm of ∂_xu is bounded by a constant times $\|Lu\|_{\mathcal C^{0}([0,1])}.$ We also show that this is false if the first-order term vanishes at one or both of the boundary points.

2 The Graph Closure in the $\mathcal C^{0}$-Topology

One of the goals in this paper is to describe the domain of the $\mathcal C^{0}$-graph closure of a Kimura diffusion in 1d. The key to this result is a gradient estimate for the elliptic problem. This is also a step towards proving sharp estimates on the gradient for the solution of the parabolic problem as well as proving estimates in higher dimensional cases. These analytic results are also useful in the development of spectrally accurate numerical methods for solving variants of the Kimura diffusion equation (see [4]).

In 1d, we work on the unit interval, [0,1], where the simplest Kimura diffusions are given by

$$ L_{\alpha,\beta}u = \frac{x(1-x)}{2\rho}{\partial_{x}^{2}}u + \alpha(1-x)\partial_{x} u -\beta x\partial_{x} u, \qquad (0<x<1), $$

(2.1)

with ρ > 0, α ≥ 0, and β ≥ 0. For the parabolic problem u_t = L_{α, β}u, the operator L_{α, β} can be generalized by allowing ρ, α, and β to depend on time, by imposing singular, possibly time-dependent boundary conditions (when α < 1 or β < 1), or by adding additional terms that vanish at the boundary, such as γ(x, t)x(1 − x)∂_xu. In this section, we assume that α, β, and ρ are constant. Scaling time and adjusting α and β accordingly, we may assume 2ρ = 1. The rescaled coefficients α and β are called the weights of L at the boundary of [0,1].

In this section, we consider the $\mathcal C^{0}$-graph closure of a Kimura diffusion operator in 1-dimension. The main result here is a precise characterization of the domain of the graph closure of a Kimura operator with respect to the $\mathcal C^{0}$-norm, which is a consequence of an estimate of ∂_xu in terms of Lu.

2.1 The Definition of WF-Hölder-Spaces

In this section, we collect some basic definitions and review some analytical results. We begin with the definitions of the 1d WF-Hölder spaces. A complete discussion can be found in [2], which contains a thorough treatment of the analysis of generalized Kimura diffusions in these specially adapted Hölder spaces.

We define the WF distance on the interval to be

$$ \rho_{\operatorname{WF}}(x_{1},x_{2})={\int}_{x_{1}}^{x_{2}}\frac{dx}{\sqrt{x(1-x)}}. $$

(2.2)

For $0\leq x_{1},x_{2}\leq \frac 12$, this behaves like $\rho _{\operatorname {WF}}(x_{1},x_{2})\propto |\sqrt {x_{1}}-\sqrt {x_{2}}|,$ and if $\frac 12\leq x_{1},x_{2}\leq 1,$ then $\rho _{\operatorname {WF}}(x_{1},x_{2})\propto |\sqrt {1-x_{1}}-\sqrt {1-x_{2}}|.$ For 0 < γ ≤ 1, the WF-Hölder space, $\mathcal C^{0,\gamma }_{\operatorname {WF}}([0,1]),$ consists of functions $f\in \mathcal C^{0}([0,1])$ for which the semi-norm

$$ [f]_{\operatorname{WF},\gamma}=\sup_{x_{1}\neq x_{2}\in [0,1]}\frac{|f(x_{1})-f(x_{2})|}{\rho_{\operatorname{WF}}^{\gamma}(x_{1},x_{2})} $$

is finite. The norm on $\mathcal C^{0,\gamma }_{\operatorname {WF}}([0,1])$ is defined by

$$ \|f\|_{\operatorname{WF},0,\gamma}=\|f\|_{\infty}+ [f]_{\operatorname{WF},\gamma}. $$

We also define another ladder of spaces denoted by $\mathcal C^{0,2+\gamma }_{\operatorname {WF}}([0,1]),$ with 0 < γ ≤ 1. A function $f\in \mathcal C^{1}([0,1])\cap \mathcal C^{2}((0,1))$ belongs to this space if the norm

$$ \|f\|_{\operatorname{WF},0,2+\gamma}=\|f\|_{\operatorname{WF},0,\gamma}+\|\partial_{x}f\|_{\operatorname{WF},0,\gamma}+\|x(1-x){\partial_{x}^{2}}f\|_{\operatorname{WF},0,\gamma} $$

(2.3)

is finite. These spaces play a central role in the analysis of Kimura diffusion operators, as the maps $(L_{\alpha ,\beta }-\lambda \operatorname {Id})^{-1}:\mathcal C^{0,\gamma }_{\operatorname {WF}}([0,1])\to \mathcal C^{0,2+\gamma }_{\operatorname {WF}}([0,1])$ are bounded for all 0 < γ < 1, and $\lambda \in (0,\infty )$ (see [2]). This provides a precise characterization of the domain of a Kimura operator acting on a WF-Hölder space. Missing from this treatment is the classical case of data in $\mathcal C^{0},$ which we consider below.

2.2 Domain of L Acting on C⁰([0,1])

We now analyze the precise domain of the $\mathcal C^{0}$-graph closure of a Kimura operator, L. It is defined using $\mathcal C^{2}([0,1])$ as a core and taking the closure with respect to the graph norm

$$ \|f\|_{\infty}+\|Lf\|_{\infty}. $$

We denote the domain of the closure by $\overline {\mathcal {D}}.$ It depends only on the weights, (α, β) in (2.1). The easiest case to treat is when both α and β are positive. Since it is no more difficult, we treat the case of

$$L=L_{\alpha,\beta}+x(1-x)s(x)\partial_{x}$$

for s(x) a continuous function on [0,1].

Theorem 2.1

Suppose that αand βare positive. The domain$\overline {\mathcal {D}}$of the$\mathcal C^{0}$-graph closure of the operator L = L_{α, β} + x(1 − x)s(x)∂_xis given by

$$ \overline{\mathcal{D}}=\left\{f\in\mathcal C^{2}((0,1))\cap \mathcal C^{1}([0,1]) : \lim_{x\to 0^{+},1^{-}}x(1-x){\partial_{x}^{2}}f(x)=0\right\}. $$

(2.4)

Proof

We first show that $\overline {\mathcal {D}}\subset \mathcal C^{1}([0,1]),$ which will imply that $\overline {\mathcal {D}}\subset \mathcal C^{2}((0,1)).$ To that end, suppose that $f\in \mathcal C^{2}([0,1])$ and Lf = h. This can be expressed as an equation for g(x) = ∂_xf(x):

$$ x(1-x)\partial_{x}g+(\alpha(1-x)-\beta x+x(1-x)s(x))g(x)=h(x). $$

An elementary calculation shows that this equation can be rewritten as

$$ \partial_{x}(\sigma(x) g(x))=\frac{\sigma(x)h(x)}{x(1-x)}, $$

where σ(x) = x^α(1 − x)^βS(x), with

$$ S(x)=\exp\left[{\int}_{0}^{x} s(y)dy\right]. $$

As α and β are both positive, integrating this relation gives two Volterra operators expressing g in terms of h:

$$ g(x)={{\int}_{0}^{x}}\frac{\sigma(y)h(y)dy}{\sigma(x)y(1-y)}\overset{d}{=}K_{0}h(x)\quad\text{ and }\quad g(x)=-{{\int}_{x}^{1}}\frac{\sigma(y)h(y)dy}{\sigma(x)y(1-y)}\overset{d}{=}K_{1}h(x). $$

(2.5)

Lemma 2.2

If αand βare positive, then$K_{0}:\mathcal C^{0}([0,\frac {1}{2}])\to \mathcal C^{0}([0,\frac {1}{2}])$and$K_{1}:\mathcal C^{0}([\frac {1}{2},1])\to \mathcal C^{0}([\frac {1}{2},1])$are bounded operators.

Proof

The proof follows easily from the facts that

$$ \max_{x\in [0,\frac{1}{2}]}{{\int}_{0}^{x}}\frac{\sigma(y)dy}{\sigma(x)y(1-y)}<\infty $$

and

$$ \max_{x\in [\frac{1}{2},1]}{{\int}_{x}^{1}}\frac{\sigma(y)dy}{\sigma(x)y(1-y)}<\infty. $$

□

Remark 2.3

It is interesting to note that

$$ \lim_{x\to 0^{+}}K_{0}h(x)=\frac{h(0)}{\alpha}\quad\text{ and }\quad \lim_{x\to 1^{-}}K_{1}h(x)=\frac{h(1)}{\beta}. $$

Since K₀h = K₁h for h in the range of L, it follows that

$$ {{\int}_{0}^{1}}\frac{\sigma(y)h(y)dy}{y(1-y)}=0, $$

which also follows from the fact that L1 = 0, and the Fredholm alternative. A simple calculation shows that the function η₀(x) = K₀1 is real analytic in [0,1) and η₁(x) = K₁1 is real analytic in (0,1].

From these observations, it follows easily that $\overline {\mathcal {D}}\subset \mathcal C^{1}([0,1])$: let $<f_{n}>\subset \mathcal C^{2}([0,1])$ be a sequence converging in the $\mathcal C^{0}$-norm to f, for which h_n = Lf_n converges in the $\mathcal C^{0}$-norm to h. From the continuity of K₀ (or K₁) on $\mathcal C^{0}$ it follows that

$$ \partial_{x} f_{n}(x) =\left\{\begin{array}{ll} K_{0}h_{n}(x)\text{ for }x\in [0,\frac{1}{2}],\\ K_{1}h_{n}(x)\text{ for }x\in [\frac{1}{2},1] \end{array}\right. $$

converges uniformly to some $g\in \mathcal C^{0}([0,1]).$ Since

$$ f_{n}(x)-f_{n}(0)={{\int}_{0}^{x}}\partial_{x} f_{n}(y)dy, $$

it therefore follows that

$$ f(x)-f(0)={{\int}_{0}^{x}}g(y)dy, $$

from which it is immediate that ∂_xf = g, and therefore $f\in \mathcal C^{1}([0,1]).$

For any function $f\in \mathcal C^{2}([0,1])$, the $\lim _{x\to 0^{+},1^{-}}x(1-x){\partial _{x}^{2}}f(x)=0.$ Under the limit considered in the previous paragraph, we know that the pair (∂_xf_n(0), ∂_xf_n(1)) converges to the pair (∂_xf(0), ∂_xf(1)); as (Lf_n(0), Lf_n(1)) also converges to (Lf(0), Lf(1)), and $Lf(x)\in \mathcal C^{0}([0,1]),$ it is clear that

$$ \lim_{x\to 0^{+},1^{-}}x(1-x){\partial_{x}^{2}}f(x)=0.$$

This shows that the set on the right hand side of (2.4) contains $\overline {\mathcal {D}}.$ To prove the opposite inclusion, we need to show that for a function f satisfying the conditions on the right hand side of (2.4), there is a sequence $<f_{n}>\subset \mathcal C^{2}([0,1]),$ so that (f_n, Lf_n) converges uniformly to (f, Lf). It is left to readers to show that the sequence

$$ f_{n}(x)=f\left( \frac{1+(n-2)x}{n}\right) $$

has the desired properties. This completes the proof of Theorem 2.1. □

Remark 2.4

The precise characterization of the domain of the $\mathcal C^{0}$-graph closure in higher dimensions remains an open problem.

As a corollary of the proof, we see that if α and β are both positive, then the operator f↦∂_xf is relatively bounded with respect to f↦Lf.

Corollary 2.5

If α and β are positive, then there exists a constant C _L so that, for all $f\in \overline {\mathcal {D}},$

$$ \|\partial_{x} f\|_{\mathcal C^{0}([0,1])}\leq C_{L} \|Lf\|_{\mathcal C^{0}([0,1])}. $$

(2.6)

If α is positive, but β = 0, then such an estimate holds on [0, x] for any 0 < x < 1, and if α = 0, but β is positive, then such an estimate holds on [x,1] for 0 < x < 1.

Remark 2.6

This result is very useful if α and β depend continuously on time. In this case, we can write

$$ \begin{array}{@{}rcl@{}} && L_{\alpha(t),\beta(t)}-\lambda\\ &=&\left[\operatorname{Id}+ [(\alpha(t) - \alpha(0))(1 - x)\partial_{x} - (\beta(t) - \beta(0))x\partial_{x}](L_{\alpha(0),\beta(0)} - \lambda)^{-1}\right]\cdot (L_{\alpha(0),\beta(0)}-\lambda). \end{array} $$

From the corollary, it follows that, as an operator from $\mathcal C^{0}([0,1])$ to itself, the term

$$ [(\alpha(t)-\alpha(0))(1-x)\partial_{x}-(\beta(t)-\beta(0))x\partial_{x}](L_{\alpha(0),\beta(0)}-\lambda)^{-1} $$

has norm bounded by C(|α(t) − α(0)| + |β(t) − β(0)|), suggesting a simple perturbative approach to solve the Kimura equation with time-dependent coefficients. Such a method is detailed in [4].

Remark 2.7

By analogy with Euclidean potential theory, one might expect that if Lf = h, then some WF-Hölder norm of ∂_xf would be bounded by the $\mathcal C^{0}$-norm of h. This, in fact, cannot be the case. If there were a γ > 0 and a C for which $\|\partial _{x} f\|_{\operatorname {WF},0,\gamma }\leq C\|Lf\|_{\mathcal C^{0}},$ then, using the observations in Remark 2.3, we could show that $K_{0}:\mathcal C^{0}([0,\frac {1}{2}])\to \mathcal C^{0,\gamma }_{\operatorname {WF}}([0,\frac {1}{2}])$ is a bounded operator. This would imply that K₀ is compact as a map from $\mathcal C^{0}([0,\frac {1}{2}])$ to itself. Assuming that α and β are positive and s(x) = 0, it is not difficult to show that, for $\lambda \in [0,\frac {1}{\alpha }],$ we have

$$ K_{0}\left[ x^{\frac{1-\lambda\alpha}{\lambda}}(1-x)^{\frac{1+\lambda\beta}{\lambda}}\right]= \lambda x^{\frac{1-\lambda\alpha}{\lambda}}(1-x)^{\frac{1+\lambda\beta}{\lambda}}. $$

For λ in this range, these functions belong to $\mathcal C^{0}([0,1])$; hence, K₀ does not have a discrete spectrum and is therefore not compact. A similar argument applies by using K₁ for $x\in [\frac {1}{2},1].$

This analysis leaves open the cases where one or both of the weights α and β vanish. We let $\overline {\mathcal {D}}_{\alpha ,\beta }$ denote the domain of the $\mathcal C^{0}$-graph norm closure of L_{α, β}. From the proof of Theorem 2.1, it is clear that the estimates for the first derivative at 0 or 1 only depend on the coefficient of the first-order term at the respective endpoint. The remaining question is whether or not the first derivative of $f\in \overline {\mathcal {D}}_{\alpha ,\beta }$ remains bounded even if the relevant coefficient is zero. This, in fact, is not the case. We demonstrate this for the simpler model operator $L_{0}=x{\partial _{x}^{2}},$ acting on $\mathcal C^{0}([0,\infty )).$ We let $\overline {\mathcal {D}}_{0}$ denote the $\mathcal C^{0}$-graph closure starting with the core ${\mathcal {C}}^2_c([0,\infty )).$

Choosing $\varphi (x)\in \mathcal C^{\infty }_{c}([0,1)),$ with φ(x) = 1 for $x\in [0,\frac 12],$ and 0 < ν < 1, we define the function

$$ l_{\nu}(x)=x [-\log x]^{1-\nu}\varphi(x). $$

Applying ∂_x and L₀ to l_ν for x the interval $[0,\frac 12]$ gives

$$ \begin{array}{@{}rcl@{}} \partial_{x} l_{\nu}(x)&=&[-\log x]^{1-\nu}-(1-\nu)[-\log x]^{-\nu},\\ L_{0} l_{\nu}(x)&=&-(1-\nu)[-\log x]^{-\nu}(1+\nu[-\log x]^{-1}). \end{array} $$

This shows that ∂_xl_ν(x) diverges as x → 0⁺,

$$ L_{0} l_{\nu}(x)\in\mathcal C^{0}([0,\infty)),\quad\text{ and }\quad\lim_{x\to 0^{+}} L_{0} l_{\nu}(x)=0. $$

It is not difficult to show that $(l_{\nu }(\cdot +\frac 1n),L_{0}l_{\nu }(\cdot +\frac 1n))$ converges uniformly to (l_ν, L₀l_ν), and hence $l_{\nu }\in \overline {\mathcal {D}}_{0}.$ If $f\in \overline {\mathcal {D}}_{0},$ then clearly $\lim _{x\to 0^{+}}x{\partial _{x}^{2}}f(x)=0,$ which implies that

$$ \partial_{x} f(x)=o(|\log x|). $$

(2.7)

From this analysis, it is clear that the domains $\overline {\mathcal {D}}$ depend only on the weights (α, β) and that the descriptions at the two endpoints are entirely independent. It is difficult to give a precise description of the domain at a boundary point where a weight vanishes, and so we do not state a result describing $\overline {\mathcal {D}}_{0,\beta }$,$\overline {\mathcal {D}}_{\alpha ,0}$, and $\overline {\mathcal {D}}_{0,0}$ beyond the statement that for $f\in \overline {\mathcal {D}}_{0,\beta }$ (resp. $f\in \overline {\mathcal {D}}_{\alpha ,0}$)

$$ \lim_{x\to 0^{+}}x{\partial_{x}^{2}} f(x)=0, \quad\left( \lim_{x\to 1^{-}}(1-x){\partial_{x}^{2}} f(x)=0 \text{ resp.}\right), $$

which in turn implies that (2.7) (or its analogue for x = 1) holds for $f\in \overline {\mathcal {D}}_{0,\beta }$ (or $f\in \overline {\mathcal {D}}_{\alpha ,0}$). The examples l_ν show that this estimate is essentially sharp. We leave the details of these (elementary) arguments to readers.

3 General Vector Fields as Perturbations

As noted above, in many applications, the first-order part of the Kimura operator is of the form

$$ \alpha(1-x)\partial_{x} u -\beta x\partial_{x} u+s(x)x(1-x)\partial_{x}, $$

where s(x) is a smooth function on [0,1]. In population genetics, the vector field s(x)x(1 − x)∂_x models the effects of selection. We denote this vector field by

$$ V=s(x)x(1-x)\partial_{x}. $$

In the simplest model of selection, s is a constant, which quantifies the selective difference between the two types. An examination of the arguments that follow shows that it suffices to treat the case that s(x) = 1, which we henceforth assume.

In the previous section, we proved that any vector field is a relatively bounded operator with respect to L provided that α and β are both positive. In this section, we show that a vector field with coefficient vanishing at 0 and 1 is a relatively compact perturbation of the operator

$$ L_{\alpha,\beta}=x(1-x){\partial_{x}^{2}}+ \alpha(1-x)\partial_{x} -\beta x\partial_{x}. $$

More precisely, for 0 < λ, the operators V (L_{α, β} − λ Id)^− 1 and (L_{α, β} − λ Id)^− 1V are compact maps from $\mathcal C^{0}([0,1])$ to itself. The truth of this statement only requires that α and β are non-negative. We begin with the easier result:

Proposition 3.1

For λin the resolvent set of L_{α, β}, the operator V (L_{α, β} − λ Id)^− 1defines a compact map from$\mathcal C^{0}([0,1])$ to itself. There is a constant C so that, for $\lambda \in (0,\infty )$we have the estimate

$$ \|V(L_{\alpha,\beta}-\lambda\operatorname{Id})^{-1}f\|_{\infty}\leq \frac{C}{\sqrt{\lambda}}\|f\|_{\infty}. $$

(3.1)

Proof

This result follows from the basic estimate for the 1d-Cauchy problem. There is a constant C₀ so that if u(x, t) is the solution of

$$ (\partial_{t}-L_{\alpha,\beta})u=0\quad\text{ with }\quad u(x,0)=f(x), $$

then

$$ |\partial_{x}u(x,t)|\leq\frac{C_{0}\|f\|_{\infty}}{\sqrt{tx(1-x)}}, $$

(3.2)

which follows from [2, Lemma 6.1.10], and the construction of the resolvent kernel in [1]. By integrating this estimate in x, we can easily show that there is a constant C₁ so that

$$ |u(x_{1},t)-u(x_{2},t)|\leq \frac{C_{1}}{\sqrt{t}}\rho_{\operatorname{WF}}(x_{1},x_{2})\|f\|_{\infty}, $$

(3.3)

where ρ_WF is defined in (2.2). This estimate implies that for t > 0, the operator $e^{tL_{\alpha ,\beta }}: L^{\infty }([0,1])\to \mathcal C^{0,1}_{\operatorname {WF}}([0,1])$ is bounded, with norm bounded by $C/\sqrt {t}.$

The resolvent operator can be expressed as the Laplace transform of the heat kernel

$$ R(\lambda)= (L_{\alpha,\beta}-\lambda\operatorname{Id})^{-1}={\int}_{0}^{\infty}e^{tL_{\alpha,\beta}}e^{-t\lambda}dt. $$

From this representation and the estimate in (3.2), we easily derive the estimate

$$ \begin{array}{@{}rcl@{}} |x(1-x)\partial_{x} (L_{\alpha,\beta}-\lambda\operatorname{Id})^{-1}f|&\leq& C_{0}\|f\|_{\infty}\sqrt{x(1-x)} {\int}_{0}^{\infty}\frac{e^{-t\lambda}}{\sqrt{t}}dt\\ &=&\frac{C_{0}\|f\|_{\infty}\sqrt{x(1-x)}}{\sqrt{\lambda}} {\int}_{0}^{\infty}\frac{e^{-s}}{\sqrt{s}}ds. \end{array} $$

The estimate in (3.1) follows easily from this.

From (3.2), we see that, for any fixed λ, and finite 𝜖 > 0, there is a constant C_λ so that

$$ \left\| x(1-x)\partial_{x} {\int}_{0}^{\epsilon}e^{tL_{\alpha,\beta}}e^{-t\lambda}fdt\right\|_{\infty} \leq C_{\lambda}\|f\|_{\infty} \sqrt{\epsilon}. $$

From this, it follows that to prove the compactness of x(1 − x)∂_x(L_{α, β} − λ Id)^− 1 it suffices to show that, for sufficiently large positive λ, the operator

$$ VR_{\epsilon}(\lambda)f= x(1-x)\partial_{x} {\int}_{\epsilon}^{\infty}e^{tL_{\alpha,\beta}} e^{-t\lambda}fdt $$

is compact for any 𝜖 > 0. The operator R_𝜖(λ) is defined by

$$ R_{\epsilon}(\lambda)={\int}_{\epsilon}^{\infty}e^{tL_{\alpha,\beta}} e^{-t\lambda}dt. $$

This statement follows from [2, Theorem 11.2.1], which shows that there is a k > 0 so that, for any 0 < γ < 1, we have

$$ \|e^{tL_{\alpha,\beta}}f\|_{\operatorname{WF},0,2+\gamma}\leq C_{\gamma}\left[\frac{1}{t}+e^{kt}\right]\|f\|_{\operatorname{WF},0,\gamma}. $$

(3.4)

We rewrite $VR_{\epsilon }(\lambda )f=Ve^{-\frac {\lambda \epsilon }{2}}R_{\frac {\epsilon }{2}}(\lambda ) e^{\frac {\epsilon }{2}L_{\alpha ,\beta }}f.$ The estimate in (3.3) shows that $e^{\frac {\epsilon }{2}L_{\alpha ,\beta }}$ maps the space C⁰([0,1]) into $\mathcal C^{0,\gamma }_{\operatorname {WF}}([0,1]).$ Since $V:\mathcal C^{0,2+\gamma }_{\operatorname {WF}}([0,1])\to \mathcal C^{0,\gamma }_{\operatorname {WF}}([0,1])$, it follows easily from the estimate in (3.4) that, for k < λ, the composition $VR_{\epsilon }(\lambda )=Ve^{-\frac {\lambda \epsilon }{2}}R_{\frac {\epsilon }{2}}(\lambda ) e^{\frac {\epsilon }{2}L_{\alpha ,\beta }}$ maps $\mathcal C^{0}([0,1])$ to $\mathcal C^{0,\gamma }_{\operatorname {WF}}([0,1])$ and is therefore a compact operator from $\mathcal C^{0}([0,1])$ to itself. This in turn shows that R(λ) is a compact operator if k < λ. The resolvent identity R(μ) = R(λ) + (μ − λ)R(μ)R(λ) then implies that this is true throughout the resolvent set. □

For the transition from VR(λ) to R(λ)V, we use an identity satisfied by the operators L_{α, β} and x(1 − x)∂_x. Simple calculations show that

$$ x(1-x)\partial_{x} L_{1,1}=L_{0,0}x(1-x)\partial_{x}, $$

and

$$ x(1-x)\partial_{x}[\alpha(1-x)\partial_{x}-\beta x\partial_{x}]= [\alpha(1-x)\partial_{x}-\beta x\partial_{x}]x(1-x)\partial_{x}-[\alpha(1-x)^{2}+\beta x^{2}]\partial_{x}. $$

Combining these results, we see that

$$ x(1-x)\partial_{x} L_{\alpha+1,\beta+1}=L_{\alpha,\beta}x(1-x)\partial_{x}-[\alpha(1-x)^{2}+\beta x^{2}]\partial_{x}. $$

From this identity, we easily derive that for $\lambda \in (0,\infty )$, we have the relation

$$ \begin{array}{@{}rcl@{}} (L_{\alpha,\beta}-\lambda\operatorname{Id})^{-1}x(1-x)\partial_{x}&=& x(1-x)\partial_{x} (L_{\alpha+1,\beta+1}-\lambda\operatorname{Id})^{-1}-(L_{\alpha,\beta}-\lambda\operatorname{Id})^{-1}\\ &&\times [(\alpha(1-x)^{2}+\beta x^{2}) \partial_{x}](L_{\alpha+1,\beta+1}-\lambda\operatorname{Id})^{-1}. \end{array} $$

(3.5)

With this formula and Proposition 3.1, we establish the following result.

Proposition 3.2

The operator (L_{α, β} − λId)^− 1x(1 − x)∂_xis compact on$\mathcal C^{0}([0,1]).$There is a constant C_{α, β}so that its norm, as an operator from$\mathcal C^{0}$to itself, is bounded by$C_{\alpha ,\beta }/\sqrt {\lambda },$for$\lambda \in (0,\infty ).$

Proof

Proposition 3.1 shows that the first term on the right hand side of (3.5) is compact and satisfies the desired norm bound. From Theorem 2.1, it follows that

$$[\alpha(1-x)^{2}+\beta x^{2}] \partial_{x}(L_{\alpha+1,\beta+1}-\lambda\operatorname{Id})^{-1}$$

is bounded from $\mathcal C^{0}([0,1])$ to itself; the estimate in (3.3) implies that

$$(L_{\alpha,\beta}-\lambda\operatorname{Id})^{-1}:\mathcal C^{0}([0,1])\to\mathcal C^{0,1}_{\operatorname{WF}}([0,1])$$

is bounded with norm bounded by $C/\sqrt {\lambda }.$

Using the operators K₀ and K₁ defined in (2.5), with parameters α + 1, β + 1, we see that, for $x\in [0,\frac {1}{2}],$

$$ \begin{array}{@{}rcl@{}} [\partial_{x}(L_{\alpha+1,\beta+1}-\lambda\operatorname{Id})^{-1}f](x)&=&[K_{0}L_{\alpha+1,\beta+1} (L_{\alpha+1,\beta+1}-\lambda\operatorname{Id})^{-1}f](x)\\ &=& \left( K_{0}[\operatorname{Id}+\lambda(L_{\alpha+1,\beta+1}-\lambda\operatorname{Id})^{-1}]f\right)(x), \end{array} $$

(3.6)

with a similar formula employing K₁ for $x\in [\frac {1}{2},1].$ Combining these observations, with the estimate (3.1), the observations at the end of the proof of Proposition 3.1, and the fact that (L_{α+ 1, β+ 1} − λ Id)^− 1 as a map from $\mathcal C^{0}$ to itself has norm 1/λ, we obtain the norm bound

$$ \|(L_{\alpha,\beta}-\lambda\operatorname{Id})^{-1}x(1-x)\partial_{x} f\|_{\infty}\leq \frac{C_{\alpha,\beta}}{\sqrt{\lambda}}\|f\|_{\infty}, \text{ for }\lambda\in (0,\infty). $$

□

4 Compact Resolvents in Arbitrary Dimension

The WF-Hölder theory of a Kimura diffusion, L, on an arbitrary compact manifold with corners, P, is systematically developed in [2]. A fundamental question left unanswered is whether the resolvent (L − λ Id)^− 1 is always a compact operator on $\mathcal C^{0}(P).$ This is proved in [3] under the additional assumption that the weights are strictly positive, i.e., the first order part of L is uniformly transverse to ∂P. In this final section, we see that the argument used to prove Proposition 3.1 is easily adapted to prove this statement in general. Let P denote a compact manifold with corners, as defined in [2, Chapter 2.1] and L a generalized Kimura diffusion operator, as defined in [2, Definition 2.2.1].

As above, ρ_WF denotes the incomplete metric defined on P by σ₂(L), the principal symbol of L. For 0 < γ < 1, we let $\mathcal C^{0,\gamma }_{\operatorname {WF}}(P)$ denote the subset of $\mathcal C^{0}(P)$ consisting of functions for which the semi-norm

$$ [f]_{\operatorname{WF},\gamma}=\sup_{\{x_{1}\neq x_{2}\in P}\frac{|f(x_{1})-f(x_{2})|} {\rho_{\operatorname{WF}}^{\gamma}(x_{1},x_{2})} $$

is finite. It is easy to see that the unit ball in $\mathcal C^{0,\gamma }_{\operatorname {WF}}(P)$ is a compact subset of $\mathcal C^{0}(P).$

With these preliminaries, we prove the following result:

Theorem 4.1

If Pis a compact manifold with corners and Lis a generalized Kimura diffusion operator defined onP, then the resolvent$(L-\lambda \operatorname {Id})^{-1}:\mathcal C^{0}(P)\to \mathcal C^{0}(P)$is a compact operator.

Proof

For $f\in \mathcal C^{0,\gamma }_{\operatorname {WF}}(P)$, let u(p, t) denote the unique regular solution to the initial value problem (∂_t − L)u = 0 and u(p,0) = f(p). [2, Theorem 11.2.1] shows that there are constants C and K so that

$$ \|u(\cdot,t)\|_{\operatorname{WF},0,\gamma}\leq Ce^{Kt}\|f\|_{\operatorname{WF},0,\gamma}. $$

[5, Theorem 1.5] shows that, for any 0 < γ < 1, and τ > 0, there is a constant C_{τ, γ} so that

$$ \|u(\cdot,\tau)\|_{\operatorname{WF},0,\gamma}\leq C_{\tau,\gamma}\|f\|_{\mathcal C^{0}}. $$

As before, for 𝜖, λ > 0, let

$$ R_{\epsilon}(\lambda)f={\int}_{\epsilon}^{\infty}e^{-\lambda t}e^{Lt}f dt. $$

From the estimates above, we see that, for K < λ, we have the estimate

$$ \|R_{\epsilon}(\lambda)f\|_{\operatorname{WF},0,\gamma}\leq \frac{C}{\lambda-K}\|e^{\epsilon L}f\|_{\operatorname{WF},0,\gamma} \leq \frac{CC_{\epsilon,\gamma}}{\lambda-K}\|f\|_{\mathcal C^{0}}. $$

This shows that, for any 𝜖 > 0, and λ > K, the operator $R_{\epsilon }(\lambda ):\mathcal C^{0}(P)\to \mathcal C^{0}(P)$ is compact. Since e^Lt is a contraction on $\mathcal C^{0}$, we easily derive that

$$ \|[R(\lambda)-R_{\epsilon}(\lambda)]f\|_{\mathcal C^{0}}\leq\epsilon \|f\|_{\mathcal C^{0}}. $$

Hence, for K < λ, the resolvent, R(λ), is the uniform limit of a sequence of compact operators and therefore compact. Once again, the resolvent identity implies that this is true throughout the resolvent set. □

Pop’s results in [5] easily imply that the eigenfunctions of the $\mathcal C^{0}(P)$-graph closure of L are in $C^{0,\gamma }_{\operatorname {WF}}(P)$ and are therefore automatically smooth. Hence, the spectrum of L acting on $\mathcal C^{0}(P)$ is the same as its spectrum acting on $\mathcal C^{0,\gamma }_{\operatorname {WF}}(P).$ [2, Theorem 11.1.1] shows that, other than 0, this spectrum lies in a conic neighborhood of the negative real axis, contained in the open left half plane. It is also the case that generalized eigenfunctions (the null-space of (L − λ Id)^k for k > 1) belong to $\mathcal C^{0,\gamma }_{\operatorname {WF}}(P),$ and hence are smooth.

This analysis leaves open the question of characterizing the precise domain of the $\mathcal C^{0}(P)$-graph closure of L acting on $\mathcal C^{2}(P),$ when $\dim P>1.$ The result in the 1d-case, proved above, shows that this domain can be expected to depend on the transversality properties of the first-order part of L along ∂P.