Gradient and Stability Estimates of Heat Kernels for Fractional Powers of Elliptic Operator

TL;DR

This paper derives gradient and stability estimates for heat kernels related to fractional powers of elliptic operators, and also provides $L^p$-operator norm bounds for their semigroups, advancing understanding of fractional elliptic operators.

Contribution

It introduces new gradient and stability estimates for heat kernels of fractional elliptic operators and establishes $L^p$-operator norm bounds for their semigroups.

Findings

Gradient estimates for heat kernels of fractional elliptic operators

Stability estimates for these heat kernels

Bounds on $L^p$-operator norms of associated semigroups

Abstract

Gradient and stability type estimates of heat kernel associated with fractional power of a uniformly elliptic operator are obtained. $L^p$-operator norm of semigroups associated with fractional power of two uniformly elliptic operators are also obtained.

Equations

$$e^{-u^{\alpha }}$$

$$g(\alpha ,s)$$

$${\lambda }_t(ds)=g_{{}_t}(\alpha ,s)ds:=t^{-\frac{1}{\alpha }}g(\alpha ,t^{-\frac{1}{\alpha }}s)ds.$$

$$e^{-u^{\alpha }t}$$

$$Q_{t}f(x)={\int }_0^{\infty }{P}_{s}f(x){\lambda }_t(\mathrm{d}s)={\int }_0^{\infty }{P}_{s}f(x)g_{{}_t}(\alpha ,s)\mathrm{d}s,f\in \mathbf{B}.$$

$$-(-H{)}^{\alpha }f=\mathcal{M}f={\int }_0^{\infty }(P_{s}f-f)\rho (\mathrm{d}s)=\frac{\alpha }{\Gamma (1-\alpha )}{\int }_0^{\infty }\frac{P_{s}f-f}{s^{1+\alpha }}\mathrm{d}s,f\in \mathcal{D}(L).$$

$$\left|\nabla_{x}q(t,x,y)\right|\leq{c_{1}}\Big{(}t^{-\frac{d+1}{2\alpha}}\wedge{\frac{t}{\left|x-y\right|^{d+1+2\alpha}}}\Big{)},\qquad\forall(t,x,y)\in(0,\infty)\times\mathbf{D}\times\mathbf{D},$$

$${\Vert Q_t-{\tilde{Q}}_t\Vert }_p$$

$$\left|q(t,x,y)-\tilde{q}(t,x,y)\right|$$

$${\Vert a-\tilde{a}\Vert }_{L_{loc}^2}=\underset{\mathbf{k}\in {\mathbb{Z}}^d}{\sup }\sum _{i,j=1}^d{\Vert a_{ij}-{\tilde{a}}_{ij}\Vert }_{L^2(D_{\mathbf{k}})},$$

$$D_{\mathbf{k}}=\left\{\mathbf{x}\in {\mathbb{R}}^d:\left|\mathbf{x}-\mathbf{k}\right|<2\sqrt{d}\right\}.$$

$$g(\alpha ,s)\sim K{s}^{-\frac{2-\alpha }{2-2\alpha }}\exp (-A{s}^{-\frac{\alpha }{1-\alpha }}),\text{for}s\to 0+,$$

$$g(\alpha ,s)\sim B{s}^{-1-\alpha },\text{for}s\to \infty .$$

$$q(t,x,y)={\int }_0^{\infty }p(s,x,y){\lambda }_t(\mathrm{d}s)={\int }_0^{\infty }p(t^{\frac{1}{\alpha }}s,x,y)g(\alpha ,s)\mathrm{d}s.$$

$$q(t,x,y)$$

$$=t^{-\frac{1}{\alpha }}{\int }_0^{\infty }p(s,x,y)g(\alpha ,t^{-\frac{1}{\alpha }}s)\mathrm{d}s\text{(by Eq.(refg t x))}$$

$$={\int }_0^{\infty }p(t^{\frac{1}{\alpha }}{s}^{\prime },x,y)g(\alpha ,s^{\prime })\mathrm{d}{s}^{\prime }.$$

$$\displaystyle\frac{1}{c}\Big{(}t^{-\frac{d}{2\alpha}}\wedge{\frac{t}{\left|y-x\right|^{d+2\alpha}}}\Big{)}\leq q(t,x,y)\leq{c}\Big{(}t^{-\frac{d}{2\alpha}}\wedge{\frac{t}{\left|y-x\right|^{d+2\alpha}}}\Big{)}$$

$$\displaystyle\left|q(t,x,y)-q(t,x_{1},y_{1})\right|\leqslant ct^{-\frac{2d-\gamma}{2\alpha}}\big{(}\left|x-x_{1}\right|\vee\left|y-y_{1}\right|\big{)}^{\gamma},$$

$$\frac{1}{M}{t}^{-\frac{d}{2}}\exp \left\{-\frac{M{\left|y-x\right|}^2}{t}\right\}\le p(t,x,y)\le M{t}^{-\frac{d}{2}}\exp \left\{-\frac{{\left|y-x\right|}^2}{Mt}\right\}.$$

$$\left|{\nabla }_{x}p(t,x,y)\right|\le M{t}^{-\frac{\ell }{2}}\exp \left\{-\frac{{\left|x-y\right|}^2}{Mt}\right\},\forall (t,x,y)\in (0,\infty )\times \mathbf{D}\times \mathbf{D},$$

$$\left|\nabla_{x}q(t,x,y)\right|\leq{c_{1}}\Big{(}t^{-\frac{\ell}{2\alpha}}\wedge{\frac{t}{\left|x-y\right|^{\ell+2\alpha}}}\Big{)},\qquad\forall(t,x,y)\in(0,\infty)\times\mathbf{D}\times\mathbf{D}.$$

$${\nabla }_{x}q(t,x,y)={\int }_0^{\infty }{\nabla }_{x}p(t^{\frac{1}{\alpha }}s,x,y)g(\alpha ,s)\mathrm{d}s.$$

$$\left|{\nabla }_{x}q(t,x,y)\right|$$

$$⩽M{t}^{-\frac{\ell }{2\alpha }}{\int }_0^{\infty }{s}^{-\frac{\ell }{2}}\exp \left\{-\frac{{\left|x-y\right|}^2}{M{t}^{\frac{1}{\alpha }}s}\right\}g(\alpha ,s)\mathrm{d}s.$$

$${\int }_0^{\infty }{s}^{-\frac{\ell }{2}}\exp \left\{-\frac{2{\left|x-y\right|}^2}{M{t}^{\frac{1}{\alpha }}s}\right\}g(\alpha ,s)\mathrm{d}s\le {\int }_0^{\infty }{s}^{-\frac{\ell }{2}}g(\alpha ,s)\mathrm{d}s$$

$$\le {\int }_0^1{s}^{-\frac{\ell }{2}}g(\alpha ,s)\mathrm{d}s+1$$

$$\left|{\nabla }_{x}q(t,x,y)\right|\le M{t}^{-\frac{\ell }{2\alpha }}$$

$$g(\alpha ,s)\le \hat{c}{s}^{-1-\alpha },\forall s\in [0,\infty ).$$

$$\left|{\nabla }_{x}q(t,x,y)\right|$$

$$=\check{c}M{t}^{-\frac{\ell }{2\alpha }}{\int }_0^{\infty }{r}^{\frac{\ell }{2}+\alpha -1}\exp \left\{-\frac{{\left|x-y\right|}^2}{M{t}^{\frac{1}{\alpha }}}r\right\}\mathrm{d}r\text{(let r=s1)}$$

$$\displaystyle=\check{c}Mt^{-\frac{\ell}{2\alpha}}\frac{\Gamma(\alpha+\frac{\ell}{2})}{\big{(}\left|x-y\right|^{2}/Mt^{\frac{1}{\alpha}}\big{)}^{\alpha+\frac{\ell}{2}}}$$

$$=\check{c}\tilde{M}\frac{t}{{\left|x-y\right|}^{\ell +2\alpha }},$$

$$\left|{\nabla }_{x}p(t,x,y)\right|\le M{t}^{-\frac{d+1}{2}}\exp \left\{-\frac{{\left|x-y\right|}^2}{Mt}\right\},\forall (t,x,y)\in (0,\infty )\times \mathbf{D}\times \mathbf{D}.$$

$$f(t,x,y)\le M{t}^{-\frac{\ell }{2}}\exp \left\{-\frac{{\left|x-y\right|}^2}{Mt}\right\},\forall (t,x,y)\in (0,\infty )\times \mathbf{D}\times \mathbf{D},$$

$$\int_{0}^{\infty}f(t^{\frac{1}{\alpha}}s,x,y)g(\alpha,s)\mathrm{d}s\leq{c_{1}}\Big{(}t^{-\frac{\ell}{2\alpha}}\wedge{\frac{t}{\left|x-y\right|^{\ell+2\alpha}}}\Big{)},\qquad\forall(t,x,y)\in(0,\infty)\times\mathbf{D}\times\mathbf{D}.$$

Full text

Keywords.

Gradient Estimates, Stability, Subordination, Fractional Powers.

MSC(2010):

60J35, 47D07.

1. Introduction and main conclusions

Let $\mathbf{D}$ be a domain in $\mathbb{R}^{d}$ and let $a:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d^{2}}$ be a matrix valued function with $C^{\beta}$ or measurable entries. The operator $H=\nabla(a(x)\nabla)$ generated a semigroup $P_{t}$ which is given by $P_{t}f(x)=\int_{\mathbb{R}^{d}}p_{t}(x,y)f(y)dy$. Heat kernel, gradient and stability estimates associated with this semigroup are well-studied (see [2], [3], [10]). In this paper we are concerned with the similar estimates for the semigroup generated by the fractional powers of $H$, namely, $Q_{t}=e^{-t(-H)^{\alpha}}$, where $\alpha\in(0,1)$ will be fixed throughout this paper. Our motivation is recent works on fractional diffusion in random environment (see [1, 6] and references therein) arisen from super and sub diffusion in random environment. However, we shall deal with this problem in separate project.

First let us recall a result. Using the classical Bromwich contour integral, Pollard in [7] obtained the following formula for the inverse Laplace transform of the function $e^{-u^{\alpha}}$.

[TABLE]

where

[TABLE]

is a probability density function of $s\geq 0$. This class of density functions $g(\alpha,s)$ is called strictly $\alpha$-stable law which plays important role in the theory of probability.

Denote

[TABLE]

Then

[TABLE]

From this identity we can define the semigroup $Q_{t}=e^{-t(-H)^{\alpha}}$ associated with $-(-H)^{\alpha}$ as

[TABLE]

Then, $\left\{Q_{t},\,t\geq 0\right\}$ is also a strong continuous contraction semigroup on $\mathbf{B}$ and its infinitesimal generator satisfies that

[TABLE]

Moreover, $\mathcal{D}(H)$ is a core of $\mathcal{M}$ which means that $\mathcal{D}(H)\subset\mathcal{D}(\mathcal{M})$ and the closure of $\mathcal{M}|_{\mathcal{D}(H)}$, the restriction of $\mathcal{M}$ to $\mathcal{D}(H)$, equals $\mathcal{M}$. In fact, $H$ can be replaced by a more general operator.

The transformation of $\left\{P_{t}\right\}$ to $\left\{Q_{t}\right\}$ is called subordination and we refer to [8] and the references therein.

The main results of the present paper are gradient and stability estimates of the heat kernels associated with the fractional power for uniformly elliptic operators.

Theorem 1.1**.**

Suppose that $\mathbf{D}$ is a bounded $C^{2}$ domain in $\mathbb{R}^{d}$ and $H=\nabla(a(x)\nabla)$ on $\mathbf{D}$ where the matrix $a(x)$ has $C^{\beta},\,\beta\in(0,1)$ entries, and there exists a constant $\lambda\geqslant 1$ such that $\lambda^{-1}\mathrm{Id}_{d}\leqslant a(x)\leqslant\lambda\mathrm{Id}_{d}$. Then the heat kernel $q(t,x,y)$ of the fractional power of $H$, i.e.,$\mathcal{M}=-(-H)^{\alpha}$, exists and has the following gradient estimates:

[TABLE]

where $c_{1}=c_{1}(\mathbf{D},d,\alpha,\beta)$ is a strictly positive constant.

Theorem 1.2**.**

Suppose that $\mathbf{D}=\mathbb{R}^{d}$ and $H=\nabla(a(x)\nabla),\,\tilde{H}=\nabla(\tilde{a}(x)\nabla)$ on $\mathbf{D}$ with measurable coefficients. If there exists a constant $\lambda\geqslant 1$ such that $\lambda^{-1}\mathrm{Id}_{d}\leqslant a(x)\leqslant\lambda\mathrm{Id}_{d}$ and $\lambda^{-1}\mathrm{Id}_{d}\leqslant\tilde{a}(x)\leqslant\lambda\mathrm{Id}_{d}$, then their subordinated semigroups $\left\{Q_{t}\right\},\,\left\{\tilde{Q}_{t}\right\}$ and the corresponding heat kernels $q(t,x,y),\,\tilde{q}(t,x,y)$ satisfy the following stability estimate: there exist bounded, continuous functions $F_{1}(t,z),\,F_{2}(t,z)$ on $(0,\infty)\times(0,\infty)$ with $\lim_{z\to 0}F_{i}(t,z)=0,\,i=1,2$ for each $t>0$,

[TABLE]

where

[TABLE]

See below (2.15) and (2.14) for the explicit expression of the functions $F_{1}(t,z),\,F_{2}(t,z)$.

Remark 1.3**.**

Similarly, we can show that the inequality (1.6) is still valid for $\mathbf{D}$ is a bounded $C^{1}$-smooth domain in $\mathbb{R}^{d}$.

2. Proof of the main theorems

2.1. Preliminaries: Heat kernel of the subordination semigroup

The asymptotic behaviors of $g(\alpha,s)$ when $s\to 0$ and when $s\to\infty$ have been known. See for example [8] Equality (14.35) for $s\to 0$ and Equality (14.37) for $s\to\infty$. For the convenience of readers we recall these asymptotic formulae in following proposition.

Proposition 2.1**.**

The function $g(\alpha,\,s)$ has the following asymptotic formulae:

[TABLE]

*where $A,\,K$ and $B$ are constants only depending on $\alpha$. *

It is known that if $\left\{P_{t}\right\}$ has a positive kernel then so does $\left\{Q_{t}\right\}$, see for example Lemma 3.4.1 of [3] and Lemma 5.4 of [5]. We restate it as the following proposition.

Proposition 2.2**.**

Suppose that $\mathbf{D}=\mathbb{R}^{d}$ and $H=\nabla(a(x)\nabla),\,\tilde{H}=\nabla(\tilde{a}(x)\nabla)$ on $\mathbf{D}$ with measurable coefficients and suppose there exists a constant $\lambda\geqslant 1$ such that $\lambda^{-1}\mathrm{Id}_{d}\leqslant a(x)\leqslant\lambda\mathrm{Id}_{d}$ and $\lambda^{-1}\mathrm{Id}_{d}\leqslant\tilde{a}(x)\leqslant\lambda\mathrm{Id}_{d}$. Then $\left\{Q_{t}\right\}$ also has a positive kernel $q(t,x,y)$ on $(0,\infty)\times\mathbf{D}\times\mathbf{D}$ such that

[TABLE]

Proof.

Since $p(t,x,y)$ is the positive kernel of $\left\{P_{t}\right\}$, it follows from Eq.(1.4)

[TABLE]

This completes the proof. ∎

As a direct corollary of the above two propositions, we have the following results.

Theorem 2.3**.**

Suppose that $\mathbf{D}$ is a domain in $\mathbb{R}^{d}$ and $H=\nabla(a(x)\nabla)$ on $\mathbf{D}$ with measurable coefficients. If there exists a constant $\lambda\geqslant 1$ such that $\lambda^{-1}\mathrm{Id}_{d}\leqslant a(x)\leqslant\lambda\mathrm{Id}_{d}$, then the heat kernel $q(t,x,y)$ of the fractional power of $H$, i.e.,$\mathcal{M}=-(-H)^{\alpha}$, has the following Nash’s Hölder estimates: there are constants $c=c(d,\lambda,\alpha)>1$ and $\gamma=\gamma(d,\lambda)\in(0,1)$ such that

[TABLE]

for all $t>0$ and $(x,y),\,(x_{1},y_{1})\in\mathbf{D}\times\mathbf{D}$.

Proof.

The first result is already known, see [5, Lemma 5.4]. It is a consequence of (2.3) and the following estimates

[TABLE]

The second inequality follows from Eq.(1.3) of [2]. We shall not provide details since it will be similar to the proof that we present below. ∎

2.2. Proof of the main theorems

Theorem 1.1 is a corollary of the following proposition.

Proposition 2.4**.**

Suppose that $\mathbf{D}$ is a domain in $\mathbb{R}^{d}$. If there are two constants $M>0$ and $\ell\geq 0$ such that $\left|\nabla_{x}p(t,x,y)\right|$ has an upper bound

[TABLE]

then there is a strictly positive constant $c_{1}=c_{1}(M,\ell,\alpha)$ such that

[TABLE]

Proof.

We shall divide the proof into several steps. The idea is similar to the proof of Lemma 5.4 of [5].

Step 1. It follows from Lebesgue’s dominated theorem that condition (2.7) implies that one can take the derivative under the integral sign in Eq.(2.3), i.e.,

[TABLE]

Hence inequality (2.7) imply that for all $(t,x,y)\in(0,\infty)\times\mathbf{D}\times\mathbf{D}$,

[TABLE]

Since the exponential function in the above integrand is less than one, we have that

[TABLE]

since $g(\alpha,s)$ is positive and $\int_{0}^{\infty}g(\alpha,s)ds=1$. Using the asymptotic formula Eq.(2.1) we also see that $\int_{0}^{1}s^{-\frac{\ell}{2}}g(\alpha,s)\mathrm{d}s$ is finite. Thus we have

[TABLE]

Step 2. It is easy to see from Eqs (2.1)-(2.2) that there exists a constant $\hat{c}:=\hat{c}(\alpha)>0$ such that

[TABLE]

Substituting this inequality into Eq. (2.9), we obtain that

[TABLE]

where $\Gamma(\cdot)$ is the Gamma function. By putting the inequalities (2.10) and (2.12) together, we obtain the desired gradient estimate (2.8). ∎

*Proof of Theorem 1.1. * For the uniformly elliptic operator $H$, it is known that its kernel has the following gradient estimate, see for example [11] Inequality (1.6) or [12] Inequality (1.4),

[TABLE]

Hence it follows from Proposition 2.4 that Theorem 1.1 holds. $\Box$

*Proof of Theorem 1.2. * We shall divide the proof into several steps.

Step 1. Proposition 2.4 can be rewritten as the following: for a positive function $f(t,x,y)$, if there are two constants $M>0$ and $\ell\geq 0$ such that $f(t,x,y)$ has an upper bound

[TABLE]

then there is a strictly positive constant $c_{1}=c_{1}(M,\ell,\alpha)$ such that

[TABLE]

It follows form Theorem 1.2 of [2] that

[TABLE]

where $c>0,c^{\prime}>1,\gamma\in(0,1),\delta\in(0,1)$ depend only on $d$ and $\lambda$. Hence it follows from (2.3) and the inequality (2.13) that

[TABLE]

On the other hand, the first inequality in Theorem 2.3 gives the following bound:

[TABLE]

where $c_{1}=c_{1}(d,\lambda,\alpha)$ is a constant.

Combining the above two bounds together, we prove (2.14) with the choice

[TABLE]

Step 2. It follows from (1.4) that

[TABLE]

Thus it follows from Minkowski’s integral inequality that for any $p\in[0,\infty]$,

[TABLE]

It follows from Theorem 1.1 of [2] that

[TABLE]

Hence we have that

[TABLE]

Since $Q_{t}$ is also a contraction semigroup, we prove (1.6) y the choice

[TABLE]

This completes the proof of Theorem 2. $\Box$

Acknowledgements: Y. Chen is supported by China Scholarship Council (201608430079) and Hubei Provincial NSFC (2016CFB526); Y. Hu is partially supported by Simons Foundation (209206); Z. Wang is supported by Mathematical Tianyuan Foundation of China (11526117) and Zhejiang Provincial NSFC (LQ16A010006).