Fundamental Solutions of Nonlocal Hörmander’s Operators

Research paper by Xicheng Zhang

Indexed on: 16 Sep '16Published on: 07 Sep '16Published in: Communications in Mathematics and Statistics

Abstract

Abstract Consider the following nonlocal integro-differential operator: for $$\alpha \in (0,2)$$ , \begin{aligned} {\mathcal {L}}^{(\alpha )}_{\sigma ,b} f(x):=\text{ p.v. } \int _{{\mathbb {R}}^d-\{0\}}\frac{f(x+\sigma (x)z)-f(x)}{ z ^{d+\alpha }}{\mathord {\mathrm{d}}}z+b(x)\cdot \nabla f(x), \end{aligned} where $$\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d$$ and $$b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$$ are smooth and have bounded first-order derivatives, and p.v. stands for the Cauchy principal value. Let $$B_1(x):=\sigma (x)$$ and $$B_{j+1}(x):=b(x)\cdot \nabla B_j(x)-\nabla b(x)\cdot B_j(x)$$ for $$j\in {\mathbb {N}}$$ . Under the following Hörmander’s type condition: for any $$x\in {\mathbb {R}}^d$$ and some $$n=n(x)\in {\mathbb {N}}$$ , \begin{aligned} {\mathrm {Rank}}[B_1(x), B_2(x),\ldots , B_n(x)]=d, \end{aligned} by using the Malliavin calculus, we prove the existence of the heat kernel $$\rho _t(x,y)$$ to the operator $${\mathcal {L}}^{(\alpha )}_{\sigma ,b}$$ as well as the continuity of $$x\mapsto \rho _t(x,\cdot )$$ in $$L^1({\mathbb {R}}^d)$$ as a density function for each $$t>0$$ . Moreover, when $$\sigma (x)=\sigma$$ is constant and $$B_j\in C^\infty _b$$ for each $$j\in {\mathbb {N}}$$ , under the following uniform Hörmander’s type condition: for some $$j_0\in {\mathbb {N}}$$ , \begin{aligned} \inf _{x\in {\mathbb {R}}^d}\inf _{ u =1}\sum _{j=1}^{j_0} u B_j(x) ^2>0, \end{aligned} we also show the smoothness of $$(t,x,y)\mapsto \rho _t(x,y)$$ with $$\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)$$ for each $$t>0$$ .AbstractConsider the following nonlocal integro-differential operator: for $$\alpha \in (0,2)$$ , \begin{aligned} {\mathcal {L}}^{(\alpha )}_{\sigma ,b} f(x):=\text{ p.v. } \int _{{\mathbb {R}}^d-\{0\}}\frac{f(x+\sigma (x)z)-f(x)}{ z ^{d+\alpha }}{\mathord {\mathrm{d}}}z+b(x)\cdot \nabla f(x), \end{aligned} where $$\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d$$ and $$b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$$ are smooth and have bounded first-order derivatives, and p.v. stands for the Cauchy principal value. Let $$B_1(x):=\sigma (x)$$ and $$B_{j+1}(x):=b(x)\cdot \nabla B_j(x)-\nabla b(x)\cdot B_j(x)$$ for $$j\in {\mathbb {N}}$$ . Under the following Hörmander’s type condition: for any $$x\in {\mathbb {R}}^d$$ and some $$n=n(x)\in {\mathbb {N}}$$ , \begin{aligned} {\mathrm {Rank}}[B_1(x), B_2(x),\ldots , B_n(x)]=d, \end{aligned} by using the Malliavin calculus, we prove the existence of the heat kernel $$\rho _t(x,y)$$ to the operator $${\mathcal {L}}^{(\alpha )}_{\sigma ,b}$$ as well as the continuity of $$x\mapsto \rho _t(x,\cdot )$$ in $$L^1({\mathbb {R}}^d)$$ as a density function for each $$t>0$$ . Moreover, when $$\sigma (x)=\sigma$$ is constant and $$B_j\in C^\infty _b$$ for each $$j\in {\mathbb {N}}$$ , under the following uniform Hörmander’s type condition: for some $$j_0\in {\mathbb {N}}$$ , \begin{aligned} \inf _{x\in {\mathbb {R}}^d}\inf _{ u =1}\sum _{j=1}^{j_0} u B_j(x) ^2>0, \end{aligned} we also show the smoothness of $$(t,x,y)\mapsto \rho _t(x,y)$$ with $$\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)$$ for each $$t>0$$ . $$\alpha \in (0,2)$$ $$\alpha \in (0,2)$$ \begin{aligned} {\mathcal {L}}^{(\alpha )}_{\sigma ,b} f(x):=\text{ p.v. } \int _{{\mathbb {R}}^d-\{0\}}\frac{f(x+\sigma (x)z)-f(x)}{ z ^{d+\alpha }}{\mathord {\mathrm{d}}}z+b(x)\cdot \nabla f(x), \end{aligned} \begin{aligned} {\mathcal {L}}^{(\alpha )}_{\sigma ,b} f(x):=\text{ p.v. } \int _{{\mathbb {R}}^d-\{0\}}\frac{f(x+\sigma (x)z)-f(x)}{ z ^{d+\alpha }}{\mathord {\mathrm{d}}}z+b(x)\cdot \nabla f(x), \end{aligned} $$\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d$$ $$\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d$$ $$b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$$ $$b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$$ $$B_1(x):=\sigma (x)$$ $$B_1(x):=\sigma (x)$$ $$B_{j+1}(x):=b(x)\cdot \nabla B_j(x)-\nabla b(x)\cdot B_j(x)$$ $$B_{j+1}(x):=b(x)\cdot \nabla B_j(x)-\nabla b(x)\cdot B_j(x)$$ $$j\in {\mathbb {N}}$$ $$j\in {\mathbb {N}}$$ $$x\in {\mathbb {R}}^d$$ $$x\in {\mathbb {R}}^d$$ $$n=n(x)\in {\mathbb {N}}$$ $$n=n(x)\in {\mathbb {N}}$$ \begin{aligned} {\mathrm {Rank}}[B_1(x), B_2(x),\ldots , B_n(x)]=d, \end{aligned} \begin{aligned} {\mathrm {Rank}}[B_1(x), B_2(x),\ldots , B_n(x)]=d, \end{aligned} $$\rho _t(x,y)$$ $$\rho _t(x,y)$$ $${\mathcal {L}}^{(\alpha )}_{\sigma ,b}$$ $${\mathcal {L}}^{(\alpha )}_{\sigma ,b}$$ $$x\mapsto \rho _t(x,\cdot )$$ $$x\mapsto \rho _t(x,\cdot )$$ $$L^1({\mathbb {R}}^d)$$ $$L^1({\mathbb {R}}^d)$$ $$t>0$$ $$t>0$$ $$\sigma (x)=\sigma$$ $$\sigma (x)=\sigma$$ $$B_j\in C^\infty _b$$ $$B_j\in C^\infty _b$$ $$j\in {\mathbb {N}}$$ $$j\in {\mathbb {N}}$$ $$j_0\in {\mathbb {N}}$$ $$j_0\in {\mathbb {N}}$$ \begin{aligned} \inf _{x\in {\mathbb {R}}^d}\inf _{ u =1}\sum _{j=1}^{j_0} u B_j(x) ^2>0, \end{aligned} \begin{aligned} \inf _{x\in {\mathbb {R}}^d}\inf _{ u =1}\sum _{j=1}^{j_0} u B_j(x) ^2>0, \end{aligned} $$(t,x,y)\mapsto \rho _t(x,y)$$ $$(t,x,y)\mapsto \rho _t(x,y)$$ $$\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)$$ $$\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)$$ $$t>0$$ $$t>0$$