论文标题
在同质Hörmander矢量场上结构的非差异操作员：热内核和全球高斯边界
Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds
论文作者
论文摘要
令$ x_ {1}，...，x_ {m} $为一个真正的平滑矢量领域的家族，以$ \ mathbb {r}^{n} $，$ 1 $ - 综合性相对于非异体膨胀家族的hörmander等级的条件，并满足$ 0 $ $ $ $} $ {n n $ n of $ nontiasotropic家族。相对于任何谎言组的结构，矢量场不被认为是不变的。让我们考虑非变化的进化运算符$$ \ mathcal {h}：= \ sum_ {i，j = 1}^{m} a_} a_ {i，j}（t，x）x_ {i} x_ {i} x_ { $（a_ {i，j}（t，x））_ {i，j = 1}^{m} $是一种对称的统一正$ m \ times m $ m $矩阵，条目$ a_ {ij} $ bounded houlder连续函数在$ \ mathbb {r}字段。我们证明了C_ {X，X，\ Mathrm {loc}}}^{2，α}（\ MathBB {r}^{1+n} \ setMinus \ setMinus \ \ $ $ $ $ $ {双面高斯边界和$ \ partial_ {t}γ，x_ {i}γ，x_ {i} x_ {j} x_ {j}γ$满足每个条带$ [0，t] \ times \ times \ times \ mathbb {r}^n $的上高斯界限。我们还证明了$ \ Mathcal {h} $的标准抛物线抛物线不平等，以及相应的固定运算符$$的标准harnack不平等 \ Mathcal {l}：= \ sum_ {i，j = 1}^{m} a_ {i，j}（x）x_ {i} x_ {j {j}。 $$ 与HölderContinuos系数。
Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying Hörmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator $$ \mathcal{H}:=\sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-\partial_{t}% $$ where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}$ are bounded Hölder continuous functions on $\mathbb{R}^{1+n}$, with respect to the "parabolic" distance induced by the vector fields. We prove the existence of a global heat kernel $Γ(\cdot;s,y)\in C_{X,\mathrm{loc}}^{2,α}(\mathbb{R}^{1+n}\setminus\{(s,y)\})$ for $\mathcal{H}$, such that $Γ$ satisfies two-sided Gaussian bounds and $\partial_{t}Γ, X_{i}Γ,X_{i}X_{j}Γ$ satisfy upper Gaussian bounds on every strip $[0,T]\times\mathbb{R}^n$. We also prove a scale-invariant parabolic Harnack inequality for $\mathcal{H}$, and a standard Harnack inequality for the corresponding stationary operator $$ \mathcal{L}:=\sum_{i,j=1}^{m}a_{i,j}(x)X_{i}X_{j}. $$ with Hölder continuos coefficients.
