论文标题
与具有复杂系数的发散形式运算符相关的半群最大运算符
On semigroup maximal operators associated with divergence-form operators with complex coefficients
论文作者
论文摘要
令$ l_ {a} = - {\ rm div}（a \ nabla）$为椭圆形发散形式运算符，具有有限的复杂系数在任意开放集合集合$ω\ subseteq \ subseteq \ mathbb {r}^{d} $上受到混合边界条件。我们证明，最大运算符$ {\ mathscr m}^{a} f = \ sup_ {t> 0} | \ exp（-tl_ {a}）f | $在$ l^{p}（ω）$中以$ l^{p}（ω）$限制，只要$ a $ a $ a $ a $ p $ - p $ -p $ - eelliptic of of of [10]。该结果的相关性是，通常，由$ -l_ {a} $生成的半群既不是$ l^{\ infty} $也不是积极的，因此既不是hopf--dunford-schwartz-schwartz maximal ergodic定理[15，chap。我们还表明，如果$ d \ geq 3 $以及与$ l_ {a} $相关的sesquilinear形式的域，则嵌入$ l^{2^{*}}}（ω）$，$ 2^{*} = 2d/（dd/（dd/（d-d/divry））的范围$ l^{p} $ - 间隔$（RD/（（R-1）D+2），RD/（D-2））$，其中$ r \ geq 2 $是$ a $ a $ as $ r $ elliptic。通过我们的方法，我们还能够研究两参数最大操作员$ \ sup_ {s，t> 0} | t^{a_ {1}} _ {s} t^t^{a_ {2}} _ {t} _ {t} f | $ $。
Let $L_{A}=-{\rm div}(A\nabla)$ be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set $Ω\subseteq\mathbb{R}^{d}$. We prove that the maximal operator ${\mathscr M}^{A} f=\sup_{t>0}|\exp(-tL_{A})f|$ is bounded in $L^{p}(Ω)$, whenever $A$ is $p$-elliptic in the sense of [10]. The relevance of this result is that, in general, the semigroup generated by $-L_{A}$ is neither contractive in $L^{\infty}$ nor positive, therefore neither the Hopf--Dunford--Schwartz maximal ergodic theorem [15, Chap.~VIII] nor Akcoglu's maximal ergodic theorem [1] can be used. We also show that if $d\geq 3$ and the domain of the sesquilinear form associated with $L_{A}$ embeds into $L^{2^{*}}(Ω)$ with $2^{*}=2d/(d-2)$, then the range of $L^{p}$-boundedness of ${\mathscr M}^{A}$ improves into the interval $(rd/((r-1)d+2),rd/(d-2))$, where $r\geq 2$ is such that $A$ is $r$-elliptic. With our method we are also able to study the boundedness of the two-parameter maximal operator $\sup_{s,t>0}|T^{A_{1}}_{s}T^{A_{2}}_{t}f|$.
