论文标题
最大$ l_p $ - 型号理论到初始价值问题，并由添加剂过程生成的时间可测量的非局部运算符
A maximal $L_p$-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes
论文作者
论文摘要
令$ z =（z_t）_ {t \ geq0} $是一个附加过程，具有有限的三重态$（0,0，λ_T）_ {t \ geq0} $。假设对于任何schwartz函数$ \ mathbb {r}^d $，其傅立叶变换的位于$ c_c^{\ infty}中（b_ {c_s} \ setMinus b_s {c_s^{c_s^{ - 1}}）$ \ int _ {\ Mathbb {r}^d} | \ Mathbb {e} [φ（x+r^{ - 1} Z_T）] | dx \ leq N_0 E^{ - \ frac { - \ frac {n_1 t} （r，t）\ in（0,1）\ times [0，t]，\ end {equation*}和$$ \ |ψ^^μ（r^{ - 1} d）φ\ | _ {l_1（l_1（\ mathbb {\ mathbb {r}^d）} r \ in（0,1）。 $$其中$ s $是缩放函数（定义2.4），$ c_s $是与$ s $相关的正常常数，$μ$是$ \ mathbb {r}^d $，$ψ^^^^^^^^^^^^^^^^{ - 1} d $（x） ψ^^μ（r^{ - 1}ξ）\ Mathcal {f} [φ] \ right]（x）$和$$ψ^^^^^^^^^^^^^^^^^^^^^^^^^^= \ int _ {\ Mathbb {r}^d}（e^} 1}）μ（dy）。$$在本文中，我们建立了$ l_p $ -solvicability to初始值问题\ begin {equination} \ frac {\ partial u} {\ partial t}（t partial t}（t，t，x，x）= \ nathcal = \ nathcal {a} _z（a} _z（a} _z（t） （t，x）\ in（0，t）\ times \ times \ mathbb {r}^d，\ end end {equation}，换句话说，存在一个唯一的解决方案$ u $ to ekation to equation满足$ \ | u \ | | ___ {l_q（（0，t） N \ | U_0 \ | _ {b_ {p，q}^{s;γ-\ frac {2} {q}}}}}}}（\ mathbb {r}^d）}，$ n $在$ n $中属于$ u $和$ u_0 $，以及这些空间$ b_ {p，q}^{s;γ-\ frac {2} {q}}}（\ Mathbb {r}^d）$和$ h_p^{μ;γ}（\ mathbb {r}^d）$ scalove besov besov besov besov besov besov space（请参阅定义2.8）和常规bess bess space（请参阅bessel ofcess（space）（参见3）。
Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,Λ_t)_{t\geq0}$. Suppose that for any Schwartz function $φ$ on $\mathbb{R}^d$ whose Fourier transform is in $C_c^{\infty}(B_{c_s} \setminus B_{c_s^{-1}} )$, there exist positive constants $N_0$, $N_1$, and $N_2$ such that \begin{equation*} \int_{\mathbb{R}^d}|\mathbb{E}[φ(x+r^{-1}Z_t)]|dx\leq N_0 e^{- \frac{N_1 t}{s(r)}},\quad \forall (r,t)\in(0,1)\times[0,T], \end{equation*} and $$ \|ψ^μ(r^{-1}D)φ\|_{L_1(\mathbb{R}^d)}\leq \frac{N_2}{s(r)},\quad \forall r\in(0,1). $$ where $s$ is a scaling function (Definition 2.4), $c_s$ is a positive constant related to $s$, $μ$ is a symmetric Lévy measure on $\mathbb{R}^d$, $ψ^μ(r^{-1}D)φ(x)= \mathcal{F}^{-1} \left[ ψ^μ(r^{-1}ξ) \mathcal{F}[φ]\right](x)$ and $$ψ^μ(ξ):=\int_{\mathbb{R}^d}(e^{iy\cdotξ}-1-iy\cdotξ1_{|y|\leq 1})μ(dy).$$ In this paper, we establish the $L_p$-solvability to the initial value problem \begin{equation} \frac{\partial u}{\partial t}(t,x)=\mathcal{A}_Z(t)u(t,x),\quad u(0,\cdot)=u_0,\quad (t,x)\in(0,T)\times\mathbb{R}^d, \end{equation} In other words, there exists a unique solution $u$ to equation satisfying $$ \|u\|_{L_q((0,T);H_p^{μ;γ}(\mathbb{R}^d))}\leq N\|u_0\|_{B_{p,q}^{s;γ-\frac{2}{q}}(\mathbb{R}^d)}, $$ where $N$ is independent of $u$ and $u_0$, and the spaces $B_{p,q}^{s;γ-\frac{2}{q}}(\mathbb{R}^d)$ and $H_p^{μ;γ}(\mathbb{R}^d)$ are scaled Besov spaces (see Definition 2.8) and generalized Bessel potential spaces (see Definition 2.3), respectively.
