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# 数学代写|计算流体力学代写Navier-Stokes方程代考|Calderon-Zygmund operators

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## 数学代写|计算流体力学代写Navier-Stokes方程代考|Preliminaries

Let $\mathbb{S}^{n-1}=\left{x \in \mathbb{R}^{n}:|x|=1\right}$ be the unit sphere in $\mathbb{R}^{n}$, where $n \in \mathbb{N}$. Let $T_{0}$,
$$\left(T_{0} f\right)(x)=\lim {\varepsilon \downarrow 0} \int{y \in \mathbb{R}^{n},|y| \geq \varepsilon} \frac{\Omega(y /|y|)}{|y|^{n}} f(x-y) \mathrm{d} y, \quad x \in \mathbb{R}^{n}$$
with
$$\Omega \in L_{\infty}\left(\mathbb{S}^{n-1}\right), \quad \int_{\mathbb{S}^{n-1}} \Omega(\sigma) \mathrm{d} \sigma=0$$

be the classical Calder´on-Zygmund operator where f 2 domT0 D D.Rn/ DC 10 .Rn/ (domain of definition). If n D 1 then T0 refers to the Hilbert transform.
Then T0 admits a unique linear and bounded extension from D.Rn/ (or likewiseS.Rn/) to Lp.Rn/, 1<p< 1. This had been originally proved under some mild additional smoothness assumptions for , [CaZ52], [Ste70, Chapter II], [Ste93, Chapters VI, VII] and [Tor86, Chapter XI]. But according to [DuR86, Corollary4.2, p. 552] and [Duo01, Theorem 8.38, p. 192] this assertion remains valid under the weaker natural condition (2.149). We are interested whether T0 has linear and bounded extensions to local and global Morrey spaces and what these extensions (if they exist) look like. The (unique) extension T of T0 to Lp.Rn/, 1<p< 1 can be done by completion. But this does not mean immediately that T in Lp.Rn/ can be represented analytically in the same way as T0: The usual measure theoretical arguments, convergence a.e., Fubini theorem and so on, do not apply directly because the integral in (2.148) is singular. But rescue comes (under mild additional continuity conditions for in (2.149)) from the so-called maximal Calder´on Zygmund operators ensuring that the right-hand side of (2.148) with f 2 Lp.Rn/, 1<p< 1,converges a.e. to .Tf /.x/. In connection with Proposition 2.10 it will be of interest for us that this assertion can be extended to Lp.Rn; /, 1<p< 1, according to (2.72) with the measure D w.x/L, where L is the Lebesgue measure in Rn and w 2 Ap.Rn/ belongs to the Muckenhoupt class as introduced in (2.71). This applies both to the extension by completion of T0 with domT0 D D.Rn/ to T with domT D Lp.Rn; / and also its pointwise representation

$$(T f)(x)=\lim {\varepsilon \downarrow 0} \int{y \in \mathbb{R}^{n},|y| \geq \varepsilon} \frac{\Omega(y /|y|)}{|y|^{n}} f(x-y) \mathrm{d} y, \quad x \in \mathbb{R}^{n}, \text { a.e. }$$

## 数学代写|计算流体力学代写Navier-Stokes方程代考|Main assertions

Let $X\left(\mathbb{R}^{n}\right)$ be one of the spaces $\mathcal{L}{p}^{r}\left(\mathbb{R}^{n}\right), \mathcal{L}{p}^{r}\left(\mathbb{R}^{n}\right), L_{p}^{r}\left(\mathbb{R}^{n}\right), L_{p}^{r}\left(\mathbb{R}^{n}\right)$ and $\mathcal{H}^{e} L_{p}\left(\mathbb{R}^{n}\right)$, $H^{\varrho} L_{p}\left(\mathbb{R}^{n}\right)$ as introduced in the Definitions $2.1$ and $2.3$ with
$$1<p<\infty, \quad-n<\varrho<-n / p<r<0 .$$
A linear and bounded operator $T$ acting in $X\left(\mathbb{R}^{n}\right)$, hence $T: X\left(\mathbb{R}^{n}\right) \hookrightarrow X\left(\mathbb{R}^{n}\right)$, is called an extension of $T_{0}$ according to $(2.148),(2.149)$ if it coincides on dom $T_{0}=$ $D\left(\mathbb{R}^{n}\right)$ with (2.148).

## 数学代写|计算流体力学代写NAVIER-STOKES方程代考|Distinguished representations

In connection with Theorem 2.22 and the discussion in the preceding Section 2.5.2 one may ask several questions. The extensions T of T0 in (2.154), (2.155) are unique (because D.Rn/ is dense in LV r p.Rn/ and H%Lp.Rn/). The situation in (2.156) is different. Using the so-called Cotlar decomposition Peetre proved in [Pee66, Theorem 1.1, p. 296] that there are linear and bounded extensions T of T0 to Lr p.Rn/. But this covers neither uniqueness (which actually does not hold) nor a representation of .Tf /.x/ as in (2.150) with f 2 Lr p.Rn/ a.e..One can see quite easily that extensions T of T0 from D.Rn/ to Lr p.Rn/ are by no means unique. Let

$$f_{1} \in L_{p}^{r}\left(\mathbb{R}^{n}\right) \backslash \stackrel{L}{L}{p}^{r}\left(\mathbb{R}^{n}\right), \quad\left|f{1} \mid L_{p}^{r}\left(\mathbb{R}^{n}\right)\right|=1$$
and let
$$G=\left{g=f_{0}+\lambda f_{1}: f_{0} \in \stackrel{\circ}{L}_{p}^{r}\left(\mathbb{R}^{n}\right), \lambda \in \mathbb{C}\right}$$

## 数学代写|计算流体力学代写NAVIER-STOKES方程代考|PRELIMINARIES

$$\left(T_{0} f\right)(x)=\lim {\varepsilon \downarrow 0} \int{y \in \mathbb{R}^{n},|y| \geq \varepsilon} \frac{\Omega(y /|y|)}{|y|^{n}} f(x-y) \mathrm{d} y, \quad x \in \mathbb{R}^{n}$$
with
$$\Omega \in L_{\infty}\left(\mathbb{S}^{n-1}\right), \quad \int_{\mathbb{S}^{n-1}} \Omega(\sigma) \mathrm{d} \sigma=0$$

$$(T f)(x)=\lim {\varepsilon \downarrow 0} \int{y \in \mathbb{R}^{n},|y| \geq \varepsilon} \frac{\Omega(y /|y|)}{|y|^{n}} f(x-y) \mathrm{d} y, \quad x \in \mathbb{R}^{n}, \text { a.e. }$$

## 数学代写|计算流体力学代写NAVIER-STOKES方程代考|MAIN ASSERTIONS

$$1<p<\infty, \quad-n<\varrho<-n / p<r<0 .$$
A linear and bounded operator $T$ acting in $X\left(\mathbb{R}^{n}\right)$, hence $T: X\left(\mathbb{R}^{n}\right) \hookrightarrow X\left(\mathbb{R}^{n}\right)$, is called an extension of $T_{0}$ according to $(2.148),(2.149)$ if it coincides on dom $T_{0}=$ $D\left(\mathbb{R}^{n}\right)$ with (2.148).

## 数学代写|计算流体力学代写NAVIER-STOKES方程代考|DISTINGUISHED REPRESENTATIONS

$$f_{1} \in L_{p}^{r}\left(\mathbb{R}^{n}\right) \backslash \stackrel{L}{L}{p}^{r}\left(\mathbb{R}^{n}\right), \quad\left|f{1} \mid L_{p}^{r}\left(\mathbb{R}^{n}\right)\right|=1$$
and let
$$G=\left{g=f_{0}+\lambda f_{1}: f_{0} \in \stackrel{\circ}{L}_{p}^{r}\left(\mathbb{R}^{n}\right), \lambda \in \mathbb{C}\right}$$

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