Minklin-Hormander

1. Minklin-Hormander type symbols

Let ${X,Y}$ be vector-spaces of measurable functions from ${\mathbb{R}^{n}}$ to itself and let ${T}$ be a bounded linear operator from ${X}$ to ${Y}$ . Recall that ${T}$ is called translation invariant if ${T(\tau_{y}f)=\tau_{y}(T(f))}$ for all ${y\in\mathbb{R}^{n}}$ and ${f\in X}$ . Let ${X=L^p}$ and ${Y=L^q}$ with ${1\leq q\leq p<\infty}$ , we found that each such operator ${T}$ is determined by a certain tempered distribution ${K}$ such that ${Tf=K\ast f}$ for every ${f\in\mathcal{S}}$ (Schwartz space). So taking Fourier transform ${\mathcal{F}}$ into ${Tf}$ we have ${\mathcal{F}(Tf)=\mathcal{F}(K)\mathcal{F}(f)}$ . This motivate us to define a Fourier multiplier as a map ${T_{\sigma}:\mathcal{S}(\mathbb{R}^n)\rightarrow\mathcal{S}'(\mathbb{R}^n)}$ given by

$\displaystyle \begin{array}{rcl} \widehat{T_{\sigma}f}(\xi)=\sigma(\xi) \widehat{f}(\xi), \end{array}$

where ${\sigma}$ is a tempered distribution ${\mathcal{S}'(\mathbb{R}^n)}$ and ${\,\widehat{}\,}$ denotes the Fourier transform ${\mathcal{F}}$ . We refer to ${\sigma}$ as symbol of ${T_{\sigma}}$ , sometimes one writes ${T_{\sigma}}$ as ${\sigma(D)}$ to relate it with more general operators ${\sigma(D,X)}$ so-called pseudo-differential operators. No standard example of such symbols is, for ${\delta>0}$ ,

$\displaystyle \sigma_{\delta}(\xi)=(1-\vert\xi\vert^2)^{\delta}\text{ if }\vert \xi\vert\leq 1\text{ and }\;\sigma_{\delta}(\xi) =0 \text{ otherwise}. \ \ \ \ \ (1)$

In the limit case, ${\delta=0}$ , the above symbol can be written as ${\mathcal{X}_{\mathbb{D}}}$ , where ${\mathcal{X}_{\mathbb{D}}}$ denotes the characteristic function of unit disk ${\mathbb{D}}$ . It’s well-known that the condition ${n\geq2}$ and ${2n/(n+1)<p<2n/(n-1)}$ is necessary for ${T_{\sigma_{0}}}$ be a Fourier multiplier on ${L^p(\mathbb{R}^{n})}$ (see e.g., [1]). However, Fefferman (see [2],[3]) gave an intricate proof which show us that this condition is not sufficient, that is, he showed that the operator ${T_{\sigma_{0}}}$ does not extend to a bounded operator on ${L^p(\mathbb{R}^{n})}$ for any ${p\neq 2}$ and ${n\geq2}$ . This result give us a negative answer to the famous disk conjecture which states that ${T_{\sigma_0}}$ is bounded on ${L^p(\mathbb{R}^{2})}$ for ${4/3\leq p\leq 4}$ . In this post we will work with symbols more regular than (1) such as Minklin symbols ${\Sigma_{1}^{0}(\mathbb{R}^{n})=\{\sigma\in C^{k}(\mathbb{R}^n\backslash\{0\}); \vert D^{\gamma}\sigma(x)\vert \leq C \vert x\vert ^{-\vert\gamma\vert}, \vert\gamma\vert\leq k\}}$ .

In a few months ago, based on Littlewood-Paley theorem, we gave a proof that the operator ${T_{\sigma}}$ is a Fourier multiplier from ${L^{p}(\mathbb{R}^{n})}$ to itself (see Theorem 7) provided that ${1<p<\infty}$ and ${\sigma}$ satisfies

$\displaystyle \sup_{j\in\mathbb{Z}}\Vert \widehat{\psi}_{j}\sigma\Vert_{L^2_{s}(\mathbb{R}^{n})}\leq L \ \ \ \ \ (2)$

for ${s>n/2}$ and ${n\geq1}$ . In this post will be showed that if ${\sigma\in \Sigma_{1}^0}$ , then its satisfies the inequality (2). Hence, by Theorem 7 one has the following classical Minklin-Hormander theorem.

Theorem 1 (Minklin-Hormander theorem) Let ${k>n/2}$ and ${\sigma\in C^{k}(\mathbb{R}^{n})}$ away from the origin. If for ${\vert\gamma\vert\leq k}$ we have

$\displaystyle \sup_{r>0}r^{\vert \gamma\vert}\left(\frac{1}{r^n}\int_{\frac{r}{2}<\vert\xi\vert<2r}\vert (D^{\gamma}_{\xi}\sigma)(\xi)\vert^{2}d\xi\right)^{\frac{1}{2}}\leq L \ \ \ \ \ (3)$

then ${T_{\sigma}}$ is a Fourier multiplier on ${L^p}$ , ${1<p<\infty}$ . In particular, ${T_{\sigma}}$ is a Fourier multiplier on ${L^p}$ if ${\sigma\in\Sigma_1^0(\mathbb{R}^n)}$ , that is,

$\displaystyle \vert D^{\gamma}\sigma(\xi)\vert \leq C \vert \xi\vert ^{-\vert\gamma\vert}. \ \ \ \ \ (4)$

Let us recall some important definitions. Let ${1\leq p\leq \infty}$ and ${k\in\mathbb{Z}_{+}}$ , a function ${f}$ lies in Sobolev spaces ${L^{p}_k(\mathbb{R}^{n})}$ if for every ${\gamma\in (\mathbb{N}\cup\{0\})^n}$ with ${\vert \gamma\vert\leq k}$ there exists ${g_{\gamma}\in L^{p}(\mathbb{R}^{n}) }$ such that

$\displaystyle \int_{\mathbb{R}^{n}}f(x)D^{\gamma}\varphi(x)dx= (-1)^{\vert \gamma\vert}\int_{\mathbb{R}^{n}}g_{\gamma}(x)\varphi(x)dx, \;\;\; \forall\varphi\in C^{\infty}_{0}(\mathbb{R}^{n}). \ \ \ \ \ (5)$

Here, we use the standard notations

$\displaystyle \begin{array}{rcl} \vert \gamma\vert=\sum_{i=1}^{n}\gamma_i \text{ and } D^{\gamma}\varphi=\frac{\partial^{\vert\gamma\vert} \varphi}{\partial^{\gamma_1} _{x_1}\partial^{\gamma_2} _{x_2}\cdots \partial^{\gamma_n} _{x_n}} \end{array}$

and we say that ${g_{\gamma}}$ is the derivative of ${f}$ in distribution sense (more precisely in ${\mathcal{D}'(\mathbb{R}^{n})}$ ) and we write ${D^{\gamma}f=g_{\gamma}}$ in ${\mathcal{D}'(\mathbb{R}^{n})}$ to mean (5). The space ${L^p_k}$ equipped with the norm

$\displaystyle \begin{array}{rcl} \vert f\vert_{L^p_k}=\sum_{\vert\gamma\vert\leq k}\vert D^{\gamma}f\vert_{L^p} \end{array}$

is a Banach space. Also, notice that Sobolev spaces ${L^{2}_k(\mathbb{R}^{n})}$ coincide with inhomogeneous fractional Sobolev spaces ${L^{2}_a(\mathbb{R}^{n})}$ because of the norm equivalence

$\displaystyle \begin{array}{rcl} \vert f\vert_{L^2_k}^2&\approx&\sum_{\vert\gamma\vert\leq k}\vert D^{\gamma}f\vert_{L^2}^2=\sum_{\vert\gamma\vert\leq k}\vert \xi^{\gamma}\widehat{f}\vert_{L^2}^2\\ &=&\int_{\mathbb{R}^{n}} \sum_{\vert\gamma\vert\leq k}\vert\xi^{2\gamma}\vert \vert\widehat{f}(\xi)\vert^2d\xi\\ &\approx& \int_{\mathbb{R}^{n}} (1+\vert\xi\vert^2)^{k}\vert\widehat{f}(\xi)\vert^2d\xi= \Vert f\Vert _{L^2_k}^{2}. \end{array}$

It follows that

$\displaystyle \begin{array}{rcl} \Vert \widehat{\psi}_{j}\sigma\Vert_{L^2_k}=\Vert \sigma(2^{j}\cdot)\widehat{\psi}\Vert_{L^2_k}=\sum_{\vert\gamma\vert \leq k}\vert D^{\gamma}(\sigma(2^{j}\cdot)\widehat{\psi})\vert_{L^2}. \end{array}$

Using Leibniz’s formula we written the term ${D^{\gamma}}$ as

$\displaystyle \begin{array}{rcl} D^{\gamma}(\sigma(2^{j}\cdot)\widehat{\psi})=\sum_{\vert \nu\vert\leq\vert \gamma\vert}\binom{\nu}{\gamma}(D^{\nu}\sigma(2^{j}\cdot))(D^{\gamma-\nu}\widehat{\psi}). \end{array}$

By observing that ${\vert D^{\gamma-\nu}\widehat{\psi}\vert\leq C}$ on ${supp(\widehat{\psi})\subset \{\xi\,:\, 1/2\leq \vert\xi\vert\leq2\}}$ and zero otherwise, we get

$\displaystyle \Vert \widehat{\psi}_{j}\sigma\Vert_{L^2_k}\leq C\sum_{\vert\gamma\vert \leq k}\sum_{\vert \nu\vert\leq\vert \gamma\vert}C_{\gamma,\nu}\vert D^{\nu}\sigma(2^{j}\cdot)\vert_{L^2(\{\xi\,:\,\frac{1}{2}< \vert\xi\vert<2\})}. \ \ \ \ \ (6)$

Now making the change of variable ${\xi\mapsto r\xi}$ one has ${D^{\nu}_{\xi}\sigma(r\cdot) =r^{\vert\nu\vert}(D_{\xi}^{\nu}\sigma)(r\xi)}$ . Hence, by (3) it follows that

$\displaystyle \sup_{r>0}\left(\int_{\frac{1}{2}<\vert\xi\vert<2}\vert D^{\nu}_{\xi}\sigma(r\cdot)\vert^{2}d\xi\right)^{\frac{1}{2}}\leq L. \ \ \ \ \ (7)$

Let ${r=2^{j}}$ . Therefore, inserting (7) into (6) easily gets

$\displaystyle \sup_{j}\Vert \widehat{\psi}_{j}\sigma\Vert_{L^2_k}\leq C\sum_{\vert\gamma\vert \leq k}\sum_{\vert \nu\vert\leq\vert \gamma\vert}C_{\gamma,\nu} L=C_{k}L \ \ \ \ \ (8)$

and Theorem 1 is a consequence of the Theorem 7 as we desired. Notice that if ${\vert D_{\xi}^{\gamma}\sigma (\xi)\vert\leq L\vert\xi\vert^{-\vert\gamma\vert}}$ ,

$\displaystyle \begin{array}{rcl} \vert D^{\nu}_{\xi}\sigma(2^j\cdot)\vert = 2^{j\vert\nu\vert} \vert (D^{\nu}_{\sigma}\sigma )(2^j\xi)\vert \leq L\vert\xi\vert^{-\vert\nu\vert}. \end{array}$

Hence ${\vert D^{\nu}\sigma(2^{j}\cdot)\vert_{L^2(\{\xi\,:\,\frac{1}{2}< \vert\xi\vert<2\})}\leq C}$ which implies (8).

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