H-distributions with unbounded multipliers

Abstract

We investigate H-distributions for sequences in the dual pairs of Bessel spaces, $(H^q_s , H^{p}_{-s}), s\in \mathbb {R}, q>1$ and $q=p/(p-1),$ by the use of unbounded multipliers, with the finite regularity, as test functions. The results relating weak convergence, H-distributions and strong convergence are applied in the analysis of strong convergence for a sequence of approximated solutions to a class of differential equations $P(x,D)u_n=f_n$, where P(x, D) is a differential operator of order k with coefficients in the Schwartz class and $(f_n)$ is a strongly convergent sequence in an appropriate Bessel potential space.

A \(L^p\)-boundedness theorem

Theorem 13

Let N be an integer such that ${N > 2d}$, ${1<p < \infty }$ and $\displaystyle T: S_{N}^{0}\times L^p\left( \mathbb {R}^d\right) \rightarrow L^p\left( \mathbb {R}^d\right)$ be defined by

$$\begin{aligned} T(\sigma ,u)=T_\sigma u. \end{aligned}$$

Then, T is a continuous bilinear operator and there exists $c_N > 0$ such that the following estimate holds

$$\begin{aligned} \left\| T_{\sigma }u\right\| _{L^p}\le c_{N} \left| \sigma \right| _{S_N^0}\left\| u\right\| _{L^p}. \end{aligned}$$

(25)

Proof

Steps of the proof are the same as in the proof of Theorem 10.7 from [17]. For the sake of completeness, we will focus ourselves on constants appearing in the estimates of Theorem 10.7, which will imply continuity with respect to $\sigma$. We represent $\displaystyle \mathbb {R}^d$ as a union of cubes, i.e. $\mathbb {R}^d=\bigcup _{l \in \mathbb {Z}^d} Q_l$, where $Q_l$ is the cube with centre at l, with edges parallel to coordinate axes and of length one. Further on, we introduce $\displaystyle \eta \in \displaystyle C_{c}^{\infty }\left( \mathbb {R}^d\right)$ such that $\eta (x)=1$ for $\displaystyle x \in Q_0$ and define $\sigma _{l}(x,\xi )=\eta (x-l)\sigma (x,\xi )$, $x,\xi \in \mathbb {R}^d, \; l \in \mathbb {Z}^d$. Then $\displaystyle T(\sigma _l,\cdot )=T_{\sigma _l}=\eta (x-l)T_\sigma$ and

$$\begin{aligned} \int _{Q_l}\big |\left( T_{\sigma }\varphi \right) (x)\big |^p dx \le \int _{\mathbb {R}^d}\left| \left( T_{\sigma _l}\varphi \right) (x)\right| ^p dx,\ \ \varphi \in {{\mathscr {S}}}\left( \mathbb {R}^d\right) . \end{aligned}$$

(26)

Next,

$$\begin{aligned} (T_{\sigma _l}\varphi )(x)=(2\pi )^{-d}\int _{\mathbb {R}^d}e^{ i x \lambda }\left[ (2\pi )^{-d}\int _{\mathbb {R}^d} e^{ ix \xi }\hat{\sigma }_{l}(\lambda ,\xi )\hat{\varphi }(\xi )d\xi \right] d\lambda , \end{aligned}$$

(27)

where $\hat{\sigma }_l(\lambda ,\xi )=\int _{\mathbb {R}^d}e^{- i \lambda x} \sigma _l(x,\xi )dx$ for $\lambda ,\xi \in \mathbb {R}^d$. The proof of Lemma 10.9 in [17] gives that for all $\displaystyle \alpha$, $\beta \in \mathbb {N}_{0}^{d},$

$$\begin{aligned} \left| (-i\lambda )^\beta \partial _{\xi }^{\alpha } \hat{\sigma }_l(\lambda ,\xi )\right| \le c_\beta \langle \xi \rangle ^{-\left| \alpha \right| }\sup _{\gamma \le \beta , x,\xi \in \mathbb {R}^d}\left| \partial _{\xi }^{\alpha } \partial _{x}^{\gamma }\sigma (x,\xi )\right| \langle \xi \rangle ^{\left| \alpha \right| }. \end{aligned}$$

Moreover, for all $\displaystyle \alpha \in {\mathbb {N}}_0^{d}$ and for all positive integers n there is a $c_{n}>0$ such that

$$\begin{aligned} \left| \partial _{\xi }^{\alpha }\hat{\sigma }_l(\lambda ,\xi )\right| \le c_{n}\langle \xi \rangle ^{-\left| \alpha \right| }\left( 1+\left| \lambda \right| \right) ^{-n} \left( \sup _{\left| \beta \right| \le n, x,\xi \in \mathbb {R}^d} \left| \partial _{\xi }^{\alpha }\partial _{x}^{\beta }\sigma (x,\xi )\right| \langle \xi \rangle ^{\left| \alpha \right| }\right) . \end{aligned}$$

(28)

Hence, for any integer $N > d/2$ we conclude from (28) that $|\partial _{\xi }^{\alpha }\hat{\sigma }_l(\lambda ,\xi )|\le B |\xi |^{-|\alpha |}$ for $|\xi |>\xi _0$ and $|\alpha | \le N$, where

$$\begin{aligned} B=c_N\left( 1+\left| \lambda \right| \right) ^{-N} \max _{\left| \alpha \right| , \left| \beta \right| \le N}\sup _{x,\xi \in \mathbb {R}^d}\left| \partial _{\xi }^{\alpha }\partial _{x}^{\beta }\sigma (x,\xi )\right| \langle \xi \rangle ^{\left| \alpha \right| }. \end{aligned}$$

Therefore, we can use Theorem 1 with $\psi (\xi )= \hat{\sigma }_{l}(\lambda ,\xi )$ and $B=c_{N} \left( 1+|\lambda |\right) ^{-N} |\sigma |_{S_{N}^{0}}$ to conclude that the operator $\left( \tilde{T}_{l,\lambda }\varphi \right) (x)=(2\pi )^{-d}\int _{\mathbb {R}^d}e^{ i x \xi }\hat{\sigma }_{l}(\lambda ,\xi )\hat{\varphi }(\xi )d\xi$, $\varphi \in {{\mathscr {S}}}\left( \mathbb {R}^d\right)$ can be extended to a bounded operator on $\displaystyle L^{p}\left( \mathbb {R}^d\right)$ so that with a suitable $c>0$

$$\begin{aligned} \Vert \tilde{T}_{l,\lambda }\varphi \Vert _{p} \le c c_N\left( 1+|\lambda |\right) ^{-N} |\sigma |_{S_{N}^{0}}\Vert \varphi \Vert _p,\ \varphi \in L^p\left( \mathbb {R}^d\right) . \end{aligned}$$

(29)

Then, by (27), there exists (new) $c>0$ such that

$$\begin{aligned} \left\| T_{\sigma _l}\varphi \right\| _{p}\le c c_N\left| \sigma \right| _{S_{N}^{0}} \left\| \varphi \right\| _p\int _{\mathbb {R}^d}\left( 1+\left| \lambda \right| \right) ^{-N}d\lambda . \end{aligned}$$

(30)

Then, (30) and (26), for integer $N > d$, imply that there exists $c>0$, independent on l, so that

$$\begin{aligned} \int _{Q_l}\left| \left( T_{\sigma }\varphi \right) (x)\right| ^{p}dx \le c c_N^{p} \left( \left| \sigma \right| _{S_{N}^{0}}\right) ^p\left\| \varphi \right\| _{p}^{p},\ \varphi \in {{\mathscr {S}}}\left( \mathbb {R}^d\right) . \end{aligned}$$

(31)

According to [17], Lemma 10.10, for $\varphi \in {{\mathscr {S}}}\left( \mathbb {R}^d\right)$ vanishing in a neighborhood of fixed $x\in \mathbb {R}^d,$ we have that $(T_\sigma \varphi )(x)=(2\pi )^{-d/2}\int _{\mathbb {R}^d}K(x,x-z)\varphi (z)dz,$ where ${ K(x,z)=(2\pi )^{-d/2}\int _{\mathbb {R}^d}e^{iz \xi }\sigma (x,\xi )d\xi , x,z\in \mathbb {R}^d}$, in the sense of distributions. Following the proof of Lemma 10.10, we have that for every integer $k >d$ there exists $C_{k} > 0$ such that

$$\begin{aligned} \big |K(x,z)\big |\le C_{k} \left| z\right| ^{-k}\left| \sigma \right| _{S_{k}^{0}},\ z\not =0. \end{aligned}$$

(32)

Next, we construct cubes $Q_l^{*}$ and $Q_l^{**}$ as in the proof of Theorem 10.7 in [17]. More precisely, $Q_l^{**}$ is the double of $Q_l$ and $Q_l^{*}$ has the same center l as $Q_l$ and $Q_l^{**}$ and $Q_l \subset Q_l^{*} \subset Q_l^{**}$. Then $\psi \in C_{c}^{\infty }\left( \mathbb {R}^d\right)$ is introduced so that its support is in $Q_l^{**}$, $0 \le \psi (x)\le 1$ and $\psi (x)=1$ in a neighborhood of $Q_l^*$. Then we write $T_{\sigma }\varphi = T_{\sigma }\varphi _1+T_{\sigma }\varphi _2$, where $\varphi _1=\psi \varphi$ and $\varphi _2=(1-\psi )\varphi$. We introduce notation $I_l = \int _{Q_l}|(T_{\sigma }\varphi )(x)|^p dx$ and $J_l = \int _{Q_l}\left| \left( T_{\sigma }\varphi _2\right) (x)\right| ^p dx$. Using (31) we get

$$\begin{aligned} I_l \le c 2^p c_N^p \left( \left| \sigma \right| _{S_{N}^{0}}\right) ^p\left\| \varphi _1\right\| _{p}^{p}+2^p J_l. \end{aligned}$$

(33)

By (32) we have that for every integer $k > d$ there is a $C>0$ such that

$$\begin{aligned} \left| \left( T_{\sigma }\varphi _2\right) (x)\right| \le C C_{k}\left| \sigma \right| _{S_{k}^0}\int _{\mathbb {R}^d\backslash Q_l^*}\left| x-z\right| ^{-k}\left| \varphi _{2}(z)\right| dz,\ x \in Q_l, z \in \mathbb {R}^d \backslash Q_l^*. \end{aligned}$$

Next, following [17] ((10.13), (10.14) and (10.15) , Theorem 10.7) and taking ${1/p+1/q=1}$, we obtain, with a new constant $C>0$:

$$\begin{aligned} \left| (T_{\sigma }\varphi _2)(x)\right| \le C C_{k}\left| \sigma \right| _{S_{k}^{0}}\int _{\mathbb {R}^d\backslash Q_l^*}\frac{(\mu +\left| x-z\right| )^{-k/2}\left| \varphi _2(z)\right| }{(\mu + \left| l-z\right| )^{\frac{k}{2} \left( \frac{1}{p}+\frac{1}{q}\right) }}dz, \end{aligned}$$

where $x \in Q_l,\ z \in \mathbb {R}^d\backslash Q_l^*$, ${\mu = \sqrt{d}/2+1}$. Then, by Minkowski’s and H$\ddot{\text{ o }}$lder’s inequality:

$$\begin{aligned} \left( \int _{Q_l}\left| T_{\sigma }\varphi _2\right| ^p dx\right) ^{\frac{1}{p}}\le & {} C C_{k}\left| \sigma \right| _{S_{k}^{0}} \left( \int _{\mathbb {R}^d\backslash Q_l^*}\frac{dz}{(\mu + \left| l-z\right| )^{k /2}}\right) ^{\frac{1}{q}}\\&\times \left( \int _{\mathbb {R}^d\backslash Q_l^*}\frac{\left| \varphi _2(z)\right| ^{p}dz}{(\mu +\left| l-z\right| )^{k/2}}\right) ^{\frac{1}{p}} \end{aligned}$$

We conclude, with a new constant C and for $k/2 > d$, that

$$\begin{aligned} J_l \le C C_{k}^{p}\left( \left| \sigma \right| _{S_{k}^{0}}\right) ^p\int _{\mathbb {R}^d\backslash Q_l^*}\frac{\left| \varphi _2(z)\right| ^{p}dz}{\left( \mu +\left| l-z\right| \right) ^{k/2 }}. \end{aligned}$$

(34)

By (33) and (34), there exists $C_1>0$ such that:

$$\begin{aligned} \int _{Q_l}\left| (T_{\sigma }\varphi )(x)\right| ^p dx\!\le \!C_1 C_{k}^{p}\left( \left| \sigma \right| _{S_{k}^{0}}\right) ^p \left( \int _{Q_l^{**}}\left| \varphi (x)\right| ^p dx \!+\! \int _{\mathbb {R}^d\backslash Q_l^*}\frac{\left| \varphi _2(z)\right| ^{p}dz}{\left( \mu +\left| l-z\right| \right) ^{k /2}}\right) . \end{aligned}$$

Summing over all $l \in \mathbb {Z}^d$, we get:

$$\begin{aligned} \int _{\mathbb {R}^d}\left| (T_{\sigma }(\varphi )(x))\right| ^p dx \le C_2(\left| \sigma \right| _{S_{k}^{0}})^p\left( 1+\sum _{l \in \mathbb {Z}^d}\frac{1}{(1+\left| l\right| )^{k/2}}\right) \int _{\mathbb {R}^d}\left| \varphi (x)\right| ^{p}dx. \end{aligned}$$

Therefore, with $k=N >2d$, we obtain the desired estimate:

$$\begin{aligned} \int _{\mathbb {R}^d}\left| \left( T_{\sigma }(\varphi )(x)\right) \right| ^p dx \le C_N \left( \left| \sigma \right| _{S_{N}^{0}}\right) ^p \int _{\mathbb {R}^d}\left| \varphi (x)\right| ^{p}dx. \end{aligned}$$

Extending by density both sides to $\displaystyle u \in L^p(\mathbb {R}^d)$, we obtain (25). $\square$