Domains of type 1,1 operators: a case for Triebel--Lizorkin spaces

TL;DR

This paper proves the boundedness of type 1,1 pseudo-differential operators from Triebel--Lizorkin spaces to Lebesgue spaces, extending classical conditions and establishing optimal domain results.

Contribution

It establishes the continuity of type 1,1 pseudo-differential operators on Triebel--Lizorkin spaces and extends H"ormander's twisted diagonal condition within this framework.

Findings

Operators are continuous from $F^d_{p,1}$ to $L_p$ for $1 extless p extless \infty$.

This is the largest possible domain for such operators among Besov and Triebel--Lizorkin spaces.

Extension of H"ormander's condition to this setting using a support rule.

Abstract

Pseudo-differential operators of type 1,1 are proved continuous from the Triebel--Lizorkin space $F^d_{p,1}$ to $L_p$ for $1\le p<\infty$, when of order d, and this is the largest possible domain among the Besov and Triebel--Lizorkin spaces. H\"ormander's condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators.

Equations

$$a(x,D)=\mathrm{OP}(a)=(2\pi {)}^{-n}\int e^{ix\cdot \xi }a(x,\xi )\stackrel{\wedge }{u}(\xi )d\xi$$

$$a(x,D):{\mathcal{S}}^{\prime }({\mathbb{R}}^n)\to {\mathcal{S}}^{\prime }({\mathbb{R}}^n).$$

$$\stackrel{\wedge }{a}(\xi ,\eta )=0\text{for}C(|\xi +\eta |+1)\le |\eta |,\text{for some}C\ge 1.$$

$$a(x,D)$$

$$a(x,D):F_{p,q}^{s+d}({\mathbb{R}}^n)\to F_{p,q}^s({\mathbb{R}}^n),\text{for}p<\infty .$$

$$\operatorname{supp}\cal F(b(x,D)v)\subset\bigl{\{}\,\xi+\eta\bigm{|}(\xi,\eta)\in\operatorname{supp}\overset{{\scriptscriptstyle\wedge}}{b}(\cdot,\cdot),\eta\in\operatorname{supp}\overset{{\scriptscriptstyle\wedge}}{v}\,\bigr{\}}.$$

$$a(x,D)u(x)=a^{(1)}(x,D)u(x)+a^{(2)}(x,D)u(x)+a^{(3)}(x,D)u(x),$$

$$a^{(1)}(x,D)u$$

$$a^{(2)}(x,D)u$$

$$a_{j,k}(x,D)u_k={\int }_{{\mathbb{R}}^n}{K}_{j,k}(x,y)u_k(y)dy,\text{for}u\in {\mathcal{S}}^{\prime }({\mathbb{R}}^n).$$

$$|b(x,D)v(x)|\le c\Vert b(x,2^k\cdot )|{\dot{B}}_{1,t}^{n/t}({\mathbb{R}}^n)\Vert M_{t}v(x).$$

$$\mathinner{\bigl{\|}\,\sum_{j=0}^{k-2}a_{j,k}(x,2^{k}\cdot)\,\big{|}\dot{B}^{n/t}_{1,t}\bigr{\|}}\leq\sum_{|\alpha|\leq n+1}\mathinner{\bigl{\|}\,D^{\alpha}_{\xi}(\Psi_{k-2}(D_{x})a(x,2^{k}\cdot)\tilde{\Phi})\,\big{|}L_{1,\xi}\bigr{\|}}\leq c2^{kd},$$

$$\begin{split}\bigl{\|}\sum_{k}\sum_{j=0}^{k-2}a_{j,k}(x,D)u_{k}\bigr{\|}_{p}^{p}&\leq\int|\sum_{k}2^{kd}M_{t}u_{k}(x)|^{p}\,dx(\sup_{x,k}2^{-kd}\mathinner{\bigl{\|}\,\sum_{j=0}^{k-2}a_{j,k}(x,2^{k}\cdot)\,\big{|}\dot{B}^{n/t}_{1,t}\bigr{\|}})^{p}\\ &\leq c\int(\sum_{k}2^{kd}|u_{k}(x)|)^{p}\,dx.\end{split}$$

$$\displaystyle\operatorname{supp}\cal F\bigl{(}\sum_{j=0}^{k-2}a_{j,k}(x,D)u_{k}\bigr{)}\cup\operatorname{supp}\cal F\bigl{(}\sum_{j=0}^{k-2}a_{k,j}(x,D)u_{j}\bigr{)}\subset\bigl{\{}\,\tfrac{1}{5}2^{k}\leq|\xi|\leq 5\cdot 2^{k}\,\bigr{\}},$$

$$\displaystyle\operatorname{supp}\cal F\bigl{(}\sum_{j,l=0,1\ j+l\leq 1}a_{k-j,k-l}(x,D)u_{k-l}\bigr{)}\subset\bigl{\{}\,|\xi|\leq 4\cdot 2^{k}\,\bigr{\}}.$$

$$\operatorname{supp}\cal F\bigl{(}\sum_{j,l=0,1\ j+l\leq 1}a_{k-j,k-l}(x,D)u_{k-l}\bigr{)}\subset\bigl{\{}\,\xi\bigm{|}\tfrac{1}{4C}2^{k}\leq|\xi|\leq 4\cdot 2^{k}\,\bigr{\}}.$$

$$|\xi +\eta |\ge \frac{1}{C}|\eta |-1\ge \frac{11}{20C}{2}^{k-l}-1\ge (\frac{11}{40C}-2^{-k})2^k>\frac{1}{4C}{2}^k.$$

Full text

Domains of type $1,1$ operators: a case for

Triebel–Lizorkin spaces

Domaines des opérateurs de type $1,1$ et espaces de Triebel–Lizorkin

Jon Johnsen

Department of Mathematics, Aalborg University, Fredrik Bajers Vej 7G; DK-9220 Aalborg Øst, Denmark.

jjohnsen@math.aau.dk

 

Appeared in Comptes Rendus Academie de Sciences Paris, Serie I, 339 (2004), no. 2, 115--118.

Abstract. Pseudo-differential operators of type $1,1$ are proved continuous from the Triebel–Lizorkin space $F^{d}_{p,1}$ to $L_{p}$, $1\leq p<\infty$, when of order $d$, and this is in general the largest possible domain among the Besov and Triebel–Lizorkin spaces. Hörmander’s condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators.

**Résumé. On démontre que les opérateurs pseudo-différentiels de type $1,1$ et d’ordres $d$ sont continus de l’espace $F^{d}_{p,1}$ de Triebel–Lizorkin dans $L_{p}$, $1\leq p<\infty$, et que parmi les espaces de Besov et Triebel–Lizorkin, ces domaines sont en général le plus grand possible. La condition de Hörmander sur la diagonale tordu est établie pour ce cadre, en utilisant un resultat général sur le support de la transformation de Fourier d’un operateur pseudo-différentiel. **

1. Introduction

Recall that for symbols $a\in S^{d}_{\rho,\delta}({{\mathbb{R}}}^{n}\times{{\mathbb{R}}}^{n})$, ie $|D^{\alpha}_{\xi}D^{\beta}_{x}a(x,\xi)|\leq C_{\alpha\beta}(1+|\xi|)^{d-\rho|\alpha|+\delta|\beta|}$,

[TABLE]

map the Schwartz space $\cal S({{\mathbb{R}}}^{n})$ continuously into itself, say for $0\leq\delta\leq\rho\leq 1$. And for $(\rho,\delta)\neq(1,1)$ these operators extend to continuous, ‘globally’ defined maps

[TABLE]

But for $\rho=\delta=1$ Ching [2] proved existence of $a\in S^{0}_{1,1}$ such that $a(x,D)\notin{{\mathbb{B}}}(L_{2}({{\mathbb{R}}}^{n}))$. That every $A\in\operatorname{OP}(S^{0}_{1,1})$ is bounded on $C^{s}$ and $H^{s}$ for $s>0$ was first proved by Stein (unpublished); Meyer [6] proved continuity from $H^{s+d}_{p}$ to $H^{s}_{p}$ for $s>0$, $1<p<\infty$.

For $s\leq 0$, Hörmander [4] gave a condition on the twisted diagonal $\{\,(\xi,\eta)\mid\xi+\eta=0\,\}$: $a(x,D)$ is bounded $H^{s+d}\to H^{s}$ for all $s\in{{\mathbb{R}}}$ if $\overset{{\scriptscriptstyle\wedge}}{a}(\xi,\eta):=\cal F_{x\to\xi}a(x,\eta)$ fulfils

[TABLE]

For $s\geq 0$ and $1\leq p\leq\infty$, the next result gives a maximal domain by means of the Triebel–Lizorkin spaces $F^{s}_{p,q}({{\mathbb{R}}}^{n})$ (albeit with a Besov space for $p=\infty$).

Theorem 1.1**.**

Every $a\in S^{d}_{1,1}({{\mathbb{R}}}^{n}\times{{\mathbb{R}}}^{n})$, $d\in{{\mathbb{R}}}$, gives a bounded operator

[TABLE]

$\operatorname{OP}(S^{d}_{1,1})$* contains $a(x,D)\colon\cal S({{\mathbb{R}}}^{n})\to\cal D^{\prime}({{\mathbb{R}}}^{n})$, that are discontinuous when $\cal S({{\mathbb{R}}}^{n})$ is given the induced topology from any $F^{d}_{p,q}({{\mathbb{R}}}^{n})$ or $B^{d}_{p,q}({{\mathbb{R}}}^{n})$ with $p\in[1,\infty]$ and $q\in\,]1,\infty]$.*

So for fixed $p\in[1,\infty[\,$, every $A\in\operatorname{OP}(S^{0}_{1,1})$ is bounded $F^{0}_{p,1}\to L_{p}$ and everywhere defined, but not so on any larger $B^{s}_{p,q}$- or $F^{s}_{p,q}$-space (regardless of the codomain).

In comparison with Besov spaces, arguments in favour of Triebel–Lizorkin spaces have, perhaps, been less convincing. Indeed, $F^{s}_{p,2}=H^{s}_{p}$ for $1<p<\infty$, cf. [9], but this doesn’t necessarily make the $F^{s}_{p,q}$ a useful extension of the $H^{s}_{p}$-scale. However, Theorem 1.1 shows that also $F^{s}_{p,q}$-spaces with $q\neq 2$ are indispensable for a natural $L_{p}$-theory.

The next result extends Hörmander’s condition in (3) to $F^{s}_{p,q}$.

Theorem 1.2**.**

Any $a(x,D)\in\operatorname{OP}(S^{d}_{1,1}({{\mathbb{R}}}^{n}\times{{\mathbb{R}}}^{n}))$ is continuous, for $s>0$, $p$, $q\in[1,\infty]$,

[TABLE]

If (3) holds, (6) does so for $s\in{{\mathbb{R}}}$. (The result extends to $B^{s}_{p,q}$ and $p,q\in\,]0,\infty]$).

The proofs of Theorem 1.1–1.2 treat the symbols directly without approximation by elementary symbols, so it is crucial to control the spectra of the terms appearing in the paradifferential splitting of $a(x,D)$, and for this purpose the following was established.

Proposition 1.3** (the support rule).**

If $b\in S^{d}_{1,0}({{\mathbb{R}}}^{n}\times{{\mathbb{R}}}^{n})$ and $v\in\cal F^{-1}\cal E^{\prime}({{\mathbb{R}}}^{n})$, then

[TABLE]

Proposition 1.4**.**

Any $A$ in $\operatorname{OP}(S^{\infty}_{1,1})$ extends to a map $\cal F^{-1}\cal E^{\prime}({{\mathbb{R}}}^{n})\to\cal S^{\prime}({{\mathbb{R}}}^{n})$, that coincides with the usual one for $A\in\operatorname{OP}(S^{\infty}_{1,0})$.

The support rule generalises to $b\in S^{\infty}_{1,1}$, for all $v\in\cal F^{-1}\cal E^{\prime}$, using Proposition 1.4.

2. On the proofs

With $1=\sum_{j=0}^{\infty}\Phi_{j}$ so that $\Phi_{j}(\xi)=1\iff|\xi|\sim 2^{j}$ ($j>0$), set $\tilde{\Phi}_{j}=\Phi_{j-1}+\Phi_{j}+\Phi_{j+1}$, $a_{j,k}(x,\eta)=\cal F^{-1}_{\xi\to x}(\Phi_{j}\overset{{\scriptscriptstyle\wedge}}{a}(\cdot,\eta))\tilde{\Phi}_{k}(\eta)$ and $u_{j}=\Phi_{j}(D)u$. One can then make the ansatz

[TABLE]

when the pair $(a,u)$ is such that the following series converge in $\cal D^{\prime}({{\mathbb{R}}}^{n})$:

[TABLE]

Here $a\in S^{\infty}_{1,1}({{\mathbb{R}}}^{n}\times{{\mathbb{R}}}^{n})$ implies $a_{j,k}\in S^{-\infty}$, and if $K_{j,k}$ denotes the distribution kernel,

[TABLE]

This definition of $a(x,D)$ extends other ones, eg (1). And Prop. 1.4 follows, for if $\overset{{\scriptscriptstyle\wedge}}{u}\in\cal E^{\prime}$ both $a^{(1)}(x,D)u$, $a^{(2)}(x,D)u$ exist as finite sums; with $K_{k}(x,y):=\cal F^{-1}_{\xi\to y}(a\tilde{\Phi}_{k})(x,x-y)$ one can sum over $j\leq N$ in (11) and majorise to show $\cal S^{\prime}$-convergence to $\int K_{k}(x,\cdot)u_{k}\,dy$.

To exploit the ansatz further, the ‘pointwise’ estimate in the next lemma is useful.

Lemma 2.1**.**

Let $v\in\cal S^{\prime}({{\mathbb{R}}}^{n})$ and $b\in S^{\infty}_{1,1}({{\mathbb{R}}}^{n}\times{{\mathbb{R}}}^{n})$ such that $\operatorname{supp}\cal Fv\cup\bigcup_{x\in{{\mathbb{R}}}^{n}}\operatorname{supp}b(x,\cdot)$ is contained in a ball $B(0,2^{k})$, $k\in{\mathbb{N}}$. Then there exists a $c>0$ such that

[TABLE]

Here $M_{t}f(x)=\sup_{r>0}(\tfrac{1}{|B(x,r)|}\int_{B(x,r)}|f(y)|^{t}\,dy)^{\frac{1}{t}}$ is the maximal function; $0<t\leq 1$.

Lemma 2.1 is similar to [5, Prop. 5(a)], except that $b\in S^{\infty}_{1,1}$ replaces the vague assumption of being a ‘symbol ${{\mathbb{R}}}^{n}\times{{\mathbb{R}}}^{n}\to{{\mathbb{C}}}$’ ([5, Prop. 5(a)] itself is not easy to read, as it is extracted from an earlier proof with another set-up. But $b\in S^{\infty}_{1,1}$ implies that $b(x,D)v$ is given by an integral like (11), and estimates in [5, Prop. 4] apply to this.)

The proof of Theorem 1.1 combines (12) with $L_{p}(\ell_{1})$-boundedness of $M_{t}$ for $t<1$, so that $\tfrac{n}{t}<n+1$. Further estimates of $a$ follow from the embeddings $W^{n+1}_{1}\hookrightarrow B^{n+1}_{1,\infty}\hookrightarrow\dot{B}^{n/t}_{1,t}$: since $\tfrac{1}{4}\leq|\eta|\leq 4$ on $\operatorname{supp}\tilde{\Phi}$, so eg $2^{kd}\sim(1+|2^{k}\eta|)^{d}$, then if $\Psi_{k}=\Phi_{0}+\dots+\Phi_{k}$,

[TABLE]

where $c=c^{\prime}\mathinner{\|}\tilde{\Phi}\,|W^{n+1}_{1}\|\|\check{\Psi}\|_{1}\sup_{x,\xi;|\alpha|\leq n+1}(1+|\xi|)^{-(d-|\alpha|)}|D^{\alpha}_{\xi}a(x,\xi)|$. Using (12),

[TABLE]

For $k$ in finite sets, it now follows that the $a^{(1)}(x,D)u$-series is fundamental in $L_{p}$ when $u\in F^{d}_{p,1}({{\mathbb{R}}}^{n})$ for $1\leq p<\infty$, and (14) gives that $a^{(1)}(x,D)$ is bounded. The sum $\sum_{j=0}^{k-2}$ may then be replaced by the one pertinent for $a^{(2)}$, with a similar argument. To handle $a^{(3)}$, one may further invoke Taylor’s formula and [10, Lem. 3.8]. The case $B^{d}_{\infty,1}({{\mathbb{R}}}^{n})$ is analogous, and the counterexamples of [2] adapts easily to give the sharpness.

In the proof of Theorem 1.2, the key point is to obtain (with $\Phi_{j}$ as in [10])

[TABLE]

If (3) holds, then (16) may be supplemented by the property that, for $k$ large enough,

[TABLE]

By Proposition 1.3, (15)–(16) are easy. (17) is seen thus: given (3), Proposition 1.3 implies that any $\xi+\eta$ in $\operatorname{supp}\cal F(a_{k-j,k-l}(x,D)u_{k-l})$ for large $k$ fulfils

[TABLE]

To complete the proof of Theorem 1.2 one can modify the estimates (14) ff. into $L_{p}(\ell^{s}_{q})$ estimates; then convergence criteria for series of distributions, eg Theorems 3.6–3.7 of [10], apply by (15)–(16) (like arguments used in [6, 10, 5] etc.). The ball on the r.h.s. of (16) only yields estimates of $\mathinner{\|}a^{(2)}(x,D)u\,|F^{s}_{p,q}\|$ for $s>0$, as is well known. But if (3) holds, one can, by (17), use the criteria for series with spectra in dyadic annuli, like for $a^{(1)}$ and $a^{(3)}$ (the finitely many other terms of $a^{(2)}$ are in $\bigcap_{s>0}F^{s}_{p,q}$).

Remark 1**.**

The class $\operatorname{OP}(S^{d}_{1,1}({{\mathbb{R}}}^{n}\times{{\mathbb{R}}}^{n}))$ was first treated in $F^{s}_{p,q}$-spaces by Runst [7], but unfortunately the proofs are somewhat flawed, since in Lemma 1 there the spectral estimates require a support rule under rather weak assumptions, like in Prop. 1.3 above. This was seemingly overlooked in [7] and by Marschall [5]. Using the $\varphi$-decomposition of Frazier and Jawerth [3], Torres [8] extended the $H^{s}_{p}$-continuity of [6] to the $F^{s}_{p,q}$-scale. The borderline $s=0$ was treated by Bourdaud [1, Thm. 1]; his result on $B^{0}_{p,1}$ is improved by Thm. 1.1 above. Thm. 1.2 is a novelty concerning (3).

Références

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Bourdaud, G., Une algèbre maximale d’opérateurs pseudo-différentiels, Comm. Part. Diff. Equations 13 (1988), no. 9, 1059–1083.

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Ching, C.-H., Pseudo-differential operators with nonregular symbols, J. Diff. Equations 11 (1972), 436–447.

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Frazier, M. and Jawerth, B., A discrete transform and decomposition of distribution spaces, J. Func. Anal. 93 (1990), 34–170.

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Hörmander, L., Pseudo-differential operators of type $1,1$, Comm. Part. Diff. Equations 13 (1988), no. 9, 1085–1111.

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Marschall, J., Nonregular pseudo-differential operators, Z. Anal. Anwendungen 15 (1996), no. 1, 109–148.

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Meyer, Y., Régularité des solutions des équations aux dérivées partielles non linéaires (d’après J.-M. Bony), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin, 1981, pp. 293–302.

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Runst, T., Pseudodifferential operators of the “exotic” class $L^{0}_{1,1}$ in spaces of Besov and Triebel-Lizorkin type, Ann. Global Anal. Geom. 3 (1985), no. 1, 13–28.

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Triebel, H., Theory of function spaces, Monographs in mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983.

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Yamazaki, M., A quasi-homogeneous version of paradifferential operators, I. Boundedness on spaces of Besov type, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 33 (1986), 131–174.