Fundamental Results for Pseudo-Differential Operators of Type $\mathbf{1},\mathbf{1}$

This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type $1,1$ in H\"ormander's sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type $1,1$-operators are defined and continuous on the full space of temperate distributions, if they fulfil H\"ormander's twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool the paradifferential decomposition is derived for type $1,1$-operators, and to confirm a natural hypothesis the symmetric term is shown to cause the domain restrictions; whereas the other terms are shown to define nice type $1,1$-operators fulfilling the twisted diagonal condition. The decomposition is analysed in the type $1,1$-context by combining the Spectral Support Rule and the factorisation inequality, which gives pointwise estimates of pseudo-differential operators in terms of maximal functions.


INTRODUCTION
Pseudo-differential operators a(x, D) of type 1, 1 have long been known to have peculiar properties, almost since their invention by Hörmander [16]. This is due to initial investigations in 1972 in the thesis of Ching [10] and in lecture notes of Stein (made available in [42, Ch. VII §1.3]); and again in 1978 by Parenti and Rodino [39].
However, the first formal definition of general type 1, 1-operators was put forward in 2008 by the author [26]. It would not be unjust to view this as an axiomatization of the type 1, 1-theory, for whereas the previous contributions did not attempt to crystallise what a type 1, 1-operator is or how it can be characterised in general, the definition from [26] has been a fruitful framework for raising questions and seeking answers about type 1, 1-operators.
Indeed, being based on an operator theoretical approach, mimicking unbounded operators in Hilbert space, the definition gave from the outset a rigorous discussion of, e.g., unclosability, pseudo-locality, non-preservation of wavefront sets and the Spectral Support Rule [26]. This was followed up with a systematic L p -theory of type 1, 1-operators in [29], where a main theorem relied on a symbol analysis proved in full detail in the present paper.
The present paper goes into a deeper, systematic study of type 1, 1-operators on S ′ (R n ) and its subspaces. Indeed, the definition in [26] is shown here to give operators always defined on the maximal smooth subspace C ∞ ∩ S ′ , generalising results of Bourdaud [8] and David and Journé [12] -and shown to be defined on the entire S ′ if they belong to the self-adjoint subclass, by an extension of Hörmander's analysis of this class [18,19]. Moreover, the pointwise estimates in [28] are applied to the paradifferential decompositions, which are analysed in the type 1, 1-context here. The decomposition gives 3 other type 1, 1-operators, of which the socalled symmetric term is responsible for the possible domain restrictions, which occur when its infinite series diverges.
Altogether this should bring the theory of type 1, 1-operators to a rather more mature level.
1.1. Background. Recall that the symbol a(x, η) of a type 1, 1-operator of order d ∈ R fulfils (1.1) Classical pseudo-differential operators are, e.g., partial differential operators ∑ |α|≤d a α (x)D β , having such symbols simply with d − |α| as exponents. The presence of |β | allows for a higher growth with respect to η , which has attracted attention for a number of reasons. But for u ∈ S ′ \ S it requires another definition to settle whether u belongs to the domain of a(x, D) or not. This is indeed a main subject of the present paper, which exploits the general definition of a(x, D) presented in [26]; it is recalled in (1.10) below. The non-triviality of the above task was discovered already by Ching [10], who showed unboundedness on L 2 for certain a θ (x, D) with d = 0; cf. Example 2.5 below. As the adjoint a θ (x, D) * of Ching's operator does not leave S invariant, as can be seen explicitly e.g. from the proof of [26,Lem. 3.1], the usual extension to S ′ by duality is not possible for OP(S d 1,1 ). In general the pathologies of type 1, 1-operators are without doubt reflecting that, most interestingly, this operator class has important applications to non-linear problems: This was first described around 1980 by Meyer [35,36], who discovered that a composition operator u → F • u = F(u) with F ∈ C ∞ , F(0) = 0, can be decomposed in its action on functions u ∈ s>n/p H s p (R n ), by means of a specific u-dependent type 1, 1 symbol a u (x, η) ∈ S 0 1,1 , as F(u(x)) = a u (x, D)u(x). (1.3) He also showed that a u (x, D) extends to a bounded operator on H t r for t > 0, so the fact that u → F(u) sends H s p into itself can be seen from (1.3) by taking t = s and r = p -indeed, this proof method is particularly elegant for non-integer s > n/p. It was carried over rigorously to the present type 1, 1-framework in [26,Sec. 9], with continuity of u → F • u as a corollary. Some applications of (1.3) were explained by Taylor [44,Ch. 3].
Secondly, it was shown in [36] that type 1, 1-operators play a main role in the paradifferential calculus of Bony [2] and the microlocal inversion of nonlinear partial differential equations of the form G(x, (D α x u(x)) |α|≤m ) = 0. (1.4) This was explicated by Hörmander, who devoted [20,Ch. 10] to the subject. The resulting setup was used e.g. by Hérau [15] in a study of hypoellipticity of (1.4). Moreover, it was used for propagation of singularities in [20,Ch. 11], with special emphasis on non-linear hyperbolic equations. Recently paradifferential operators, and thus type 1, 1-operators, were also exploited for non-linear Schrödinger operators in constructions of solutions, parametrices and propagation of singularities in global wave front sets; cf. works of e.g. Tataru [43], Delort [13], Nicola and Rodino [38]. Thirdly, both type 1, 1-theory as such and Bony's paradifferential techniques played a crucial role in the author's work on semi-linear elliptic boundary problems [25].
Because of the relative novelty of this application, a sketch is given using a typical example. In a bounded C ∞ -region Ω ⊂ R n with normal derivatives γ j u = ( n · ∇) j u at the boundary ∂ Ω, and ∆ := ∂ 2 x 1 + · · · + ∂ 2 x n , let u(x) solve the perturbed ℓ-harmonic Dirichlét problem (−∆) ℓ u + u 2 = f in Ω, γ j u = ϕ j on ∂ Ω for 0 ≤ j < ℓ. (1.5) Without u 2 , the linear problem has a well-known solution u 0 = R ℓ f + K 0 ϕ 0 + · · · + K ℓ−1 ϕ l−1 , with operators belonging to the pseudo-differential boundary operator class of Boutet de Monvel [9]. For the non-linear problem in (1.5), the parametrix construction of [25] yields the solution formula u = P (N) u (R ℓ f + K 0 ϕ 0 + · · · + K ℓ−1 ϕ ℓ−1 ) + (R ℓ L u ) N u, (1.6) where the parametrix P (N) u is the linear map given by the finite Neumann series P (N) u = I + R ℓ L u + · · · + (R ℓ L u ) N−1 (1.7) in terms of the exact paralinearisation L u of u 2 with the sign convention −L u (u) = u 2 ; cf. [25]. One merit of (1.6) is to show why u's regularity is unchanged by the non-linear term u 2 : each parametrix P (N) u is of order 0, hence does not change Sobolev regularity when applied to u 0 ; while in (1.6) the remainder (R ℓ L u ) N u will be in C k (Ω) for every fixed k if N is taken large enough. Indeed, R ℓ L u has a fixed negative order if u is given with just the weak a priori regularity necessary to make sense of the boundary condition and make u 2 defined and a priori more regular than (−∆) ℓ u.
Type 1, 1-operators are important for the fact that (1.6) easily implies that extra regularity properties of f in subregions Ξ ⋐ Ω carry over to u; e.g. if f | Ξ is C ∞ so is u| Ξ . Indeed, such implications boil down to the fact that the exact paralinearisation L u factors through an operator A u of type 1, 1, that is, if r Ω denotes restriction to Ω and ℓ Ω is a linear extension to R n \ Ω, A u ∈ OP(S ∞ 1,1 ). (1.8) Now, by inserting (1.8) into (1.7)-(1.6) for a large N and using cut-off functions supported in Ξ in a well-known way, cf. [25,Thm. 7.8], the regularity of u locally in Ξ is at once improved to the extent permitted by the data f by using the pseudo-local property of A u : sing suppAu ⊂ sing suppu for u ∈ D(A). (1.9) However, the pseudo-local property of general type 1, 1-operators was only proved recently in [26], inspired by the application below (1.8). Yet, pseudo-locality was anticipated more than three decades ago by Parenti and Rodino [39], who gave an inspiring but incomplete indication, as they relied on the future to bring a specific meaning to a(x, D)u for u ∈ S ′ \C ∞ 0 . A rigorous definition of general type 1, 1 operators was first given in [26]. In a way the definition abandons Fourier analysis (temporarily) and mimicks the theory of unbounded operators in Hilbert spaces. This is by viewing a type 1, 1-operator as a densely defined, unbounded operator a(x, D) : S ′ → D ′ between the two topological vector spaces S ′ (R n ) and D ′ (R n ); thus the graph of a(x, D) may be closed or unclosed in S ′ × D ′ etc. Indeed, it was proposed in [26] to stipulate that u ∈ S ′ belongs to the domain D(a(x, D)) of a(x, D) and to set (1.10) whenever this limit does exist in D ′ (R n ) for every ψ ∈ C ∞ 0 (R n ) with ψ = 1 in a neighbourhood of the origin, and does not depend on such ψ .
In passing it is noted that, beyond the definition, operator theory is also felt in the rules of calculus, since as shown in Proposition 2.4 below the well-known commutator identity is replaced for type 1, 1-operators by an operator theoretical inclusion, (1.11) The unconventional definition in (1.10), by vanishing frequency modulation, is a rewriting of the usual one, which is suitable for the present general symbols: clearly (1.10) gives back the integral in (1.2) if u ∈ S . In case a ∈ S d 1,0 this identification extends further to u ∈ S ′ by duality and the calculus of classical pseudo-differential operators. Note that the above integral should be interpreted as the operator OP(ψ(2 −m D x )a(x, η)ψ(2 −m η)) in OP(S −∞ ) acting on u.
Clearly (1.10) is reminiscent of oscillatory integrals, now with the addition that u ∈ D(a(x, D)) when the regularisation yields a limit independent of the integration factor. Of course it is not a conventional integration factor that is used here, but rather the Fourier multiplier ψ(2 −m D x ) that modifies the frequencies of a(·, η). While the necessity of this modification was amply elucidated in [26], it is moreover beneficial because the use of ψ(2 −m D x ) gives easy access to Littlewood-Paley analysis of a(x, D).
The definition (1.10) was investigated in [26] from several other perspectives, of which some will be needed below. But mentioned briefly (1.10) was proved to be maximal among the definitions of A = a(x, D) that gives back the usual operators in OP(S −∞ ) and is stable under the limit in (1.10); A is always defined on F −1 E ′ ; it is pseudo-local but does change wavefront sets in certain cases (even if A is defined on H s ); and A transports supports via the distribution kernel, i.e. supp Au ⊂ supp K • supp u when u ∈ D(A) E ′ , with a similar spectral support rule for supp ∧ u; cf. (1.24) below and Appendix B. For the Weyl calculus, Hörmander [18] noted that type 1, 1-operators do not fit well, as Ching's operator can have discontinuous Weyl-symbol. Conversely Boulkhemair [3,4] showed that the Weyl operator e i(x−y)·η a( x+y 2 , η)u(y) dy dη/(2π) n may give peculiar properties by insertion of a(x, η) from S d 1,1 . E.g., already for Ching's symbol with d = 0, the real or imaginary part gives a Weyl operator that is unbounded on H s for every s ∈ R.
For more remarks on the subject's historic development the reader may refer to Section 2; or consult the review in the introduction of [26].
1.2. Outline of Results. The purpose of this paper is to continue the foundational study in [26] and support the definition in (1.10) with further consequences.
First of all this concerns the hitherto untreated question: under which conditions is a given type 1, 1-operator a(x, D) an everywhere defined and continuous map For this it is shown in Proposition 4.2 and Theorem 6.3 below to be sufficient that a(x, η) fulfils Hörmander's twisted diagonal condition, i.e. the partially Fourier transformed symbol should vanish in a conical neighbourhood of a non-compact part of the twisted diagonal T given by ξ + η = 0 in R n × R n . More precisely this means that for some It should perhaps be noted that it is natural to consider ∧ a(ξ , η), as it is related (cf. [26,Prop. 4.2]) both to the kernel K of a(x, D) and to the kernel K of F −1 a(x, D)F , (1.15) More generally the S ′ -continuity (1.12) is obtained in Theorems 4.6 and 6.5 below for the a(x, η) in S d 1,1 that merely satisfy Hörmander's twisted diagonal condition of order σ for all σ ∈ R. These are the symbols which for some c α,σ and 0 < ε < 1 fulfil In this asymptotic formula ∧ a χ,ε denotes a specific localisation of ∧ a(x, η) to the conical neighbourhood |ξ + η| + 1 ≤ 2ε|η| of the twisted diagonal T .
Details on the cut-off function χ in (1.16) are recalled in Section 2.3, in connection with an account of Hörmander's fundamental result that validity of (1.16) for all σ ∈ R is equivalent to extendability of a(x, D) to a bounded map H s+d → H s for all s ∈ R, as well as equivalent to the adjoint a(x, D) * being of type 1, 1.
Of course these results of Hörmander make it natural to expect that the above two conditions (namely (1.14) and (1.16) for all σ ) are sufficient for the S ′ -continuity in (1.12), but this has not been addressed explicitly in the literature before. As mentioned they are verified in Theorem 6.3, respectively in Theorem 4.6 by duality and in Theorem 6.5 by exploiting (1.16) directly.
In the realm of smooth functions the situation is fundamentally different. Here there is a commutative diagram for every type 1, 1-operator a(x, D): The first column is just the integral (1.2); the second an extension from [24,26]. Column three is an improvement given below of the early contribution of Bourdaud [8] that a(x, D) extends to a map O M → D ′ , whereby O M denotes Schwartz' space of slowly increasing smooth functions. However, the fourth column restates the full result that a type 1, 1-operator is always defined on the maximal space of smooth functions C ∞ S ′ . More precisely, according to Theorem 2.8 below, it restricts to a strongly continuous map (1.18) It is noteworthy that this holds without any of the conditions (1.14) and (1.16). Another point is that, since C ∞ ⊂ S ′ , it was necessary to ask for a limit in the topology of D ′ in (1.10).
Perhaps it could seem surprising that the described results on (1.12) and (1.18) have not been established in their full generality before. However, it should be emphasised that these properties are valid for the operator defined in (1.10), so they go much beyond the mere extendability discussed by Meyer [36], Bourdaud [7], Hörmander [18,19,20], Torres [46], Stein [42].
The definition in (1.10) is also useful because it easily adapts to Littlewood-Paley analysis of type 1, 1-operators. Here the systematic point of departure is the well-known paradifferential splitting based on dyadic coronas (cf. Section 5 for details), as used by e.g. Bony [2], Yamazaki [48], Marschall [33]: a(x, D) = a (1) ψ (x, D) + a (2) ψ (x, D) + a (3) ψ (x, D  [5,6,7], Marschall [33], Runst [41], and the author in [26,29], and in [23,24] where the Lizorkin-Triebel spaces F s p,1 were shown to be optimal substitutes for the Sobolev spaces H s p at the borderline s = d for the domains of operators in OP(S d 1,1 ). It is known that the decomposition (1.19) follows from the bilinear way ψ enters (1.10), and that one finds at once the three infinite series in (5.8)-(5.10) below, which define the a ( j) ψ (x, D). But it is a main point of Sections 5 and 6 to verify that each of these series gives an operator a ( j) ψ (x, D) also belonging to OP(S d 1,1 ); which is non-trivial because of the modulation function ψ in (1.10).
As general properties of the type 1, 1-operators a (1) ψ (x, D) and a (3) ψ (x, D), they are shown here to satisfy the twisted diagonal condition (1.14), so (1.19) can be seen as a main source of such operators. Consequently these terms are harmless as they are defined on S ′ because of (1.12) ff.
Therefore, it is the so-called symmetric term a (2) ψ (x, D) which may cause a(x, D)u to be undefined, as was previously known e.g. for functions u in a Sobolev space; cf. [24]. This delicate situation is clarified in Theorem 6.4 with a natural identification of type 1, 1-domains, namely D(a(x, D)) = D(a (2) ψ (x, D)). (1.20) This might seem obvious at first glance, but really is without meaning before the a (2) ψ -series has been shown to define a type 1, 1-operator. Hence (1.20) is a corollary to the cumbersome bookkeeping needed for this identification of a (2) ψ (x, D). In fact, the real meaning of (1.20) is that both domains consist of the u ∈ S ′ for which the a (2) ψ -series converges; cf. Theorem 6.4. In comparison, convergence of the series for a (1) ψ (x, D)u and a (3) ψ (x, D)u is in Theorem 6.1 verified explicitly for all u ∈ S ′ , a ∈ S ∞ 1,1 , and these operators are proved to be of type 1, 1. Thus (1.19) is an identity among type 1, 1-operators. It was exploited for estimates of arbitrary a ∈ S d 1,1 in e.g. Sobolev spaces H s p and Hölder-Zygmund spaces C s * in [29], by giving full proofs (i.e. the first based on (1.10)) of the boundedness for all s > 0, 1 < p < ∞, This was generalised in [29] to all s ∈ R when a fulfills the twisted diagonal condition of order σ in (1.16) for all σ ∈ R. This sufficient condition extends results for p = 2 of Hörmander [18,19] to 1 < p < ∞, s ∈ R. The special case s = 0 = d was considered recently in [21]. The present results on a ( j) ψ (x, D) are of course natural, but they do rely on two techniques introduced rather recently in works of the author. One ingredient is a pointwise estimate Although u → u * is non-linear, it is useful for convergence of series: e.g. in H s p since it is L pbounded, and as shown here also in S ′ since it has polynomial bounds u * (x) ≤ c(1 + R|x|) N .
The second ingredient is the Spectral Support Rule from [26]; cf. also [23,24]. It provides control of supp F (a(x, D)u) in terms of the supports of The simple case in which u ∈ S was covered by Metivier [34,Prop. 4.2.8]. A review of (1.24) is given in Appendix B, including an equally easy proof for arbitrary ∧ u ∈ E ′ and a ∈ S d 1,1 . A main purpose of (1.24) is to avoid a cumbersome approximation by elementary symbols. These were introduced by Coifman and Meyer [11] to reduce the task of bounding the support of F (a(x, D)u): indeed, elementary symbols have the form a(x, η) = ∑ m j (x)Φ j (η) for multipliers m j ∈ L ∞ and a Littlewood- is a finite sum when ∧ u ∈ E ′ ; whence the rule for convolutions yields (1.24) for such symbols. However, approximation by elementary symbols is not just technically redundant because of (1.24), it would also be particularly cumbersome to use for a type 1, 1-symbol, as (1.10) would then have to be replaced by a double-limit procedure. Moreover, in the proof of (1.19), as well as in the L p -theory based on it in [29], (1.24) also yields a significant simplification.
Remark 1.1. The Spectral Support Rule (1.24) shows clearly that Hörmander's twisted diagonal condition (1.14) ensures that a(x, D) cannot change (large) frequencies in supp ∧ u to 0. In fact, the support condition in (1.14) implies that ξ cannot be close to −η when (ξ , η) ∈ supp ∧ a, which by (1.24) means that η ∈ supp ∧ u will be changed by a(x, D) to the frequency ξ + η = 0.
Contents. Notation is settled in Section 2 along with basics on operators of type 1, 1 and the C ∞ -results in (1.17) ff. In Section 3 some pointwise estimates are recalled from [28] and then extended to a version for frequency modulated operators. Section 4 gives a precise analysis of the self-adjoint part of S d 1,1 , relying on the results and methods from Hörmander's lecture notes [20,Ch. 9]; with consequences derived from the present operator definition. Littlewood-Paley analysis of type 1, 1-operators is developed in Section 5. In Section 6 the operators resulting from the paradifferential splitting (1.19) is further analysed, especially concerning their continuity on S ′ (R n ) and the domain relation (1.20). Section 7 contains a few final remarks.

PRELIMINARIES ON TYPE 1, 1-OPERATORS
Notation and notions from Schwartz' distribution theory, such as the spaces C ∞ 0 , S , C ∞ of smooth functions and their duals D ′ , S ′ , E ′ of distributions, and the Fourier transformation F , will be as in Hörmander's book [17] with these exceptions: u, ϕ denotes the value of a distribution u on a test function ϕ . The Sobolev space of order s ∈ R based on L p is written H s p , and H s = H s 2 . The space O M (R n ) consists of the slowly increasing f ∈ C ∞ (R n ), i.e. the f that for each multiindex α and some N > 0 fulfils |D α f (x)| ≤ c(1 + |x|) N .
As usual t + = max(0,t) is the positive part of t ∈ R whilst [t] denotes the greatest integer ≤ t . In general, c will denote positive constants, specific to the place of occurrence.
2.1. The General Definition of Type 1, 1-Operators. For type 1, 1-operators the reader may consult [26] for an overview of previous results. The present paper is partly a continuation of [23,24,26], but it suffices to recall just a few facts.
By standard quantization, each operator a(x, D) is defined on the Schwartz space S (R n ) by Hereby its symbol a(x, η) is required to be in C ∞ (R n × R n ), of order d ∈ R and type 1, 1, which means that for all multiindices α , β ∈ N n 0 it fulfils (1.1), or more precisely has finite seminorms: The Fréchet space of such symbols is denoted by S d 1,1 (R n × R n ), or just S d 1,1 for brevity, while as usual S −∞ = d S d 1,1 . Basic estimates yield that the bilinear map (a, u) → a(x, D)u is continuous 3) The distribution kernel K(x, y) = F −1 η→z a(x, η) z=x−y is well known to be C ∞ for x = y also in the type 1, 1 context; cf. [26,Lem. 4.3]. It fulfils a(x, D)u, ϕ = K, ϕ ⊗ u for all u, ϕ ∈ S .
For arbitrary u ∈ S ′ \ S it is a delicate question whether or not a(x, D)u is defined. The general definition of type 1, 1-operators in [26] uses a symbol modification, exploited throughout below, namely b(x, η) = ψ(2 −m D x )a(x, η), or more precisely is an arbitrary cut-off function equal to 1 in a neighbourhood of the origin, let When for each such ψ the limit a ψ (x, D)u exists in D ′ (R n ) and moreover is independent of ψ , then u belongs to the domain D(a(x, D)) by definition and This was called definition by vanishing frequency modulation in [26], because the removal of high frequencies in x and η achieved by ψ(2 −m D x ) and ψ(2 −m η) disappears for m → ∞. Note that the action on u is well defined for each m in (2.5) as the modified symbol is in S −∞ . Occasionally the function ψ will be referred to as a modulation function.
The frequency modulated operator OP(ψ(2 −m D x )a(x, η)ψ(2 −m η)) has, by the comparison made in [26,Prop. 5.11], its kernel K m (x, y) conveniently given as a convolution, up to conjugation by the involution M : However, this requires verification because the commutator of the convolution (2.7) and pointwise multiplication by ϕ ⊗ (1 − χ) is a nontrivial pseudo-differential, hence non-local operator. A proof of (2.8) based on the Regular Convergence Lemma can be found in [26,Prop. 6.3].
In general the calculus of type 1, 1-operators is delicate, cf. [18,19,20], but the following result from [29] is just an exercise (cf. the proof there). It is restated here for convenience. 1 1,1 (R n × R n ) and a symbol with constant coefficients b(η) belongs to S d 2 1

Proposition 2.3. When a(x, η) is in S d
In particular D(c(x, D)) = D(a(x, D)b(D)), so the two sides are simultaneously defined.
This result applies especially to differential operators, say b(D) = D j for simplicity. But as a minor novelty, the classical commutator identity needs an atypical substitute: By classical calculations, any modulation function ψ gives the following formula for u ∈ S , hence for all u ∈ S ′ as the symbols are in S −∞ , When both terms on the left have ψ -independent limits for m → ∞, so has the right-hand side. As the first term then is a(x, D)D j u, cf. Proposition 2.3, this entails that the common domain is contained in that of D j a(x, D), with the same actions.
The inclusion (2.11) is strict in some cases, for the domains are not always invariant under differentiation. This is a well-known consequence of the classical counterexamples, which are recalled below for the reader's convenience: Example 2.5. The classical example of a symbol of type 1, 1 results from an auxiliary function Here a θ ∈ C ∞ (R n × R n ), since the terms are disjointly supported, and it clearly belongs to S d 1,1 . These symbols were used both by Ching [10] and Bourdaud [7] to show L 2 -unboundedness for d = 0, |θ | = 1. Refining this, Hörmander [18] established that continuity H s → D ′ with s > −r holds if and only if θ is a zero of A of order r ∈ N 0 . [26] gave an extension to d ∈ R, θ = 0.
The non-preservation of wavefront sets discovered by Parenti and Rodino [39] also relied on a θ (x, η). Their ideas were extended to all n ≥ 1, d ∈ R in [26, Sect. 3.2] and refined by ap- 0 is an analytic function controlling the spectrum, whilst the highly oscillating f is Weierstrass' nowhere differentiable function for orders d ∈ ]0, 1], in a complex version with its wavefront set along a half-line. (Nowhere differentiability was shown with a small microlocalisation argument, explored in [27].) Moreover, a θ (x, D) is unclosable in S ′ when A is supported in a small ball around θ , as in [26,Lem. 3.2]. Hence Definition 2.1 cannot in general be simplified to a closure of the graph in S ′ × D ′ .
As a basic result, it was shown in [26,Sec. 4 (2.14) It is a virtue of (2.14) that a(x, D) is compatible with for example OP(S ∞ 1,0 ); cf. [26] for other compatibility questions. Therefore some well-known facts extend to type 1, 1-operators: Example 2.7. Also when A = a(x, D) is of type 1, 1, one can recover its symbol from the formula Here F e i ·, ξ = (2π) n δ ξ (η) has compact support, so again it follows from (2.14) that (via a suppressed cut-off) one has A(e i ·, ξ ) = δ ξ , e i x, · a(x, ·) = e i x·ξ a(x, ξ ).

General Smooth Functions.
To go beyond the smooth functions in (2.14), it is shown in this subsection how one can extend a remark by Bourdaud [8] on singular integral operators, which shows that every type 1, 1 symbol a(x, η) of order d = 0 induces a mapÃ : is given by the following, using the distribution kernel K of a(x, D) and an auxiliary function χ ∈ C ∞ 0 (R n ) equal to 1 on a neighbourhood of supp ϕ , However, to free the discussion from the slow growth in O M , one may restate this in terms of the One advantage here is that both terms obviously make sense as long as f is smooth and temperate, i.e. for every f ∈ C ∞ (R n ) S ′ (R n ).
Moreover, for ϕ with support in the interior C • of a compact set C ⊂ R n and χ = 1 on a neighbourhood of C , the right-hand side of (2.19) gives the same value for anyχ ∈ C ∞ 0 equal to 1 around C , for in the difference of the right-hand sides equals 0 since a(x, D) Crude estimates of (2.19) now show thatÃ f yields a distribution in D ′ (C • ), and the above There is also a more explicit formula forÃ f : whenφ ∈ C ∞ 0 is chosen so thatφ ≡ 1 around C while suppφ has a neighbourhood where χ = 1, then ϕ =φϕ in (2.19) gives, for (2.20) NowÃ f ∈ C ∞ follows, for the first term is in S , and the second coincides in C • with a function in S , as a corollary to the construction of which is a basic continuity property ofÃ.
By settingÃ in relation to Definition 2.1, the above gives the new result that a(x, D) always is a map defined on the maximal set of smooth functions, i.e. on C ∞ S ′ : which locally is given by formula (2.20). The map (2.21) is continuous when C ∞ (R n ) has the usual Fréchet space structure and S ′ (R n ) has the strong dual topology.
The intersection C ∞ S ′ is topologised by enlarging the set of seminorms on C ∞ by those on S ′ . Here the latter have the form f → sup ψ∈B | f , ψ | for an arbitrary bounded set B ⊂ S . , D)). With f ∈ C ∞ S ′ and ϕ, χ as above, this is the case for u = χ f ∈ C ∞ 0 . Exploiting the convergence in Remark 2.2 in (2.19), it is seen that is contained in C ∞ now follows from the remarks to (2.20).
It is also straightforward to see that (φ(x)(1 − χ(·)))K(x, ·) stays in a bounded set in S (R n ) as x runs through C . Therefore, when f → 0 also in the strong dual topology on S ′ , then the second term in (2.20) tends to 0 uniformly with respect to x ∈ C . As x-derivatives may fall on K(x, ·), the same argument gives that sup , which gives the stated continuity property.
Remark 2.9. It is hardly a drawback that continuity in Theorem 2.8 holds for the strong dual topology on S ′ , as for sequences weak and strong convergence are equivalent (a well-known consequence of the fact that S is a Montel space).
In view of Theorem 2.8, the difficulties for type 1, 1-operators do not stem from growth at infinity for C ∞ -functions. Obviously the codomain C ∞ is not contained in S ′ , but this is not just made possible by the use of D ′ in Definition 2.1, it is indeed decisive for the above construction.
In the proof above, the fact thatÃ f ∈ C ∞ also follows from the pseudo-local property of a(x, D); cf [26,Thm. 6.4]. The direct argument above is rather short, though. In addition to the smoothness, the properties of a(x, D) f can be further sharpened by slow growth of f : . Moreover, c = max(1, 1/r) 2N /(2π) n can be chosen uniformly for x ∈ R n as it suffices to have (2.20) with 0 ≤ χ ≤ 1 and C = B(0, j), suppφ = B(0, j + 1) and χ −1 ({1}) = B(0, j + 2) for an arbitrarily large j ∈ N, which yields r = 1, c ≤ 1.
Example 2.11. The space C ∞ (R n ) S ′ (R n ) clearly contains functions of non-slow growth, e.g.
f (x) = e x 1 +···+x n cos(e x 1 +···+x n ). (2.25) Remark 2.12. Prior to the T 1-theorem, David and Journé explained in [12] how a few properties of the distribution kernel of a continuous map T : [8] used this construction forÃ, so by Theorem 2.8 this extension of T ∈ OP(S 0 1,1 ) from [12] is contained in Definition 2.1. 2.3. Conditions along the Twisted Diagonal. As the first explicit condition formulated for the symbol of a type 1, 1-operator, Hörmander [18] proved that a(x, D) has an extension by continuity In detail this means that the partially Fourier transformed symbol ∧ a(ξ , η) := F x→ξ a(x, η) is trivial in a conical neighbourhood of a non-compact part of the twisted diagonal (2.28) Localisations to certain conical neighbourhoods of T were also used by Hörmander [18,19,20] as (2.32) Using this, Hörmander introduced and analysed a milder condition than the strict vanishing in (2.27). Namely, for some σ ∈ R the symbol should satisfy an estimate, for all multiindices α and 0 < ε < 1, This is an asymptotic formula for ε → 0. It always holds for σ = 0, cf. [20, Lem. 9.3.2]: The map a → a χ,ε is continuous in S d 1,1 . The last remark on continuity has been inserted here for later reference. It is easily verified by observing in the proof of [20, Lem. 9.3.2] (to which the reader is referred) that the constant C α,β (a) is a continuous seminorm in S d 1,1 . In case σ > 0 there is a faster convergence to 0 in (2.33). In [19] this was proved to imply that a(x, D) is bounded as a densely defined map clearly a χ,ε ≡ 0 for 2ε < 1/B then.

POINTWISE ESTIMATES
A crucial technique in this paper will be to estimate |a(x, D)u(x)| at an arbitrary point x ∈ R n . Some of recent results on this by the author [28] are recalled here and further elaborated in Section 3.2 with an estimate of frequency modulated operators.  a(x, D) can be separated from u at the cost of an estimate, which is the factorisation inequality Hereby u * (x) = u * (N, R; x) denotes the maximal function of Peetre-Fefferman-Stein type, The parameter N is often chosen to satisfy N ≥ order ∧ u.
The a-factor F a , also called the symbol factor, only depends on u in a vague way, i.e. only through the N and R in (3.2). It is related to the distribution kernel of a(x, D) as Here it is first recalled that every u ∈ S ′ has finite order since, for ψ ∈ S , To avoid a discussion of the converse, it will throughout be convenient to call the least integer N fulfilling (3.5) the temperate order of u, written N = order S ′ (u).
Returning to (3.4), when the compact spectrum of u results from Fourier multiplication, then the below O(2 kN )-information on the constant will be used repeatedly in the present paper.
for a constant C independent of k.
Proceeding as before the lemma, the claim follows with C = c ′′ max(1, R −N ).
Secondly, for the a-factor in (3.3) one has F a ∈ C(R n ) ∩ L ∞ (R n ) and an estimate highly reminiscent of the Mihlin-Hörmander conditions for Fourier multipliers: and let F a (N, R; x) be given by (3.3) for parameters R, N > 0, with the auxiliary function taken as χ = ψ(R −1 ·) for ψ ∈ C ∞ 0 (R n ) equalling 1 in a set with non-empty interior. Then one has for all x ∈ R n that For the elementary proof the reader may consult [28]; cf. Theorem 4.1 and Section 6 there. A further analysis of how F a depends on a(x, η) and R is a special case of [28,Cor. 4 (3.10) If ψ(η) = 0 only holds in a corona 0 < θ 0 ≤ |η| ≤ Θ 0 , and ψ(η) = 1 holds for This can be improved when a(x, η) is modified by removing the low frequencies in the x-variable (cf. the a (3) -term in Section 5 below). In fact, with a second spectral quantity Q > 0, the following is contained in [28,Cor. 4.4]:

Estimates of Frequency Modulated Operators.
The results in the previous section easily give the following, which later in Sections 5 and 6 will be used repeatedly.
Proposition 3.6. For a(x, η) in S d 1,1 (R n × R n ), v ∈ S ′ (R n ) and arbitrary Φ, Ψ ∈ C ∞ 0 (R n ), for which Ψ is constant in a neighbourhood of the origin and with its support in B(0, R) for some R ≥ 1, there is for k ∈ N 0 and N ≥ order S ′ (F v), cf. (3.6), a polynomial bound For 0 / ∈ supp Ψ this may be sharpened to C(k) = c2 k(N+d) for all values of N + d.
Proof. In this proof it is convenient to let a k ( Then the factorisation inequality (3.1) gives In case 0 / ∈ supp Ψ, the auxiliary function χ = ψ(·/(R2 k )) used in F a k , cf. Theorem 3.2, can be so chosen that it fulfils the corona condition in Corollary 3.3; e.g. it is possible to have Θ 1 = 1 and θ 1 = r/R when Ψ ≡ 0 on B(0, r). Since Remark 3.5 implies p(a k ) ≤ p(a) |F −1 Φ(y)| dy, (3.16) When combined with the above, this inequality yields the claim in case 0 / ∈ Ψ. In the general case one has does not have 0 in its support, the above shows that for j = 1, . . . , k one has 1 yields polynomial bounds of v * j , say with a constant C ′ , so the sum on the right-hand side of (3.17) is estimated, for d + N = 0, by

ADJOINTS OF TYPE 1, 1-OPERATORS
For classical pseudo-differential operators a(x, D) : S → S ′ it is well known that the adjoint a(x, D) * : S → S ′ has symbol a * (x, η) = e i D x ·D η a(x, η), and that a → a * sends e.g. S d 1,0 into itself.

2)
for certain seminorms C αβ that are continuous on S d 1,1 (R n × R n ) and do not depend on B. Proof. When ψ ∈ C ∞ 0 (R n ) is such that ψ = 1 in a neighbourhood of the origin, a simple convolution estimate (cf. [26, Lem. 2.1]) gives that in the topology of S d+1 Since the supports of the partially Fourier transformed symbols Combining this with the fact that b(x, D) as an operator on the Schwartz space depends continuously on the symbol, one has for u ∈ S ′ (R n ), ϕ ∈ S (R n ), As the left-hand side is independent of ψ the limit in (2.5) is so, hence the definition of a(x, D) gives that every u ∈ S ′ (R n ) is in D(a(x, D)) and a(x, D)u = b(x, D) * u as claimed.
The mere extendability to S ′ under the twisted diagonal condition (2.27) could have been observed already in [18,20], but the above result seems to be the first sufficient condition for a type 1, 1-operator to be defined on the entire S ′ (R n ). 11 . Proposition 4.2 shows that (2.27) suffices for D(a(x, D)) = S ′ . But (2.27) is too strong to be necessary; a vanishing to infinite order along T should suffice.

The Self-Adjoint Subclass S d
In this section, the purpose is to prove that a(x, D) : S ′ → S ′ is continuous if more generally the twisted diagonal condition of order σ , that is (2.33), holds for all σ ∈ R.
It should be observed from (i) that a(x, η) fulfils condition (ii) or (iii) if and only if a * (x, η) does so -whereas neither (ii) nor (iii) make this obvious. But (ii) immediately gives the (expected) inclusion S d 1,1 ⊂ S d ′ 1,1 for d ′ > d. Condition (iii) is close in spirit to the Mihlin-Hörmander multiplier theorem, and it is useful for the estimates shown later in Section 6.
Remark 4.4. Conditions (ii), (iii) both hold either for all χ satisfying (2.33) or for none, for (i) does not depend on χ . It suffices to verify (ii) or (iii) for 0 < ε < ε 0 for some convenient ε 0 ∈ ]0, 1[ . This is implied by Lemma 2.13 since every power ε p is bounded on the interval [ε 0 , 1]. Theorem 4.3 was undoubtedly known to Hörmander, for he stated the equivalence of (i) and (ii) explicitly in [18,Thm. 4.2] and [20, Thm. 9.4.2] and gave brief remarks on (iii) in the latter. Equivalence with continuous extensions H s+d → H s for all s ∈ R was also shown. However, the exposition there left a considerable burden of verification to the reader.

4.2.4.
Consequences for the Self-Adjoint Subclass. One can set Theorem 4.3 in relation to the definition by vanishing frequency modulation, simply by elaborating on the above proof:

adjoint operation is stable with respect to vanishing frequency modulation in the sense that, when a
holds in the topology of S d+1 1,1 (R n × R n ).
Proof. For brevity b m (x, η) = ψ(2 −m D x )a(x, η)ψ(2 −m η) denotes the symbol that is frequency modulated in both variables. The proof consists in insertion of a(x, η) − b m (x, η) into (4.18), where the first sum tends to 0 for m → ∞ by majorised convergence. Note that for each ν ≥ 0 in the first sum of (4.18) one must control C α,β (e m 2 −ν ) for m → ∞ and η). (4.27) To do so, a convolution estimate first gives Here both terms on the right-hand side tend to 0 for m → ∞, in view of the continuity of a → a χ,ε on S d+1 1,1 ; cf. Lemma 2.13. Hence C α,β (e m 2 −ν ) → 0 for m → ∞. It therefore suffices to replace d by d +1 in (4.18) and majorise. However, a → a χ,ε commutes with a → b m as maps in S ′ (R n × R n ), so since a ∈ S d+1 1,1 , it follows from (ii) that Using this in the previous inequality, C α,β (e m 2 −ν ) ≤ C2 −νN is obtained for C independent of m ∈ N. Now it follows from (4.18) that b m (x, η) * → a(x, η) * in S d+1 1,1 as desired. Thus prepared, the proof of Proposition 4.2 can now be repeated from (4.5) onwards, which immediately gives the first main result of the paper: Theorem 4.6. When a symbol a(x, η) of type 1, 1 belongs to the classS d 1,1 (R n × R n ), as characterised in Theorem 4.3, then a(x, D) is everywhere defined and continuous It equals the adjoint of OP(e i D x ·D ηā (x, η)) : S → S .
Like for Proposition 4.2, there seems to be no previous attempts in the literature to obtain this clarification (Theorem 4.6 was stated without proof in [29]). However, it seems to be open whether (4.30) conversely implies that a ∈S d 1,1 .

DYADIC CORONA DECOMPOSITIONS
This section adopts Littlewood-Paley techniques to provide a passage to auxiliary operators a ( j) (x, D), j = 1, 2, 3, which may be easily analysed with the pointwise estimates of Section 3.
For fixed ψ it is convenient to take an integer h ≥ 2 so large that 2R < r2 h .

Inserting the relation (5.3) twice in (2.5), bilinearity gives
Of course the sum may be split in three groups having j ≤ k − h, | j − k| < h and k ≤ j − h. For m → ∞ this yields the well-known paradifferential decomposition a ψ (x, D)u = a (1) ψ (x, D)u + a (2) ψ (x, D)u + a (3) ψ (x, D)u, (5.7) whenever a and u fit together such that the three series below converge in D ′ (R n ): η)) etc. Using this and the index convention, the so-called symmetric term in (5.9) has the brief form a (2) In the following the subscript ψ is usually dropped because this auxiliary function will be fixed (ψ was left out already in a j and a j ; cf. (5.4)-(5.5)). Note also that the above a ( j) (x, D) for now is just a convenient notation for the infinite series. The full justification of this operator notation will first result from Theorems 6.1-6.4 below.
Indeed, while this is trivial for the right-hand side of (5.12), where the symbol is in S −∞ , it is for the type 1, 1-operator on the left-hand side of (5.12) a fact that follows at once from (2.14). Thus the inclusion F −1 E ′ ⊂ D(a(x, D)) in (2.14) is crucial for the simple formulae in the present paper. Analogously Definition 2.1 may be rewritten briefly as a(x, D)u = lim m a m (x, D)u m .
The importance of the decomposition in (5.8)-(5.10) lies in the fact that the summands have localised spectra. E.g. there is a dyadic corona property: Proposition 5.2. If a ∈ S d 1,1 (R n × R n ) and u ∈ S ′ (R n ), and r, R are chosen as in (5.1) for each auxiliary function ψ , then every h ∈ N such that 2R < r2 h gives whereby R h = r 2 − R2 −h > 0. Proof. By (5.2) and the Spectral Support Rule, cf. the last part of Theorem B.1, So by the triangle inequality every ζ = ξ + η in the support fulfils, as h ≥ 2, This shows (5.14), and (5.15) follows analogously.
To achieve simpler constants one could take h so large that 4R ≤ r2 h , which instead of R h would allow r/4 (and 9R/8). But the present choice of h is preferred in order to reduce the number of terms in a (2) In comparison the terms in a (2) (x, D)u only satisfy a dyadic ball condition. Previously this was observed e.g. for functions u ∈ H s in [24], as was the fact that when the twisted diagonal condition (2.27) holds, then the situation improves for large k. This is true for arbitrary u: Proposition 5.3. When a ∈ S d 1,1 (R n × R n ), u ∈ S ′ (R n ), and r, R are chosen as in (5.1) for each auxiliary function ψ , then every h ∈ N such that 2R < r2 h gives If (2.27) holds for some B ≥ 1, then the support is contained in the annulus is seen to be a subset of Thence any ζ in the support fulfils |ζ | ≤ R2 k + R2 k−1 = (3R/2)2 k . If (2.27) holds, then one has B(1 + |ξ + η|) ≥ |η| on supp F x→ξ a, so for all k larger than the given limit The term (a k − a k−h )(x, D)u k is analogous but will cause 3R/2 to be replaced by 2R.
Remark 5.4. The inclusions in Propositions 5.2 and 5.3 have been a main reason for the introduction of the paradifferential splitting (5.7) in the 1980's, but they were then only derived for elementary symbols; cf. [6,7,48]. With the Spectral Support Rule, cf. Theorem B.1, this restriction is redundant; cf. also the remarks to (1.24) in the introduction.

Polynomial Bounds.
In the treatment of a (1) (x, D)u and a (3) (x, D)u in (5.8) and (5.10) one may conveniently commence by observing that, according to Proposition 5.2, the terms in these series fulfil condition (A.1) in Lemma A.1 for θ 0 = θ 1 = 1. So to deduce their convergence from Lemma A.1, it remains to obtain the polynomial bounds in (A.2). For this it is natural to use the efficacy of the pointwise estimates in Section 3: Proof. The second inequality follows by taking the two cut-off functions in Proposition 3.6 as Φ = ϕ and Ψ = ψ(2 −h ·). The first claim is seen by interchanging their roles, i.e. for Φ = ψ(2 −h ·) and Ψ = ϕ ; the latter is 0 around the origin so N + d is obtained without the positive part. Clearly similar estimates hold for the terms in a (2) (x, D)u. E.g., taking ψ − ψ(2 −h ·) and ϕ , respectively, as the cut-off functions in Proposition 3.6, one finds for k ≥ h the estimate in (5.24). Note that the positive part can be avoided for 0 ≤ k < h by using a sufficiently large constant.
The difference in the above estimates appears because u k in (5.22) has spectrum in a corona. However, one should not confound this with spectral inclusions like (A.1) that one might obtain after application of a k−h (x, D), for these are irrelevant for the pointwise estimates here. 5.3. Induced Paradifferential Operators. Although (5.8)-(5.10) yield a well-known splitting, the operator notation a ( j) (x, D) requires justification in case of type 1, 1-operators.
Departing from the right hand sides of (5.8)-(5.10) one is via (5.12) led directly to the symbols In addition, letting δ k≥h stand for 1 when k ≥ h and for 0 in case k < h, These three series converge in the Fréchet space S d+1 1,1 (R n × R n ), for the sums are locally finite. Therefore it is clear that a(x, η) = a (1) (x, η) + a (2) (x, η) + a (3) (x, η), (5.29) where some of the partially Fourier transformed symbols have conical supports, This well-known fact follows from the supports of ψ and ϕ . But a sharper exploitation gives Proposition 5.6. For each a ∈ S d 1,1 and every modulation function ψ ∈ C ∞ 0 (R n ) in (5.1), the associated symbols a (1) ψ (x, η) and a (3) ψ (x, η) fulfil the twisted diagonal condition (2.27).
Anyhow, it is a natural programme to verify that u ∈ S ′ belongs to the domain of the operator a ( j) (x, D) precisely when the previously introduced series denoted a ( j) (x, D)u converges; cf. (5.8)-(5.10). In view of the definition by vanishing frequency modulation in (2.5) ff, this will necessarily be lengthy because a second modulation function Ψ has to be introduced.
To indicate the details for a (1) (x, η), let ψ, Ψ ∈ C ∞ 0 (R n ) be equal to 1 around the origin, and let ψ be used as the fixed modulation function entering a (1) (x, D) = a (1) ψ (x, D) in (5.8); and set ϕ = ψ − ψ(2·). The numbers r, R and h are then chosen in relation to ψ as in (5.1).
Consequently, with µ = [log 2 (λ /R)] and k = m + l, for l ∈ Z, one has for u ∈ S ′ (R n ) that A similar reasoning applies to a (3) (x, η). The main difference is that the possible inclusion of supp ϕ(2 − j ·), into the level sets where Ψ(2 −m ·) equals 1 or 0, in this case applies to the symbol , η)). Therefore one has for the same µ , Treating a (2) ψ (x, D) analogously, it is not difficult to see that once again the central issue is whether supp ϕ(2 −k ·) is contained in the set where Ψ(2 −m ·) = 1 or = 0. So when m ≥ h for simplicity, one has for the same µ , and with primed sums over the same integers l as above, The programme introduced after Proposition 5.6 is now completed by letting m → ∞ in (5.35)-(5.37) and observing that the infinite series in (5.8)-(5.10) reappear in this way. Of course, this relies on the fact that the remainders in the primed sums over l can be safely ignored: Proposition 5.7. When a(x, η) is given in S d 1,1 (R n × R n ) and Ψ, ψ ∈ C ∞ 0 (R n ) equal 1 in neighbourhoods of the origin, then it holds for every u ∈ S ′ (R n ) that each term (with l fixed) in the primed sums in (5.35)- (5.36) This is valid for (5.37) too, if a(x, η) in addition fulfils the twisted diagonal condition (2.27).
Proof. To show that each remainder term tends to 0 for m → ∞ and fixed l, it suffices to verify (A.1) and (A.2) in view of Remark A.2. For a (1) ψ (x, D), note that by repeating the proof of Proposition 5.2 (ignoring Ψ) each remainder in (5.35) has ξ in its spectrum only when (R h 2 l )2 m ≤ |ξ | ≤ 5·2 l 4 R2 m .
An extension of the proposition's remainder analysis to general a (2) ψ (x, D) without a condition on the behaviour along the twisted diagonal does not seem feasible. But such results will follow in Section 6 from a much deeper investigation of a(x, D) itself; cf. Theorem 6.4.
Since supp F x→ξ a (2) ⊂ supp F x→ξ a it follows that a (2) (x, D) satisfies (2.27), when a(x, η) does so. Hence the preceding argument also applies to a (2) (x, D), so that it is continuous on S ′ with its adjoint being of type 1, 1.
Moreover, the series ∑ ∞ k=0 a k−h (x, D)u k in ( It should be emphasized that duality methods and pointwise estimates contribute in two different ways in Theorem 6.1: once the symbol a (1) (x, η) has been introduced, continuity on S ′ (R n ) of the associated type 1, 1-operator a (1) (x, D) is obtained by duality through Proposition 4.2. However, the pointwise estimates in Section 3 yield (vanishing of the remainder terms, hence) the identification of a (1) (x, D)u with the series in (5.8). Furthermore, the pointwise estimates also give an explicit proof of the fact that a (1) (x, D) is defined on the entire S ′ (R n ), for the right-hand side of (5.8) does not depend on the modulation function Ψ. Similar remarks apply to a (3) (x, D). Thus duality methods and pointwise estimates together lead to a deeper analysis of type 1, 1-operators. Remark 6.2. Theorem 6.1 and its proof generalise a result of Coifman and Meyer [37,Ch. 15] in three ways. They stated Lemma A.1 for θ 0 = θ 1 = 1 and derived a corresponding fact for paramultiplication, though only with a treatment of the first and third term.
Going back to the given a(x, D), one derives from Theorem 6.1 and (5.7) the following Theorem 6.3. When a ∈ S d 1,1 (R n × R n ) fulfils the twisted diagonal condition (2.27), then the associated type 1, 1-operator a(x, D) defined by vanishing frequency modulation is an everywhere defined continuous linear map
To extend the discussion to general a(x, D) without vanishing along the twisted diagonal, note that Theorem 6.1 at least shows that a (1) (x, D)u and a (3) (x, D)u are always defined and that (5.8) and (5.10) are operator identities.
It remains to justify the operator notation a (2) (x, D) in (5.9) and to give its precise relation to a(x, D) itself. The point of departure is of course the symbol splitting (5.29); the corresponding type 1, 1-operators are still denoted by a ( j) (x, D). However, to avoid ambiguity the series in (5.8)-(5.10) will now be temporarily written as A ( j) ψ u, whence (5.7) amounts to Here the left-hand side exists if and only if the series A (2) ψ u converges, as A (1) ψ u, A (3) ψ u always converge by Theorem 6.1. This strongly indicates that (6.5) below is true. In fact, this main result of the analysis is obtained by frequency modulation: Theorem 6.4. When a(x, η) ∈ S d 1,1 (R n × R n ) and a (2) (x, η) denotes the type 1, 1-symbol in (5.28), derived from the paradifferential decomposition (5.7), then D(a (2) D(a(x, D)) (6.5) and u ∈ S ′ (R n ) belongs to these domains if and only if the series (5.9), or equivalently (5.11), converges in D ′ (R n ) -in which case (also) formulae (5.9), (5.11) are operator identities.
Proof. A variant of (6.4) follows at once from (5.29), using a second modulation function Ψ and the brief notation from Remark 5.1, OP(Ψ(2 −m D x )a (l) (x, η)Ψ(2 −m η))u. (6.6) Here the terms with l = 1 and l = 3 always have Ψ-independent limits for m → ∞ according to Theorem 6.1, so it is clear from the definition by vanishing frequency modulation that u ∈ D (2) (a(x, D)) is equivalent to u ∈ D(a(x, D)), hence to convergence of A (2) Ψ u, cf. (6.4) ff. As for the last claim, whenever u ∈ D(a(x, D)), then passage to the limit (m → ∞) in the above equation yields the following, when (5.35), (5.37) and (5.36) are applied, now with the remainders in the primed sums there denoted by R m u for brevity: Note that convergence of R (2) m follows from that of the other six terms; cf. Proposition 5.7. Compared to (6.4) this yields lim m R (2) m = 0, which via (5.37) gives that a (2) (x, D)u = A (2) ψ u. a(x, η) is in the self-adjoint subclass S d 1,1 , then it follows Theorem 4.6 that the domains in (6.5) equal S ′ . However, it is interesting to give an explicit proof that the domains in (6.5) equal S ′ whenever a ∈ S d 1,1 . This can be done in a natural way by extending the proof of Theorem 6.4, where the special estimates in (2.33) enter the convergence proof for a (2) (x, D)u directly, because they are rather close to the symbol factors from the factorisation inequalities in Section 3. The full generality with θ 0 < θ 1 in the corona criterion Lemma A.1 is also needed now. Proof. The continuity on S ′ is assured by Theorem 4.6. For the convergence of the series in the paradifferential splitting, it is convenient to write, in the notation of (2.33) ff, a(x, η) = (a(x, η) − a χ,1 (x, η)) + a χ,1 (x, η), (6.8) where a − a χ,1 satisfies (2.27) for B = 1, so that Theorem 6.3 applies to it. As a χ,1 is in S d 1,1 like a and a − a χ,1 (the latter by Proposition 4.2), one may reduce to the case in which ∧ a(x, η) = 0 =⇒ max(1, |ξ + η|) ≤ |η|. (6.9) Continuing under this assumption, it is according to Theorems 6.1 and 6.4 enough to show for all u ∈ S ′ that there is convergence of the two contributions to a (2)
When k fulfils 2 k(1−θ ) > 2 h+2 /r, so that the last right-hand side is > 1, these inequalities give This shows that the corona condition (A.1) in Lemma A.1 is fulfilled for θ 0 = 1 − θ = 1/2 and θ 1 = 1, and the growth condition (A.2) is easily checked since both a k,χ,ε (x, D)v k and a k (x, D)v k are estimated by 2 k(N+d + ) (1 + |x|) N+d + , as can be seen from (6.17) and Proposition 3.6, respectively. Hence ∑ b k (x, D)v k converges rapidly. For the series ∑ ∞ k=0 | (a k − a k−h )(x, D)u k , φ | it is not complicated to modify the above. Indeed, the pointwise estimates of the v * k are easily carried over to u * k , for R2 k was used as the outer spectral radius of v k ; and r2 k−h−1 may also be used as the inner spectral radius of u k . In addition the symbol a k − a k−h can be treated by replacing ϕ(2 −k ξ ) by ψ(2 −k ξ ) − ψ(2 h−k ξ ) in (6.11) ff., for the use of Minkowski's inequality will now give the factor |ψ − ψ(2 h ·)| dy in the constant. For the remainder b k (x, D)u k = (a k − a k−h )(x, D)u k − (a k − a k−h ) χ,ε (x, D)u k (6.22) one can apply the treatment of b k (x, D)v k verbatim.
Remark 6.6. The analysis in Theorem 6.5 is also exploited in the L p -theory of type 1, 1-operators in [29]. Indeed, the main ideas of the above proof was used in [29,Sect. 5.3] to derive certain continuity results in the Lizorkin-Triebel scale F s p,q (R n ) for p < 1, which (except for a small loss of smoothness) generalise results of Hounie and dos Santos Kapp [21].

FINAL REMARKS
In view of the satisfying results on type 1, 1-operators in S ′ (R n ), cf. Section 6, and the continuity results in the scales H s p , C s * , F s p,q and B s p,q presented in [29], their somewhat unusual definition by vanishing frequency modulation in Definition 2.1 should be well motivated.
As an open problem, it remains to characterise the type 1, 1-operators a(x, D) that are everywhere defined and continuous on S ′ (R n ). For this it was shown above to be sufficient that a(x, η) is in S d 1,1 (R n × R n ), and it could of course be conjectured that this is necessary as well. Similarly, since the works of Bourdaud and Hörmander, cf. [6, Ch. IV], [7], [18,19] and also [20], it has remained an open problem to determine B(L 2 (R n )) ∩ OP(S 0 1,1 ). (7.1) Indeed, this set was shown by Bourdaud to contain the self-adjoint subclass OP( S 0 1,1 ), and this sufficient condition has led some authors to a few unfortunate statements, for example that lack of L 2 -boundedness for OP(S 0 1,1 ) is "attributable to the lack of self adjointness". But self-adjointness is not necessary, since already Bourdaud, by modification of Ching's operator (2.13), gave an example [7, p. 1069] of an operator σ (x, D) in B(L 2 ) OP(S 0 1,1 \ S 0 1,1 ); that is, this σ (x, D) * is not of type 1, 1.

APPENDIX A. DYADIC CORONA CRITERIA
Convergence of a series ∑ ∞ j=0 u j of temperate distributions follows if the u j both fulfil a growth condition and have their spectra in suitable dyadic coronas. This is a special case of Lemma A.1, which for θ 0 = θ 1 = 1 was given by Coifman and Meyer [37,Ch. 15] without arguments.
Extending the proof given in [30], the refined version in Lemma A.1 allows the inner and outer radii of the spectra to grow at different exponential rates θ 0 < θ 1 , even though the number of overlapping spectra increases with j. This is crucial for Theorem 6.5, so a full proof is given. Lemma A.1. 1 • Let (u j ) j∈N 0 be a sequence in S ′ (R n ) fulfilling that there exist A > 1 and θ 1 ≥ θ 0 > 0 such that supp and that for suitable constants C ≥ 0, N ≥ 0, Then ∑ ∞ j=0 u j converges rapidly in S ′ (R n ) to a distribution u, for which ∧ u is of order N . 2 • For every u ∈ S ′ (R n ) both (A.1) and (A.2) are fulfilled for θ 0 = θ 1 = 1 by the functions u 0 = Φ 0 (D)u and u j = Φ(2 − j D)u when Φ 0 , Φ ∈ C ∞ 0 (R n ) and 0 / ∈ supp Φ. In particular this is the case for a Littlewood-Paley decomposition 1 = Φ 0 + ∑ ∞ j=1 Φ(2 − j ξ ).
Proof. In 2 • it is clear that Φ is supported in a corona, say { ξ | 1 A ≤ |ξ | ≤ A } for a large A > 0; hence (A.1). (A.2) follows from the proof of Lemma 3.1.
Remark A.2. The above proof yields that the conjunction of (A.1) and (A.2) implies u j , ϕ = O(2 − jN ) for all N > 0; hence there is rapid convergence of u = ∑ ∞ j=0 u j in S ′ in the sense that u − ∑ j<k u j , ϕ = ∑ j≥k u j , ϕ = O(2 −kN ) for N > 0, ϕ ∈ S .

APPENDIX B. THE SPECTRAL SUPPORT RULE
To control the spectrum of x → a(x, D)u, i.e. the support of ξ → F a(x, D)u, there is a simple rule which is recalled here for the reader's convenience.
Writing F a(x, D)F −1 ( ∧ u) instead, the question is clearly how the support of F u is changed by the conjugated operator F a(x, D)F −1 . In terms of its distribution kernel K (ξ , η), cf. (1.15), one should expect the spectrum of a(x, D)u to be contained in For supp F u ⋐ R n this was proved in [24]; but in general the closure Ξ should be used instead: Theorem B.1. Let a ∈ S ∞ 1,1 (R n × R n ) and suppose u ∈ D(a(x, D)) has the property that (2.5) holds in the topology of S ′ (R n ) for some ψ ∈ C ∞ 0 (R n ) equal to 1 around the origin. Then supp F (a(x, D)u) ⊂ Ξ, (B.2) Ξ = ξ + η (ξ , η) ∈ supp F x→ξ a, η ∈ supp F u .

(B.3)
When u ∈ F −1 E ′ (R n ) the S ′ -convergence holds automatically and Ξ is closed for such u.
The reader is referred to [26] for the deduction of this from the kernel formula. Note that whilst (1.15) yields (B.3), it suffices for (B.2) to take any v ∈ C ∞ 0 (R n ) with support disjoint from Ξ and verify that Here the middle expression makes sense as (v ⊗ ∧ u)K , 1 , as noted in [26], using the remarks to [17,Def. 3.1.1]. However, the first equality sign is in general not trivial to justify: the limit in Definition 2.1 is decisive for this.