The principal transmission condition

The paper treats pseudodifferential operators $P=Op(p(\xi ))$ with homogeneous complex symbol $p(\xi )$ of order $2a>0$, generalizing the fractional Laplacian $(-\Delta )^a$ but lacking its symmetries, and taken to act on the halfspace $R^n_+$. The operators are seen to satisfy a principal $\mu $-transmission condition relative to $R^n_+$, but generally not the full $\mu $-transmission condition satisfied by $(-\Delta )^a$ and related operators (with $\mu =a$). However, $P$ acts well on the so-called $\mu $-transmission spaces over $R^n_+$ (defined in earlier works), and when $P$ moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for $P$, leading to regularity results with a factor $x_n^\mu $ (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over $R^n_+$ for $P$ acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, for this new class of operators. Since the principal $\mu $-transmission condition has weaker requirements than the full $\mu $-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.


Introduction.
Boundary value problems for fractional-order pseudodifferential operators P , in particular where P is a generalization of the fractional Laplacian (−∆) a (0 < a < 1), have currently received much interest in applications, such as in financial theory and probability (but also in mathematical physics and differential geometry), and many methods have been used, most often probabilistic or potential-theoretic methods.
The author has studied such problems by pseudodifferential methods in [G15-G21], under the assumption that the operators satisfy a µ-transmission condition at the boundary of the domain Ω ⊂ R n , which allows to show regularity results for solutions of the Dirichlet problem in elliptic cases, to show integration by parts formulas, and much else.
In the present paper we consider translation-invariant pseudodifferential operators (ψdo's) P = Op(p(ξ)) of order 2a > 0 with homogeneous symbol p(ξ), which are only taken to satisfy the top-order equation in the µ-transmission condition (relative to the domain Ω = R n + ), we call this the principal µ-transmission condition.It is shown that they retain some of the features: The solution spaces for the homogeneous Dirichlet problem in the elliptic case equal the µ-transmission spaces from [G15] (in a setting of low-order Sobolev spaces), having a factor x µ n .The integration by parts formula holds (even when P is not elliptic): and compactly supported.We also treat nonhomogeneous local Dirichlet problems with Dirichlet trace γ 0 (u/x µ−1 n ), and show how the above formula implies a "halfways" Green's formula where one factor has nonzero Dirichlet trace.P can be of any positive order, and µ can be complex.
The results apply in particular to the operator L = Op(A(ξ) + iB(ξ)) with A real, positive and even in ξ, B real and odd in ξ, which satisfies the principal µ-transmission equation for a suitable real µ.Hereby we can compensate for an error made in the recent publication [G21] (see also [G22]), where it was overlooked that L may not satisfy the full µ-transmission condition when B = 0 (it does so for B = 0).The general L are now covered by the present work.They were treated earlier by Dipierro, Ros-Oton, Serrra and Valdinoci [DRSV21] under some hypotheses on a and µ; they come up in applications as infinitesimal generators of α-stable n-dimensional Lévy processes, see [DRSV21].(The calculations in [G21] are valid when applied to operators satisfying the full µ-transmission condition.) The study of x-independent ψdo's P on the half-space R n + serves as a model case for x-dependent operators on domains Ω ⊂ R n with curved boundary, and can be expected to be a useful ingredient in the general treatment, as carried out for the operator L in [DRSV21].
Plan of the paper: In Section 2 we give an overview of the aims and results of the paper with only few technicalities.Section 3 introduces the principal transmission condition in detail for homogeneous ψdo symbols.In Section 4, the Wiener-Hopf method is applied to derive basic decomposition and factorization formulas for such symbols.This is used in Section 5 to establish mapping properties for the operators, and regularity properties for solutions of the homogeneous Dirichlet problem in strongly elliptic cases; here µ-transmission spaces (known from [G15]) defined in an L 2 -framework play an important role.Section 6 gives the proof of the above-mentioned integration by parts formula on R n + .Section 7 treats nonhomogeneous local Dirichlet conditions, and a halfways Green's formula is established.

Presentation of the main results.
The study is concerned with the so-called model case, where the pseudodifferential operators have x-independent symbols, hence act as simple multiplication operators in the Fourier transformed space (this frees us from using the deeper composition rules needed for x-dependent symbols), and the considered open subset Ω of R n is simplest possible, namely We assume n ≥ 2 and denote x = (x 1 , . . ., x n ) = (x ′ , x n ), x ′ = (x 1 , . . ., x n−1 ).Recall the formulas for the Fourier transform F and the operator P = Op(p(ξ)): (2.1) We work in L 2 (R n ) and L 2 (R n + ) and their derived L 2 -Sobolev spaces (the reader is urged to consult (5.1) below for notation).On L 2 (R n ), the Plancherel theorem makes norm estimates of operators easy.(There is more on Fourier transforms and distribution theory e.g. in [G09].)The model case serves both as a simplified special case, and as a proof ingredient for more general cases of domains with curved boundaries, and possibly x-dependent symbols.
A typical example is the squareroot Laplacian with drift: (2.3) this is important in regularity discussions.Some results are obtained without the ellipticity hypothesis; as an example we can take the operator L 2 with symbol whose real part is zero e.g. when ξ = (1, −1, 0, . . ., 0).The operators are well-defined on the Sobolev spaces over R n : When p is homogeneous of degree m ≥ 0, there is an inequality (we say that p is of order m); then But for these pseudodifferential operators it is not obvious how to define them relative to the subset R n + , since they are not defined pointwise like differential operators, but by integrals (they are nonlocal).The convention is here to let them act on suitable linear subsets of L 2 (R n + ), where we identify L 2 (R n + ) with the set of u ∈ L 2 (R n ) that are zero on R n − , i.e., have their support supp u ⊂ R n + .(The support supp u of a function or distribution u is the complement of the largest open set where u = 0.The operator that extends functions on R n + by zero on R n − is denoted e + .)Then we apply P and restrict to R n + afterwards; this is the operator r + P .(r + stands for restriction to R n + .)Aiming for the integration by parts formula mentioned in the start, we have to clarify for which functions u, u ′ the integrals make sense.It can be expected from earlier studies ([RS14], [G16], [DRSV21]) that the integral will be meaningful for solutions of the so-called homogeneous Dirichlet problem on R n + , namely the problem (2.7) (where the latter condition can also be written supp u ⊂ R n + ).This raises the question of where r + P lands; which f can be prescribed?Or, if f is given in certain space, where should u lie in order to hit the space where f lies?
Altogether, we address the following three questions on P : (1) Forward mapping properties.From which spaces does r + P map into an H s -space for f ?(2) Regularity properties.If u solves (2.7) with f in an H s -space for a high s, will u then belong to a space with a similar high regularity?(3) Integration by parts formula for functions in spaces where r + P is well-defined.
It turns out that the answers to all three points depend profoundly on the introduction of so-called µ-transmission spaces.To explain their importance, we turn for a moment to the fractional Laplacian which has a well-established treatment: For the case of (−∆) a , 0 < a < 1, it was shown in [G15] that the following space is relevant: Here E ′ (R n ) is the space of distributions with compact support, so the intersection with this space means that we consider functions in E a that are zero outside a compact set.
For Sobolev spaces, it was found in [G15] that the good space for u is the so-called a-transmission space H a(t) (R n + ); here ).The definition of the space H a(t) (R n + ) is recalled below in (2.15) and in more detail in Section 5.3; let us for the moment just mention that it is the sum of the space Ḣt (R n + ) and a certain subspace of x a n H t−a (R n + ).This also holds when a is replaced by a more general µ.
For (−∆) a , the a-transmission spaces provide the right answers to question (1), and they are likewise right for question (2) (both facts established in [G15]), and there are integration by parts formulas for (−∆) a applied to elements of these spaces, [G16, G18].
The key to the proofs is the so-called a-transmission condition that (−∆) a satisfies; it is an infinite list of equations for p(ξ) and its derivatives, linking the values on the interior normal to R n + with the values on the exterior normal.We formulate it below with a replaced by a general µ.
Definition 2.1.Let µ ∈ C, and let p(ξ) be homogeneous of degree m.Denote the interior resp.exterior normal to the boundary of Note that µ is determined from p in (2.11) up to addition of an integer, when p(0, 1) = 0.The operators considered on smooth domains Ω in [G15] were assumed to satisfy (2.12) (for the top-order term p 0 in the symbol) at all boundary points x 0 ∈ ∂Ω, with (0, 1) replaced by the interior normal ν at x 0 , and (0, −1) replaced by −ν.The lower-order terms p j in the symbol, homogeneous of degree m − j, should then satisfy analogous rules with m − j instead of m.
The principal µ-transmission condition (2.11) is of course much less demanding than the full µ-transmission condition (2.12).What we show in the present paper is that when (2.11) holds, the µ-transmission spaces are still relevant, and provide the appropriate answers to both questions (1) and (2), however just for t (the regularity parameter) in a limited range.This range is large enough that integration by parts formulas can be established, answering (3).
By simple geometric considerations one finds: Proposition 2.2. 1 • When p(ξ) is homogeneous of degree m, there is a µ ∈ C, uniquely determined modulo Z if p(0, 1) = 0, such that (2.11) holds. 2 • If moreover, p is strongly elliptic (2.4) and m = 2a > 0, µ can be chosen uniquely to satisfy µ = a + δ with | Re δ| < 1 2 .This is shown in Section 3. From here on we work under two slightly different assumptions.The symbol p(ξ) is in both cases taken homogeneous of degree m = 2a > 0 and C 1 for ξ = 0. We pose Assumption 3.1 requiring that p is strongly elliptic and µ is chosen as in Proposition 2.2 2 • .We pose Assumption 3.2 just requiring that µ is defined according to Proposition 2.2 1 • .In all cases we write µ = a + δ, and define Here L 1 (0, 1) = 1 + ib n and L 1 (0, −1) = 1 − ib n .The angle θ in C = R 2 between the positive real axis and 1
When b n = 0, hence δ = 0, neither of these symbols satisfy the full µ-transmission condition Definition 2.1 2 • , since second derivatives remove the (ib b b • ξ)-term so that the resulting symbol is even (with µ = a + δ replaced by µ = a).
Our answer to (1) is now the following (achieved in Section 5.4): Theorem 2.4.Let P satisfy Assumption 3.2.For Re µ − 1 2 < t < Re µ + 3 2 , r + P defines a continuous linear mapping (2.13) It is important to note that r + P then also makes good sense on subsets of , any ε > 0, by (2.13).When Re δ > − 1 2 (always true under Assumption 3.1), this is assured to be contained in H 1−a (R n + ).Our answer to (2) is (cf.Section 5.4): Theorem 2.5.Let P satisfy Assumption 3.1.Then P = P + P ′ , where P ′ is of order 2a − 1, and r + P is a bijection from In other words, there is unique solvability of (2.7) with P replaced by P , in the mentioned spaces.
For r + P itself, there holds the regularity property: The last statement shows a lifting of the regularity of u in the elliptic case, namely if it solves (2.7) lying in a low-order space Ḣσ (R n + ), then it is in the best possible µ-transmission space according to Theorem 2.4, mapping into the given range space H t−2a (R n + ).In other words, the domain of the homogeneous Dirichlet problem with range in H The strategy for both theorems is, briefly expressed, as follows: The first step is to replace P = Op(p(ξ)) by P = Op( p(ξ)), where p(ξ) is better controlled at ξ ′ = 0 and p ′ (ξ) = p(ξ) − p(ξ) is O(|ξ| 2a−1 ) for |ξ| → ∞.The second step is to reduce P to order 0 by composition with "plus/minus order-reducing operators" Ξ t ± = Op(( ξ ′ ± iξ n ) t ) ((3.11), (5.2)) geared to the value µ (recall µ ′ = 2a − µ): Then the homogeneous symbol q associated with Q satisfies the principal 0-transmission condition.The third step is to decompose Q into a sum (when Assumption 3.2 holds) or a product (when Assumption 3.1 holds) of operators whose action relative to the usual Sobolev spaces Ḣs (R n + ) and H s (R n + ) can be well understood, so that we can show forward mapping properties and (in the strongly elliptic case) bijectiveness properties for Q.The fourth step is to carry this over to forward mapping properties and (in the strongly elliptic case) bijectiveness properties for P .The fifth and last step is to take P ′ = P − P back into the picture and deduce the forward mapping resp.regularity properties for the original operator P .
It is the right-hand factor Ξ −µ + in (2.14) that is the reason why the µ-transmission spaces, defined by enter.Here e + H t−Re µ (R n + ) has a jump at x n = 0 when t > Re µ + 1 2 , and then the coefficient x µ n appears.The analysis of Q is based on a Wiener-Hopf technique (cf.Section 4) explained in Eskin's book [E81], instead of the involvement of the extensive Boutet de Monvel calculus used in [G15].
An interesting feature of the results is that the µ-transmission spaces have a universal role, depending only on µ and not on the exact form of P .
Finally, we answer (3) by showing an integration by parts formula, based just on Assumption 3.2.
Theorem 2.6.Let P satisfy Assumption 3.2, and assume moreover that Re µ > −1, where ) are interpreted as dualities when needed.The basic step in the proof is the treatment of one order-reducing operator in Proposition 6.1, by an argument shown in detail in [G16, Th. 3.1,Rem. 3.2] and recalled in [G21, Th. 4.1].
In the proof of (2.16) in Section 6, the formula is first shown for the nicer operator P , and thereafter extended to P .(The formula (2.16) for (−∆) a in Ros-Oton and Serra [RS14, Th. 1.9] should have a minus sign on the boundary contribution; this has been corrected by Ros-Oton in the survey [R18, p. 350].) The theory will be carried further, to include "large" solutions of a nonhomogeneous local Dirichlet problem, and to show regularity results and a "halfways Green's formula", see Section 6, but we shall leave those aspects out of this preview.
The example and positive, and B(ξ) real and odd in ξ.There are more details below in (3.5)ff.(this stands for (3.5) and the near following text) and Examples 5.9, 6.5, 7.4.L was first studied in [DRSV21] (under certain restrictions on µ), and our results apply to it.Theorem 2.6 gives an alternative proof for the same integration by parts formula, established in [DRSV21,Prop. 1.4] by extensive real function-theoretic methods.
The result on the integral over R n + is combined in [DRSV21] with localization techniques to get an interesting result for curved domains, and it is our hope that the present results for more general strongly elliptic operators can be used in a similar way.

Analysis of homogeneous symbols.
Let p(ξ) be a complex function on R n that is homogeneous of degree m in ξ, and let ν ∈ R n be a unit vector.For a complex number µ, we shall say that p satisfies the principal µ-transmission condition in the direction ν, when When p(ν) = 0, we can rewrite (3.1) as where log is a complex logaritm.This determines the possible µ up to addition of an integer.
The (full) µ-transmission property defined in [G15] demands much more, namely that Besides assuming infinite differentiability, this is a stronger condition than (3.1) in particular because of the requirements it puts on derivatives of p transversal to ν.
To analyse this we observe that when a (sufficiently smooth) function and its derivative outside t = 0 is a function homogeneous of degree m − 1 satisfying In particular, if c 1 = 0, m = 0, In the case m = 0, f is constant for t > 0 and t < 0, and the derivative is zero there.
Thus, when p(ξ) is a (sufficiently smooth) function on R n \ {0} that is homogeneous of degree m = 0, and we consider it on a two-sided ray {tν | t ∈ R} where ν is a unit vector and p(ν) = 0, then So for example, when ν is the inward normal (0, 1 For p(ξ) satisfying (3.1), this means that when p(ν) = 0, it will also satisfy in view of (3.3).This argument can be repeated, showing that for all k ∈ N (possibly vanishing from a certain step on).On the other hand, we cannot infer that other derivatives ∂ α ξ of p have the property (3.2); an example will be given below.In general, µ takes different values for different ν.When Ω is a sufficiently smooth subset of R n with interior normal ν(x) at boundary points x ∈ ∂Ω, we say that p satisfies the principal µ-transmission condition at Ω if µ(x) is a function on ∂Ω such that (3.1) holds with this µ(x) at boundary points x ∈ ∂Ω.For Ω = R n + , the normal ν equals (0, 1) at all boundary points and µ is a constant; this is the situation considered in the present paper.
But the full µ-transmission condition need not hold.For example, the symbol and its derivatives satisfy the conditions in (3.2) for ν = (0, 1) with µ replaced by a.
The statement in [G21, Th. 3.1] that solutions of the homogeneous Dirichlet problem have a structure with the factor x µ n , was quoted from [G15] based on the full µ-transmision condition, and therefore applies to L = Op(L) when B = 0 (a case belonging to [G15]), but not in general when B = 0. Likewise, the integration by parts formulas for L derived in [G21] using details from the Boutet de Monvel calculus are justified when B = 0 or when other operators P satisfying the full µ-transmission condition are inserted, but not in general when B = 0. Fortunately, there are cruder methods that do lead to such results, on the basis of the principal µ-transmission condition alone, and that is what we show in this paper.
The treatment of L will be incorporated in a treatment of general strongly elliptic homogeneous symbols in the following.This requires that we allow complex values of µ.
Let P = Op(p(ξ)) be defined by (2.1) from a symbol p(ξ) that is C 1 for ξ = 0, homogeneous of order m = 2a > 0, and now also strongly elliptic (2.4).To fix the ideas, we shall consider the operator relative to the set R n + , with interior normal ν = (0, 1).Denote p(ξ)|ξ| −2a = p 1 (ξ); it is homogeneous of degree 0. Both p and p 1 take values in a closed subsector of {z ∈ C | Re ξ n > 0} ∪ {0}.For any ξ ′ ∈ R n−1 , one has for +1 and −1 respectively, lim With the logarithm log z defined to be positive for real z > 1, with a cut along the negative real axis, denote log p(0, ±1) = α ± ; here Re α ± = log |p(0, ±1)| and Im α ± is the argument of p(0, ±1).With this notation, p(0, −1)/p(0, 1) = e α − /e α + = e α − −α + , so (3.1) for m = 2a holds with ν = (0, 1) when this µ is the factorization index.These calculations were given in [G15,Sect. 3] (with m = 2a), and are in principle consistent with the determination of the factorization index by Eskin in [E81, Ex. 6.1] (which has different plus/minus conventions because of a different definition of the Fourier transform).Since p(ξ) takes values in {Re z > 0} for ξ = 0, both p(0, 1) and p(0, −1) lie there and the difference between their arguments is less than π, so | Im(α + − α − )/2π| < 1 2 ; in other words Note that δ is real in the case (3.5).We collect the information on P in the following description: Assumption 3.1.The operator P = Op(p(ξ)) is defined from a symbol p(ξ) that is C 1 for ξ = 0, homogeneous of order m = 2a > 0, and strongly elliptic (2.4).It satisfies the principal µ-transmission condition in the direction (0, 1): with µ equal to the factorization index µ = a + δ derived around (3.7), and In the book [E81], the case of constant-coefficient pseudodifferential operators considered on R n + is studied in § §4-17, and the calculations rely on the principal transmission condition up to and including §9.From §10 on, additional conditions on transversal derivatives are required (the symbol class D (0) α+iβ seems to correspond to our full 0-transmission condition, giving operators preserving smoothness up to the boundary).In the following, we draw on some of the points made in § §6-7 there.
For an operator A defined from a homogeneous symbol a(ξ), the behavior at zero can be problematic to deal with.In [E81, §7] there is introduced a technique that leads to a nicer operator, in the context of operators relative to R n + : One eliminates the singularity at ξ ′ = 0 by replacing the homogeneous symbol a(ξ ′ , ξ n ) by the corresponding operator denoted A. (In comparison with [E81] we have replaced the factor 1 + |ξ ′ | used there by hence is of lower order in a certain sense.Many results with Sobolev estimates are then shown primarily for the "hatted" version A = Op( a), and supplied afterwards with information on A ′ = Op(a ′ ).Indeed, we shall see that the results we are after for our operators P = Op(p), can be obtained in a manageable way for P = Op( p), and then extended to P by a supplementing analysis of P ′ .The important thing is that special properties with respect to ξ n , such as holomorphic extendability into C + or C − , are not disturbed when a is replaced by a.Some of the results that we shall show do not require ellipticity of P .We therefore introduce also a weaker assumption: Assumption 3.2.The operator P = Op(p(ξ)) is defined from a symbol p(ξ) that is C 1 for ξ = 0, homogeneous of order m = 2a > 0, and satisfies the principal µ-transmission condition in the direction (0, 1) with µ = a + δ for some δ ∈ C. Denote a − δ = µ ′ .
For the symbols p considered in the rest of the paper, we assume at least that Assumption 3.2 holds.As noted earlier, when P satisfies (3.1) for some µ, it also does so with µ replaced by µ + k, k ∈ Z.The precision in Assumption 3.1, that µ should equal the factorization index, is needed for elliptic solvability statements.
4.1 The sum decomposition.Since p(ξ) is only assumed to satisfy the principal µ-transmission condition, q(ξ) will in general only satisfy the principal 0-transmission condition, not the full one, so the techniques of the Boutet de Monvel calculus brought forward in [G15] are not available.Instead we go back to a more elementary application of the original Wiener-Hopf method Lemma 4.1.Suppose that b(ξ ′ , ξ n ) is homogeneous of degree 0 in ξ, is C 1 for ξ ′ = 0, and satisfies Then the function defined for τ < 0 by is holomorphic with respect to ξ n + iτ in C − , is homogeneous of degree 0, extends by continuity with respect to (ξ ′ , ξ n + iτ ) ∈ C − for |ξ| + |τ | > 0, τ ≤ 0, and satisfies the estimate There is an analogous statement for b − with C − replaced by C + .
The symbol q derived from p by (3.12)ff.satisfies where f is likewise homogeneous of degree 0, and has f (0, 1) = f (0, −1) = 0. We make two applications of Lemma 4.1.One is, under Assumption 3.2, to apply it directly to f to get a sum decomposition f = f + + f − where the terms extend holomorphically to C − resp.C + with respect to ξ n ; this will be convenient in establishing the forward mapping properties and integration by parts formula for the present operators.The other is, under Assumption 3.1, to apply the lemma to the function b(ξ) = log q(ξ) to get a sum decomposition of b and hence a factorization of q; this is used to show that P has appropriate solvability properties (the solutions exhibiting a singularity x µ n at the boundary).We show that f has the properties required for Lemma 4.1 as follows: To see that (4.2) is verified by f , note that the second inequality follows since ∂ j f is bounded on the unit sphere {|ξ| = 1} and homogeneous of degree −1.For the first inequality we have, when using the mean value theorem and the fact that We have obtained: Proposition 4.2.When p satisfies Assumption 3.2 and q is derived from p by (3.12)ff.then there is a sum decomposition of f = q − s 0 : where f + (ξ ′ , ξ n ) is holomorphic with respect to ξ n + iτ in C − , and continuous with respect to (ξ ′ , ξ n + iτ ) ∈ C − for |ξ| + |τ | > 0, τ ≤ 0, and satisfies estimates and f − has the analogous properties with C − replaced by C + .
For the corresponding hatted symbol, we then have q = s 0 + f + + f − , with f ± defined from f ± .They have similar holomorphy properties, and satisfy estimates as in (4.6) with |ξ ′ | replaced by ξ ′ .

The product decomposition.
In order to obtain a factorization for symbols satisfying Assumption 3.1, we shall study log q.By the strong ellipticity, q(ξ) = 0 for ξ = 0.Moreover, p(ξ)|ξ| Assume first that s 0 = 1; this can simply be obtained by dividing out q(0, 1).The function b(ξ) = log q(ξ) is homogeneous of degree 0 and has b(0, 1) = b(0, −1) = 0 and the appropriate continuity properties, and bounds on first derivatives, so the same proof as for f applies to b to give the decomposition b = b + + b − .Then we define q ± = exp(b ± ), they are homogeneous of degree 0. For example, Proposition 4.3.When p satisfies Assumption 3.1 and q is derived from p by (3.12)ff.and satisfies s 0 = 1, then there is a factorization of q: q(ξ) = q − (ξ)q + (ξ), where q + (ξ ′ , ξ n ) is holomorphic with respect to ξ n + iτ in C − , and continuous with respect to (ξ and q − , g − = q − − 1 have the analogous properties with C − replaced by C + .The symbols are homogeneous of degree 0, and q + and q − are elliptic. For general s 0 , we apply the factorization to q 0 = s −1 0 q, so that q 0 = q − 0 q + 0 ; then q = q − q + with q − = s 0 q − 0 = s 0 (1 + g − ) and q + = q + 0 = 1 + g + .The ellipticity follows from the construction as exp(b ± ), or one can observe that the product q + q − = q is elliptic (i.e., nonzero for ξ = 0).
The notation with upper index ± is chosen here to avoid confusion with the lower + used later to indicate truncation, P + = r + P e + .
Turning to the corresponding hatted symbols, we have obtained q = q − q + , with q ± , g ± defined from q ± , g ± , respectively.They have similar holomorphy properties, the q ± are elliptic, and the g ± satisfy estimates as in (4.7) with |ξ ′ | replaced by ξ ′ : 5. Mapping properties and the homogeneous Dirichlet problem.

Some function spaces.
First recall some terminology: E ′ (R n ) is the space of distributions on R n with compact support, S(R n ) is the Schwartz space of C ∞ -functions f on R n such that x β D α f is bounded for all α, β, and S ′ (R n ) is its dual space of temperate distributions.ξ stands for (1+|ξ| 2 ) 1 2 .We denote by r + the operator restricting distributions on R n to distributions on R n + , and by e + the operator extending functions on R n + by zero on The following notation for L 2 -Sobolev spaces will be used, for s ∈ R: (5.1) , the supported space, as in our earlier papers on fractional-order operators.An elaborate presentation of L pbased spaces was given in [G15].(The notation with dots and overlines stems from Hörmander [H85, App.B.2] and is practical in formulas where both types of spaces occur.There are other notations without the overline, and where the dot is replaced by a ring or twiddle.) Here H s (R n + ) identifies with the dual space of Ḣ−s (R n + ) for all s ∈ R (the duality extending the L 2 (R n + ) scalar product).When |s| < 1 2 , there is an identification of Ḣs (R n + ) with H s (R n + ) (more precisely with e + H s (R n + )).The trace operator γ 0 : u → lim x n →0+ u(x ′ , x n ) extends to a continuous mapping γ 0 : H s (R n + ) → H s− 1 2 (R n−1 ) for s > 1 2 .The order-reducing operators Ξ t ± are defined for t ∈ C by Ξ t ± = Op(χ t ± ), where χ t ± = ( ξ ′ ± iξ n ) t , cf. (3.11).These operators have the homeomorphism properties: ) is defined successively as the linear hull of first-order derivatives of elements of E µ+1 (R n + ) when Re µ ≤ −1 (then distributions supported in the boundary can occur).The spaces were introduced in Hörmander's unpublished lecture notes [H66] and are presented in [G15] (and with a different notation in [H85,Sect. 18.2]), and they satisfy for all µ (cf.[G15,Props. 1.7,4.1]): (5.4) A sharper statement follows from [G21, Lemma 6.1] (when Re µ > −1): (5.5) 5.2 Mapping properties of the zero-order operator Q in Sobolev spaces.
Let P satisfy Assumption 3.1, and consider Q ± = Op( q ± ), defined from the symbols q ± (ξ) introduced in Proposition 4.3.Since q ± are bounded symbols with bounded inverses, and extend holomorphically in the latter follows since r + Q − e + is the adjoint of Op( q − ) over R n + , where Op( q − ) defines homeomorphisms in Ḣs (R n + ) (since q − has similar properties as q + ).The inverses 2 , it follows that we also have for |s| < 1 2 : If q satisfies the full 0-transmission condition, we are in the case studied in [G15], and the bijectiveness in H s (R n + ) can be lifted to all higher s by use of elements of the Boutet de Monvel calculus, as accounted for in the proof of [G15,Th. 4.4].The symbol q presently considered is only known to satisfy the principal 0-transmission condition (and possibly a few more identities).We shall here show that a lifting is possible in general up to s < 3 2 .Proposition 5.1.Let P satisfy Assumption 3.1, and consider Q + = Op( q + ) derived from it in Section 3.
For any and the same holds for the operator (( Q + ) −1 ) + defined from its inverse ( Q + ) −1 .In fact, (5.7) is a homeomorphism, and the inverse of Q + + is (( Q + ) −1 ) + .Proof.We already have the mapping property (5.7) for |s| < 1 2 , because q + is a bounded symbol, and e + H s (R n + ) identifies with Ḣs (R n + ) then.Now let s = 3 2 − ε for a small ε > 0.Here we need to show that when u ∈ H ) for j = 1, . . ., n.For j < n, this follows simply because ∂ j can be commuted through r + , Q + and e + so that we can use that and therefore, since Q + = I + G + where G + = Op( g + (ξ)) from Proposition 4.3, In the restriction to R n + , r + I(γ 0 u ⊗ δ(x n )) drops out, so we are left with Here K g + is a potential operator (in the terminology of Eskin [E81] and , generalizing the concept of Poisson operator of Boutet de Monvel [B66,B71]), which acts as follows: . Thus Altogether, this shows the desired mapping property for s = 3 2 − ε, and the property for general 1 2 ≤ s < 3 2 follows by interpolation with the case s = 0.The mapping property (5.7) holds for the inverse ( Q + ) −1 , since its symbol (q + ) −1 equals 1 + k≥1 (−b + ) k with essentially the same structure.
The identity (( , and extends by continuity to H s (R n + ) for − 1 2 < s < 0. When P merely satisfies Assumption 3.2, we can still show a useful forward mapping property of Q, based on the decomposition in Proposition 4.2.
Proposition 5.2.Let P satisfy Assumption 3.2, and consider Q and F ± = Op( f ± ) derived from it in Section 3.
The operator F +,+ = r + F + e + is continuous (5.9) The operator Proof.Since F + has bounded symbol, it maps Ḣs (R n + ) into H s (R n ) for all s, so for |s| < 1 2 , (5.9) follows since Ḣs (R n + ) = e + H s (R n + ) then.For 1 2 < s < 3 2 , we proceed as in the proof of Proposition 5.1, using that where K f + satisfies similar estimates as K g + by Proposition 4.2.
For r + F − e + , the statement follows since it is on R n + the adjoint of Op( f − ), which preserves support in R n + and therefore maps Ḣs (R n + ) into itself for all s ∈ R. For Q, the statement now follows since it equals s 0 + F − + F + .This is as far as we get by applying Lemma 4.1 to f .To obtain the mapping property for higher s would require a control over the potential operators for j ≥ 1 as well.At any rate, the property shown in Proposition 5.2 will be sufficient for the integration by parts formulas we are aiming for.
In the elliptic case, we conclude from Proposition 5.1 for the operator Q: Corollary 5.3.Let P satisfy Assumption 3.1, and consider the operators Q, Q + , Q − with symbols q, q + , q − derived from it in Section 3. The operator

mapping continuously and bijectively
(5.11) and the inverse (continuous in the opposite direction) equals (5.12) (r Proof.We have for u ∈ Ḣs (R since r − Q + e + u = 0; this identity is also valid on the subspaces H s (R n + ) with s ≥ 1 2 .Combining the homeomorphism property of r + Q + e + shown in Proposition 5.1 with the known homeomorphism property of r + Q − e + on H s (R n + )-spaces (cf.(5.6)), we get (5.11).The inverse is pinned down by using that r + Q − e + has inverse r + ( Q − ) −1 e + on H s (R n + ) for all s, and r 5.3 Mapping properties of P using µ-transmission spaces.Now turn the attention to P , which is related to Q by (5.13) cf. (3.12)-(3.13).
We shall describe the solutions of the homogeneous Dirichlet problem (in the strongly elliptic case) (5.14) with f given in a space H s (R n + ), and u assumed a priori to lie in a space Ḣσ (Ω) for low σ, e.g. with σ = a.
First we observe for Ξ since, as accounted for in [G15,Rem. 1.1,(1.13)],the action of ) Composing the equation in (5.14) with Ξ −µ ′ −,+ to the left, we can therefore write it as (5.17) Next, we shall also replace u.Because of the right-hand factor Ξ −µ + in the expression for Q in (5.13), we need to introduce the µ-transmission spaces (5.18) defined in [G15]; they are Hilbert spaces.(For t ≤ Re µ − 1 2 , the convention is to take ), but this is rarely used.)The following properties were shown in [G15]: where Rule 1 • is shown in [G15,Prop. 1.7].Rule 2 • , shown in [G15, (1.26)], holds because of the mapping property (5.2) for Ξ −µ + and the identification of e ,Th. 5.1,Cor. 5.3,Th. 5.4]; it deals with a higher t, where e + H t−Re µ (R n + ) has a jump at x n = 0, and the coefficient x µ n appears.Let us just mention the key formula ) is connected with the factor x µ n .Besides in [G15,Sect. 5], explicit calculations are carried out e.g. in [G19, Lemma 3.3] (and [G14, Appendix]).
Here Re µ > − 1 2 since a > 0 and | Re δ| < 1 2 , so the rules in Theorem 5.4 3 • apply.Proof.In view of (5.15) and (5.16), and the mapping property of Q established in Corollary 5.3, r + P has the forward mapping property in (5.21).
To solve (5.14), let σ = Re µ − 1 2 + ε for a small ε,  ,Th. 7.3] for showing solvability in a higher-order Sobolev space, say with 1 2 < t−Re µ < 3 2 , f given in H t−2a (R n + ), is to supplement P with a potential operator K P constructed from P such that the solutions are of the form u = u + + K P ϕ with u + ∈ Ḣt (R n + ), ϕ a generalized trace derived from f .Our aim is to show that there is a universal description of the space of solutions u of (5.14) with right-hand side in H t−2a (R n + ), that depends only on µ, and applies to any P of the given type.The µtransmission spaces (5.18) serve this purpose.In [G15], they are shown to have this role for arbitrarily high t when the full µ-transmission condition holds.
One more important property of µ-transmision spaces is that the spaces with G15,Prop. 4.1] and [G21, Lemma 7.1]).Recall also (5.5), which makes the statement for e + x µ n S(R n + ) rather evident, since S(R n + ) is dense in H s (R n + ) for all s ∈ R. Hence r + P applies nicely to these spaces.
When P merely satisfies Assumption 3.2, we have at least the forward mapping part of (5.21): Theorem 5.7.Let P satisfy Assumption 3.2.For Re µ − 1 2 < t < Re µ + 3 2 , r + P maps continuously (5.23) Proof.This follows as in the preceding proof, now using the mapping property of r + Qe + established in Proposition 5.2.
5.4 Consequences for the given operator P .
The following consequences can be drawn for the original operator P : Theorem 5.8. 1 • Let P satisfy Assumption 3.2.Then P = P + P ′ , where P is defined by (3.9) and P ′ is of order 2a − 1.For Re µ − 1 2 < t < Re µ + 3 2 , r + P maps continuously (5.24) 2 • Let P satisfy Assumption 3.1.Then in the decomposition P = P + P ′ , r + P is invertible, as described in Theorem 5.5.
2 ) solve the homogeneous Dirichlet problem (5.25) Then u ∈ H µ(t) (R n + ).Proof.The original operator P equals Op(p(ξ)) with p(ξ) homogeneous on R n of degree 2a > 0; in particular it is continuous at 0. It is decomposed into where p ′ (ξ) is O( ξ 2a−1 ) for |ξ| ≥ 2 by (3.10) and continuous, hence This implies that P ′ = Op(p ′ ) maps H s (R n ) continuously into H s−2a+1 (R n ) for all s ∈ R, and hence (5.27) 1 • .The forward mapping property (5.23) holds for r + P by Theorem 5.7.To show that it holds for r ) to see that for small ε > 0, also matching the mapping property of P .Now (5.24) follows by adding the statements for P ′ and P .This shows 1 , so it follows from Theorem 5.5 that u ∈ H µ(t) (R n + ).This ends the proof of 2 • .Example 5.9.Theorem 5.8 applies to the operator L = Op(L(ξ)) described in (3.5)ff., showing that it maps H  [DRSV21].
In particular, the result provides a valid basis for applying r , mapping these spaces into ε>0 H Remark 5.10.The domain spaces H µ(t) (R n + ) entering in Theorem 5.8 can be precisely described: For |t − Re µ| < 1 2 , we already know from Theorem 5.4 2 n ). 6.The integration by parts formula.
6.1 An integration by parts formula for P .It will now be shown that the operators P satisfying merely the principal µ-transmission condition (Assumption 3.2) have an integration by parts formula over R n + , involving traces γ 0 (u/x µ n ).The study will cover the special operator L in Example 5.9 (regardless of whether a full µ-transmission condition might hold, as assumed in [G21]).It also covers more general strongly elliptic operators, and it covers operators that are not necessarily elliptic.
The basic observation is: The left-hand side is interpreted as in (6.2) below when Re µ ≤ 0.
The formula extends to w ∈ H Proof.This was proved in [G16, Th. 3.1] for µ = a > 0 (see also Remark 3.2 there with the elementary case a = 1), and in [G21, Th. 4.1] for real µ > − 1 2 , so the main task is to check that the larger range of complex µ is allowed.We write u ′ as ū′ for short.
Note that when Im µ By the mapping properties of Ξ ) and compactly supported, so the left-hand side of (6.1) makes sense as an integral of an L 1 -function.When µ is general, we observe that for any small ε > 0, so the integral I makes sense as the duality (6.2) .
For the whole analysis, it suffices that w ∈ H then, so that the duality in (6.2) is well-defined.
Note that the boundary condition in (7.1) is local.There is a different problem which is also regarded as a nonhomogeneous Dirichlet problem, namely to prescribe nonzero values of u in the exterior of Ω; it has somewhat different solution spaces (a link between this and the homogeneous Dirichlet problem is described in [G14]).

1 2 +
b b b • ∇, with symbol L 1 (ξ) = |ξ| + ib b b • ξ, where b b b = (b 1 , . . ., b n ) is a real vector.Here m = 1, a = 1 2 .It satisfies the condition for strong ellipticity, which is: (2.4) Re p(ξ) ≥ c 0 |ξ| m with c 0 > 0; 0, when the integrals have a sense.When b is suitably nice, b + is holomorphic in ξ n + iτ for τ < 0 and extends to a continuous function on C − (also denoted b + ), b − has these properties relative to C + , and b(ξ n ) = b + (ξ n ) + b − (ξ n ) on R. With the notation of spaces H, H ± introduced by Boutet de Monvel in [B71], denoted H, H ± in our subsequent works, the decomposition holds for b ∈ H with b ± ∈ H ± on R. Since we are presently dealing with functions with cruder properties, we shall instead apply a useful lemma shown in [E81, Lemma 6.1]: −2a = χ −a 0,− p(ξ)χ −a 0,+ takes values in a subsector of {z ∈ C | Re z > 0} and the multiplication by χ δ 0,− and χ −δ 0,+ gives the function q taking values in the sector {z ∈ C | | arg z| ≤ π( 1 2 + | Re δ|)} disjoint from the negative real axis.So the logarithm is well-defined with inverse exp.
Remark 5.6.This theorem differs from the strategy pursued in [E81], and gives a new insight.The technique in [E81 t−Re µ (R n + ), so u must lie in H µ(t) (R by (5.27), matching the mapping property of P .
n + • . 2 • .The first statement registers what we already know about r + P .Proof of the regularity statement: With u and f as defined there, denote σ = Re µ − 1 2 + ε; here ε > 0. Then r + P u = r + P u − r + P ′ u ∈ H and that solutions of the homogeneous Dirichlet problem with f ∈ H t−2a (R n + ) are in H µ(t) (R n + ) for these t.The appearance of the factor x µ n (cf.(5.19)) is consistent with the regularity shown in terms of Hölder spaces in