Regularity of Weakly Well-Posed Characteristic Boundary Value Problems

Abstract

We study the boundary value problem for a linear first-order partial differential system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a unique 𝐿²-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential/conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Provided that the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted conormal Sobolev spaces.

1. Introduction and Main Results

For 𝑛≥2, let ℝ^𝑛₊ denote the 𝑛-dimensional positive half-space ℝ^𝑛₊∶=𝑥=𝑥₁,𝑥^′,𝑥₁>0,𝑥^′∶=𝑥₂,…,𝑥_𝑛∈ℝ^𝑛−1. (1.1) The boundary of ℝ^𝑛₊ will be systematically identified with ℝ_𝑥^𝑛−1_′.

We are interested in the following stationary boundary value problem (BVP):𝛾+𝐿+ℬ𝑢=𝐹,inℝ^𝑛₊,𝑀𝑢=𝐺,onℝ^𝑛−1, (1.2)(1.3) where 𝐿 is the first-order linear partial differential operator𝐿=^𝑛_𝑗=1𝐴_𝑗𝜕_𝑗; (1.4) for each 𝑗=1,…,𝑛, the short notation 𝜕_𝑗∶=𝜕/𝜕𝑥_𝑗 is used.

The coefficients 𝐴_𝑗 (𝑗=1,…,𝑛) of 𝐿 are 𝑁×𝑁 matrix-valued functions in 𝐶^∞₍₀₎(ℝ^𝑛₊), the space of restrictions to ℝ^𝑛₊ of functions of 𝐶^∞₀(ℝ^𝑛). In (1.2), ℬ stands for a lower-order term whose form and nature will be specified later; compare to Theorem 1.1 and Section 3.2.

The source term 𝐹, as well as the unknown 𝑢, is a ℝ^𝑁-valued function of 𝑥; we may assume that they are both supported in the unitary positive half-ball 𝔹⁺∶={𝑥=(𝑥₁,𝑥^′)∶𝑥₁≥0,|𝑥|<1}.

The BVP has characteristic boundary of constant multiplicity 1≤𝑟<𝑁 in the following sense; the coefficient 𝐴₁ of the normal derivative in 𝐿 displays the blockwise structure𝐴₁(𝑥)=𝐴₁^𝐼,𝐼𝐴₁^{𝐼,𝐼𝐼}𝐴₁^{𝐼𝐼,𝐼}𝐴₁^{𝐼𝐼,𝐼𝐼}, (1.5) where 𝐴₁^𝐼,𝐼, 𝐴₁^{𝐼,𝐼𝐼}, 𝐴₁^{𝐼𝐼,𝐼}, 𝐴₁^{𝐼𝐼,𝐼𝐼} are, respectively, 𝑟×𝑟, 𝑟×(𝑁−𝑟), (𝑁−𝑟)×𝑟, (𝑁−𝑟)×(𝑁−𝑟) submatrices, such that𝐴^{𝐼,𝐼𝐼}_1∣𝑥1=0=0,𝐴^{𝐼𝐼,𝐼}_1∣𝑥1=0=0,𝐴^{𝐼𝐼,𝐼𝐼}_1∣𝑥1=0=0, (1.6) and 𝐴₁^𝐼,𝐼 is invertible over 𝔹⁺. According to the representation above, we split the unknown 𝑢 as 𝑢=(𝑢^𝐼,𝑢^𝐼𝐼); 𝑢^𝐼∈ℝ^𝑟 and 𝑢^𝐼𝐼∈ℝ^𝑁−𝑟 are said to be, respectively, the noncharacteristic and the characteristic components of 𝑢.

Concerning the boundary condition (1.3), 𝑀 is assumed to be the matrix (𝐼_𝑑0), where 𝐼_𝑑 denotes the identity matrix of order 𝑑, 0 is the zero matrix of size 𝑑×(𝑁−𝑑), and 𝑑 is a given positive integer ≤𝑟. The datum 𝐺 is a given ℝ^𝑑-valued function of 𝑥^′=(𝑥₂,…,𝑥_𝑛) and is supported in the unitary (𝑛−1)-dimensional ball 𝐵(0,1)∶={|𝑥^′|<1}.

Section 4 will be devoted to prove the following regularity result.

Theorem 1.1. Let 𝑘,𝑟,𝑠 be fixed nonnegative integer numbers such that 𝑠≥𝑟≥0, 𝑠>0, and suppose that the coefficients 𝐴_𝑗 (𝑗=1,…,𝑛) of the operator 𝐿 in (1.4) are given in 𝐶^∞₍₀₎(ℝ^𝑛₊) and 𝐴₁ fulfils conditions (1.5), (1.6). One assumes that for any ℎ>0 there exist some constants 𝐶₀=𝐶₀(ℎ)>0, 𝛾₀=𝛾₀(ℎ)≥1 such that for every 𝛾≥𝛾₀, for every operator ℬ=Op^𝛾_♯(𝑏), whose symbol 𝑏 belongs to Γ⁰ and satisfies |𝑏|_0,𝑘≤ℎ, and for all functions 𝐹∈ℋ^𝑠,𝑟_tan,𝛾(ℝ^𝑛₊), and 𝐺∈𝐻^𝑠_𝛾(ℝ^𝑛−1), the corresponding BVP (1.2)-(1.3) admits a unique solution 𝑢∈𝐿²(ℝ^𝑛₊), with 𝑢^𝐼_∣𝑥1=0∈𝐿²(ℝ^𝑛−1), and the following a priori energy estimate is satisfied: 𝛾‖𝑢‖²_{𝐿2ℝ𝑛+}+𝑢^𝐼_∣𝑥1=0²_{𝐿2ℝ𝑛−1}≤𝐶₀1𝛾^2𝑠+1‖𝐹‖²_ℋ^𝑠,𝑟_tan,𝛾ℝ^𝑛₊+1𝛾^2𝑠‖𝐺‖²_{𝐻𝑠𝛾ℝ𝑛−1}. (1.7) Then, for every 𝑚∈ℕ and any matrix-valued function 𝐵∈𝐶^∞₍₀₎(ℝ^𝑛₊), there exist some constants 𝐶_𝑚>0, 𝛾_𝑚 (with 𝛾_𝑚≥𝛾_𝑚−1≥1) such that if 𝛾≥𝛾_𝑚 and we are given arbitrary functions 𝐹∈ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾(ℝ^𝑛₊) and 𝐺∈𝐻_𝛾^𝑠+𝑚(ℝ^𝑛−1), the unique 𝐿²-solution 𝑢 of (1.2)-(1.3) (with data 𝐹, 𝐺, and lower-order term ℬ≡multiplicationby𝐵) belongs to 𝐻^𝑚_tan,𝛾(ℝ^𝑛₊), 𝑢^𝐼_∣𝑥1=0∈𝐻^𝑚_𝛾(ℝ^𝑛−1) and the a priori estimate of order 𝑚𝛾‖𝑢‖²_𝐻^𝑚_tan,𝛾ℝ^𝑛₊+𝑢^𝐼_∣𝑥1=0²_{𝐻𝑚𝛾ℝ𝑛−1}≤𝐶_𝑚1𝛾^2𝑠+1‖𝐹‖²_ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾ℝ^𝑛₊+1𝛾^2𝑠‖𝐺‖²_{𝐻𝛾𝑠+𝑚ℝ𝑛−1} (1.8) is satisfied.

The function spaces involved in the statement of Theorem 1.1, as well as the norms appearing in (1.7), (1.8), will be described in Section 2. The kind of lower-order operator ℬ involved in (1.2), that is allowed in Theorem 1.1, will be introduced in Section 3.2.

The BVP (1.2)-(1.3), with the aforedescribed structure, naturally arises from the study of a mixed evolution problem for a symmetric (or Friedrichs'symmetrizable) hyperbolic system, with characteristic boundary. The analysis of the regularity of the stationary problem, presented in this work, plays an important role for the study of the regularity of time-dependent hyperbolic problems, constituting the final goal of our investigation and developed in [1]. In view of the well-posedness property that problems (1.2)-(1.3) enjoy in the statement of Theorem 1.1, here we do not need to assume the hyperbolicity of the linear operator 𝐿 in (1.4); the only condition required on the structure of 𝐿 is that expressed by conditions (1.5) and (1.6). In the hyperbolic problems, the number 𝑑 of the scalar boundary conditions prescribed in (1.3) equals the number of positive eigenvalues of 𝐴₁ on {𝑥₁=0}∩𝔹⁺ (the so-called incoming characteristics of problem (1.2)-(1.3)), compare to [1]; this value 𝑑 remains constant along the boundary, as a combined effect of the hyperbolicity and the fact that 𝐴_{1∣{𝑥1=0}∩𝔹}⁺ has constant rank.

In [2], the regularity of weak solutions to the characteristic BVP (1.2)-(1.3) was studied, under the assumption that the problem is strongly 𝐿²-well posed, namely, that a unique 𝐿²-solution exists for arbitrarily given 𝐿²-data and the solution obeys an a priori energy inequality without loss of regularity with respect to the data; this means that the 𝐿²-norms of the interior and boundary values of the solution are measured by the 𝐿²-norms of the corresponding data 𝐹,𝐺.

The statement of Theorem 1.1 extends the result of [2, Theorem 15], to the case where only a weak well posedness property is assumed on the BVP (1.2)-(1.3). Here, the 𝐿²-solvability of (1.2)-(1.3) requires an additional regularity of the corresponding data 𝐹,𝐺; the integer 𝑠 represents the minimal amount of regularity, needed for data, in order to estimate the 𝐿²-norm of the solution 𝑢 in the interior of the domain, and its trace on the boundary, by the energy inequality (1.7).

Several problems, appearing in a variety of different physical contexts, such as fluid dynamics and magneto-hydrodynamics, exhibit a finite loss of derivatives with respect to the data, as considered by estimate (1.7) in the statement of Theorem 1.1. This is the case of some problems that do not satisfy the so-called uniform Kreiss-Lopatinskiĭ condition; see, for example, [3, 4]. For instance, when the Lopatinskiĭ determinant associated to the problem has a simple root in the hyperbolic region, estimating the 𝐿²-norm of the solution makes the loss of one tangential derivative with respect to the data; see, for example, [5, 6]. In [7], Coulombel and Guès show that, in this case, the loss of regularity of order one is optimal; they also prove that the weak well posedness, with the loss of one derivative, is independent of Lipschitzean lower-order terms, but not independent of bounded lower-order terms. This is a major difference with the strongly well-posed case, where there is no loss of derivatives and one can treat lower-order terms as source terms in the energy estimates. Also, this yields that the techniques we used in [2], for studying the regularity of strongly 𝐿²-well-posed BVPs, cannot be successfully performed in the case of weakly well-posed problems (see Section 4 for a better explanation).

The paper is organized as follows. In Section 2 we introduce the function spaces to be used in the following and the main related notations. In Section 3 we collect some technical tools, and the basic concerned results, that will be useful for the proof of the regularity of BVP (1.2)-(1.3), given in Section 4.

A final Appendix contains the proof of the most of the technical results used in Section 4.

2. Function Spaces

The purpose of this section is to introduce the main function spaces to be used in the following and collect their basic properties.

For 𝑗=1,2,…,𝑛, we set 𝑍₁∶=𝑥₁𝜕₁,𝑍_𝑗∶=𝜕_𝑗,for𝑗≥2. (2.1) Then, for every multi-index 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛, the conormal derivative 𝑍^𝛼 is defined by 𝑍^𝛼∶=𝑍^𝛼1₁⋯𝑍^𝛼𝑛_𝑛; (2.2) we also write 𝜕^𝛼=𝜕^𝛼1₁⋯𝜕^𝛼𝑛_𝑛 for the usual partial derivative corresponding to 𝛼.

For 𝛾≥1 and 𝑠∈ℝ, we set𝜆^𝑠,𝛾(𝜉)∶=𝛾²+𝜉^2𝑠/2 (2.3) and, in particular, 𝜆^𝑠,1∶=𝜆^𝑠.

The Sobolev space of order 𝑠∈ℝ in ℝ^𝑛 is defined to be the set of all tempered distributions 𝑢∈𝒮^′(ℝ^𝑛) such that 𝜆^𝑠𝑢∈𝐿²(ℝ^𝑛), being 𝑢 the Fourier transform of 𝑢; in particular, for 𝑠∈ℕ, the Sobolev space of order 𝑠 reduces to the set of all functions 𝑢∈𝐿²(ℝ^𝑛), for which 𝜕^𝛼𝑢∈𝐿²(ℝ^𝑛) for all 𝛼∈ℕ^𝑛 with |𝛼|≤𝑠.

Throughout the paper, for real 𝛾≥1, 𝐻^𝑠_𝛾(ℝ^𝑛) will denote the Sobolev space of order 𝑠, equipped with the 𝛾-depending norm ‖⋅‖_𝑠,𝛾 defined by‖𝑢‖²_𝑠,𝛾∶=(2𝜋)^−𝑛_ℝ𝑛𝜆^2𝑠,𝛾(𝜉)𝑢(𝜉)²𝑑𝜉, (2.4) where (𝜉=(𝜉₁,…,𝜉_𝑛) are the dual Fourier variables of 𝑥=(𝑥₁,…,𝑥_𝑛)). The norms defined by (2.4), with different values of the parameter 𝛾, are equivalent to each other. For 𝛾=1 we set for brevity ‖⋅‖_𝑠∶=‖⋅‖_𝑠,1 (and, accordingly, 𝐻^𝑠(ℝ^𝑛)∶=𝐻^𝑠₁(ℝ^𝑛)).

It is clear that, for 𝑠∈ℕ, the norm in (2.4) turns out to be equivalent, uniformly with respect to 𝛾, to the norm ‖⋅‖_𝐻^𝑠_𝛾(ℝ^𝑛₎ defined by‖𝑢‖²_𝐻^𝑠_𝛾(ℝ^𝑛₎∶=_{|𝛼|≤𝑠}𝛾^{2(𝑠−|𝛼|)}𝜕^𝛼𝑢²_{𝐿2(ℝ𝑛)}. (2.5) Another useful remark about the parameter depending norms defined in (2.4) is provided by the following counterpart of the usual Sobolev imbedding inequality:‖𝑢‖_𝑠,𝛾≤𝛾^𝑠−𝑟‖𝑢‖_𝑟,𝛾, (2.6) for arbitrary 𝑠≤𝑟 and 𝛾≥1.

In Section 4, the ordinary Sobolev spaces, endowed with the weighted norms above, will be considered in ℝ^𝑛−1 (interpreted as the boundary of the half-space ℝ^𝑛₊); regardless to the different dimension, the same notations and conventions as before will be used there.

Let us introduce now some classes of function spaces of Sobolev type, defined over the half-space ℝ^𝑛₊; these spaces will be used to measure the regularity of solutions to characteristic BVPs with sufficiently smooth data (cf. Theorem 1.1 and Section 4).

Given an integer 𝑚≥1, the conormal Sobolev space of order 𝑚 is defined as the set of functions 𝑢∈𝐿²(ℝ^𝑛₊) such that 𝑍^𝛼𝑢∈𝐿²(ℝ^𝑛₊), for all multi-indices 𝛼 with |𝛼|≤𝑚. Agreeing with the notations set for the usual Sobolev spaces, for 𝛾≥1, 𝐻^𝑚_tan,𝛾(ℝ^𝑛₊) will denote the conormal space of order 𝑚 equipped with the 𝛾-depending norm‖𝑢‖²_𝐻^𝑚_tan,𝛾ℝ^𝑛₊∶=_{|𝛼|≤𝑚}𝛾^{2(𝑚−|𝛼|)}𝑍^𝛼𝑢²_{𝐿2ℝ𝑛+}, (2.7) and we again write 𝐻^𝑚_tan(ℝ^𝑛₊)∶=𝐻^𝑚_tan,1(ℝ^𝑛₊).

For later use, we need to consider also a class of mixed tangential/conormal spaces, where different orders of tangential and conormal smoothness are allowed. Namely, for every 𝑚,𝑟∈ℕ, with 𝑚≥𝑟, we let ℋ^𝑚,𝑟_tan(ℝ^𝑛₊) denote the space of all functions 𝑢∈𝐿²(ℝ^𝑛₊) such that 𝑍^𝛼𝑢∈𝐿²(ℝ^𝑛₊), whenever |𝛼|≤𝑚 and 0≤𝛼₁≤𝑟: here derivatives 𝑍^𝛼 are required belonging to 𝐿² up to the order 𝑚, but conormal derivatives (namely, derivatives involving the operator 𝑍₁) are allowed only up to the lower order 𝑟, the remaining 𝑚−𝑟 derivatives being purely tangential (i.e, involving only differentiation with respect to tangential variables 𝑥^′). This space is provided with the expected 𝛾-depending norm‖𝑢‖²_ℋ^𝑚,𝑟_tan,𝛾ℝ^𝑛₊∶=_{|𝛼|≤𝑚,0≤𝛼1≤𝑟}𝛾^{2(𝑚−|𝛼|)}𝑍^𝛼𝑢²_{𝐿2ℝ𝑛+}. (2.8) The notation ℋ^𝑚,𝑟_tan,𝛾(ℝ^𝑛₊) is used, here and below, with the same meaning as for usual and conormal Sobolev spaces (accordingly, one has ℋ^𝑚,𝑟_tan(ℝ^𝑛₊)=ℋ^𝑚,𝑟_tan,1(ℝ^𝑛₊)).

For a given Banach space 𝑌 (with norm ‖⋅‖_𝑌) and 1≤𝑝≤+∞, 𝐿^𝑝(0,+∞;𝑌) will denote the space of the 𝑌-valued measurable functions on (0,+∞) such that ∫₀^+∞‖𝑢(𝑡)‖^𝑝_𝑌𝑑𝑡<+∞.

It is easy to see that the following identities hold true for ℋ^𝑚,𝑟_tan(ℝ^𝑛₊), in the border cases 𝑟=0 and 𝑟=𝑚:ℋ^𝑚,0_tanℝ^𝑛₊=𝐿²0,+∞;𝐻^𝑚ℝ^𝑛−1,ℋ^𝑚,𝑚_tanℝ^𝑛₊=𝐻^𝑚_tanℝ^𝑛₊. (2.9) Actually all of the previously collected observations and properties of 𝛾-weighted norms on usual Sobolev spaces can be readily extended to the weighted norms defined on conormal and mixed spaces.

Remark 2.1. The above-considered tangential-conormal spaces ℋ^𝑚,𝑟_tan(ℝ^𝑛₊) can be viewed as a conormal counterpart, by the action of the ♯ mapping introduced below, of corresponding mixed spaces of Sobolev type in ℝ^𝑛, studied in Hörmander’s [8].

3. Preliminaries and Technical Tools

In this section, we collect several technical tools that will be used in the subsequent analysis (cf. Section 4).

We start by recalling the definition of two operators ♯ and ♮, introduced by Nishitani and Takayama in [9], with the main property of mapping isometrically square integrable (resp., essentially bounded) functions over the half-space ℝ^𝑛₊ onto square integrable (resp., essentially bounded) functions over the full space ℝ^𝑛.

The mappings ♯∶𝐿²(ℝ^𝑛₊)→𝐿²(ℝ^𝑛) and ♮∶𝐿^∞(ℝ^𝑛₊)→𝐿^∞(ℝ^𝑛) are, respectively, defined by𝑤^♯(𝑥)∶=𝑤𝑒^𝑥1,𝑥^′𝑒^𝑥1/2,𝑎^♮(𝑥)=𝑎𝑒^𝑥1,𝑥^′,∀𝑥=𝑥₁,𝑥^′∈ℝ^𝑛. (3.1) They are both norm preserving bijections.

It is also useful to notice that the above operators can be extended to the set 𝒟^′(ℝ^𝑛₊) of Schwartz distributions in ℝ^𝑛₊. It is easily seen that both ♯ and ♮ are topological isomorphisms of the space 𝐶^∞₀(ℝ^𝑛₊) of test functions in ℝ^𝑛₊ (resp., 𝐶^∞(ℝ^𝑛₊)) onto the space 𝐶^∞₀(ℝ^𝑛) of test functions in ℝ^𝑛 (resp., 𝐶^∞(ℝ^𝑛)). Therefore, a standard duality argument leads to define ♯ and ♮ on 𝒟^′(ℝ^𝑛₊), by setting for every 𝜑∈𝐶^∞₀(ℝ^𝑛)⟨𝑢^♯,𝜑⟩∶=⟨𝑢,𝜑^♯−1𝑢⟩,_♮,𝜑∶=𝑢,𝜑^♭ (3.2) (⟨⋅,⋅⟩ is used to denote the duality pairing between distributions and test functions either in the half-space ℝ^𝑛₊ or the full space ℝ^𝑛). In the right-hand sides of (3.2), ♯⁻¹ is just the inverse operator of ♯, while the operator ♭ is defined by𝜑^♭(𝑥)=1𝑥₁𝜑log𝑥₁,𝑥^′,∀𝑥₁>0,𝑥^′∈ℝ^𝑛−1, (3.3) for functions 𝜑∈𝐶^∞₀(ℝ^𝑛). The operators ♯⁻¹ and ♭ arise by explicitly calculating the formal adjoints of ♯ and ♮, respectively.

Of course, one has that 𝑢^♯,𝑢^♮∈𝒟^′(ℝ^𝑛); moreover the following relations can be easily verified (cf. [9]):𝜓𝑢^♯=𝜓^♮𝑢^♯,𝜕_𝑗𝑢^♮=𝑍_𝑗𝑢^♮𝜕,𝑗=1,…,𝑛,₁𝑢^♯=𝑍₁𝑢^♯+12𝑢^♯𝜕_𝑗,𝑢^♯=𝑍_𝑗𝑢^♯,𝑗=2,…,𝑛, (3.4)(3.5)(3.6)(3.7) whenever 𝑢∈𝒟^′(ℝ^𝑛₊) and 𝜓∈𝐶^∞(ℝ^𝑛₊) (in (3.4) 𝑢∈𝐿²(ℝ^𝑛₊) and 𝜓∈𝐿^∞(ℝ^𝑛₊) are even allowed).

From formulas (3.6), (3.7) and the 𝐿²-boundedness of ♯, it also follows that ♯∶𝐻^𝑚_tan,𝛾(ℝ^𝑛₊)→𝐻^𝑚_𝛾(ℝ^𝑛) is a topological isomorphism, for each integer 𝑚≥1 and real 𝛾≥1.

Following [9] (see also [2]), in the next subsection the lastly mentioned property of ♯ will be exploited to shift some remarkable properties of the ordinary Sobolev spaces in ℝ^𝑛 to the functional framework of conormal Sobolev spaces over the half-space ℝ^𝑛₊.

In the end, we observe that the operator ♯ continuously maps the space 𝐶^∞₍₀₎(ℝ^𝑛₊) into the Schwartz space 𝒮(ℝ^𝑛) of rapidly decreasing functions in ℝ^𝑛 (note also that the same is no longer true for the image of 𝐶^∞₍₀₎(ℝ^𝑛₊) under the operator ♮, which is only included into the space 𝐶^∞_𝑏(ℝ^𝑛) of infinitely smooth functions in ℝ^𝑛, with bounded derivatives of all orders).

3.1. Parameter-Depending Norms on Sobolev Spaces

We recall a classical characterization of ordinary Sobolev spaces in ℝ^𝑛, according to Hörmander’s [8], based upon the uniform boundedness of a suitable family of parameter-depending norms.

For given 𝑠∈ℝ, 𝛾≥1 and for each 𝛿∈]0,1] a norm in 𝐻^𝑠−1(ℝ^𝑛) is defined by setting‖𝑢‖²_{𝑠−1,𝛾,𝛿}∶=(2𝜋)^−𝑛_ℝ𝑛𝜆^2𝑠,𝛾(𝜉)𝜆^−2,𝛾(𝛿𝜉)𝑢(𝜉)²𝑑𝜉. (3.8) According to Section 2, for 𝛾=1 and any 0<𝛿≤1, we set ‖⋅‖_{𝑠−1,𝛿}∶=‖⋅‖_{𝑠−1,1,𝛿}; the family of 𝛿-weighted norms {‖⋅‖_{𝑠−1,𝛿}}_0<𝛿≤1 was deeply studied in [8]; easy arguments (relying essentially on a 𝛾-rescaling of functions) lead to get the same properties for the norms {‖⋅‖_{𝑠−1,𝛾,𝛿}}_0<𝛿≤1 defined in (3.8) with an arbitrary 𝛾≥1.

Of course, one has ‖⋅‖_{𝑠−1,𝛾,1}=‖⋅‖_{𝑠−1,𝛾} (cf. (2.4), with 𝑠−1 instead of 𝑠). It is also clear that, for each fixed 𝛿∈]0,1[, the norm ‖⋅‖_{𝑠−1,𝛾,𝛿} is equivalent to ‖⋅‖_{𝑠−1,𝛾} in 𝐻_𝛾^𝑠−1(ℝ^𝑛), uniformly with respect to 𝛾; notice, however, that the constants appearing in the equivalence inequalities will generally depend on 𝛿 (see (3.18)).

The next characterization of Sobolev spaces readily follows by taking account of the parameter 𝛾 into the arguments used in [8, Theorem 2.4.1].

Proposition 3.1. For every 𝑠∈ℝ and 𝛾≥1, 𝑢∈𝐻^𝑠_𝛾(ℝ^𝑛) if and only if 𝑢∈𝐻_𝛾^𝑠−1(ℝ^𝑛), and the set {‖𝑢‖_{𝑠−1,𝛾,𝛿}}_0<𝛿≤1 is bounded. In this case, one has ‖𝑢‖_{𝑠−1,𝛾,𝛿}↑‖𝑢‖_𝑠,𝛾,as𝛿↓0. (3.9)

In order to show the regularity result stated in Theorem 1.1, it is useful to provide the conormal Sobolev space 𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊), 𝑚∈ℕ, 𝛾≥1, with a family of parameter-depending norms satisfying analogous properties to those of norms defined in (3.8). Nishitani and Takayama [9] introduced such norms in the “unweighted” case 𝛾=1, just applying the ordinary Sobolev norms ‖⋅‖_{𝑚−1,𝛿} in (3.8) to the pull-back of functions on ℝ^𝑛₊, by the ♯ operator; then these norms were used in [2] to characterize the conormal regularity of functions.

Following [9], for 𝛾≥1, 𝛿∈]0,1], and all 𝑢∈𝐻^𝑚−1_tan(ℝ^𝑛₊), we set‖𝑢‖²_ℝ^𝑛_{+,𝑚−1,tan,𝛾,𝛿}∶=𝑢^♯2_{𝑚−1,𝛾,𝛿}=(2𝜋)^−𝑛_ℝ𝑛𝜆^2𝑚,𝛾(𝜉)𝜆^−2,𝛾(𝛿𝜉)𝑢^♯(𝜉)²𝑑𝜉. (3.10) Because ♯ is an isomorphism of 𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊) onto 𝐻_𝛾^𝑚−1(ℝ^𝑛), the family of norms {‖⋅‖_ℝ^𝑛_{+,𝑚−1,tan,𝛾,𝛿}}_0<𝛿≤1 keeps all the properties enjoyed by the family of norms defined in (3.8).

In particular, the same characterization of ordinary Sobolev spaces on ℝ^𝑛, given by Proposition 3.1, applies also to conormal Sobolev spaces in ℝ^𝑛₊ (cf. [2, 9]).

Proposition 3.2. For every positive integer 𝑚 and 𝛾≥1, 𝑢∈𝐻^𝑚_tan,𝛾(ℝ^𝑛₊) if and only if 𝑢∈𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊), and the set {‖𝑢‖_ℝ^𝑛_{+,𝑚−1,tan,𝛾,𝛿}}_0<𝛿≤1 is bounded. In this case, one has ‖𝑢‖_ℝ^𝑛_{+,𝑚−1,tan,𝛾,𝛿}↑‖𝑢‖_ℝ^𝑛_{+,𝑚,tan,𝛾},as𝛿↓0. (3.11)

As regards to the mixed space ℋ^𝑚,𝑟_tan(ℝ^𝑛₊), it is worthwhile noticing that it can be endowed with the 𝛾-weighted norm defined, by the Fourier transformation, as‖𝑢‖²_ℝ^𝑛_{+,𝑚,𝑟,tan,𝛾}∶=(2𝜋)^−𝑛_ℝ𝑛𝜆^2𝑟,𝛾(𝜉)𝜆𝜉^{′2(𝑚−𝑟),𝛾}𝑢^♯(𝜉)²𝑑𝜉; (3.12) here and below 𝜉^′∶=(𝜉₂,…,𝜉_𝑛) denotes the Fourier dual variables of the tangential space variables 𝑥^′=(𝑥₂,…,𝑥_𝑛), and, with a slight abuse of notation, we write 𝜆^𝑟,𝛾(𝜉^′) to mean in fact 𝜆^𝑟,𝛾(0,𝜉^′).

Of course, the norm in (3.12) is equivalent, uniformly with respect to 𝛾, to the norm (2.8).

3.2. A Class of Conormal Operators

The ♯ operator, defined at the beginning of Section 3, can be used to allow pseudodifferential operators in ℝ^𝑛  acting conormally on functions only defined over the positive half-space ℝ^𝑛₊. Then the standard machinery of pseudodifferential calculus (in the parameter depending version introduced in [10, 11]) can be rearranged into a functional calculus properly behaved on conormal Sobolev spaces described in Section 2. In Section 4, this calculus will be usefully applied to study the conormal regularity of the stationary BVP (1.2)-(1.3).

Let us introduce the pseudodifferential symbols, with a parameter, to be used later; here we closely follow the terminology and notations of [12].

Definition 3.3. A parameter-depending pseudodifferential symbol of order 𝑚∈ℝ is a real-(or complex-) valued measurable function 𝑎(𝑥,𝜉,𝛾) on ℝ^𝑛×ℝ^𝑛×[1,+∞[, such that 𝑎 is 𝐶^∞ with respect to 𝑥 and 𝜉, and for all multi-indices 𝛼,𝛽∈ℕ^𝑛 there exists a positive constant 𝐶_𝛼,𝛽 satisfying 𝜕^𝛼_𝜉𝜕^𝛽_𝑥𝑎𝑥,𝜉,𝛾≤𝐶_𝛼,𝛽𝜆^{𝑚−|𝛼|,𝛾}(𝜉), (3.13) for all 𝑥,𝜉∈ℝ^𝑛 and 𝛾≥1.

The same definition as above extends to functions 𝑎(𝑥,𝜉,𝛾) taking values in the space ℝ^𝑁×𝑁 (resp., ℂ^𝑁×𝑁) of 𝑁×𝑁 real (resp., complex) valued matrices, for all integers 𝑁>1 (where the module |⋅| is replaced in (3.13) by any equivalent norm in ℝ^𝑁×𝑁 (resp., ℂ^𝑁×𝑁)). We denote by Γ^𝑚 the set of 𝛾-depending symbols of order 𝑚∈ℝ (the same notation being used for both scalar-or matrix-valued symbols). Γ^𝑚 is equipped with the obvious norms|𝑎|_𝑚,𝑘∶=max_{|𝛼|+𝛽≤𝑘}sup_{()𝑥,𝜉∈ℝ𝑛×ℝ𝑛,𝛾≥1}𝜆^{−𝑚+|𝛼|,𝛾}(𝜉)𝜕^𝛼_𝜉𝜕^𝛽_𝑥𝑎𝑥,𝜉,𝛾,∀𝑘∈ℕ, (3.14) which turn it into a Fréchet space. For all 𝑚,𝑚^′∈ℝ, with 𝑚≤𝑚^′, the continuous imbedding Γ^𝑚⊂Γ^𝑚′ can be easily proven.

For all 𝑚∈ℝ, the function 𝜆^𝑚,𝛾 is of course a (scalar-valued) symbol in Γ^𝑚.

To perform the analysis of Section 4, it is important to consider the behavior of the weight function 𝜆^𝑚,𝛾(⋅)𝜆^−1,𝛾(𝛿⋅), involved in the definition of the parameter-depending norms in (3.8), (3.10), as a 𝛾-depending symbol according to Definition 3.3.

In order to simplify the forthcoming statements, henceforth the following short notations will be used:𝜆_𝛿^{𝑚−1,𝛾}(𝜉)∶=𝜆^𝑚,𝛾(𝜉)𝜆^−1,𝛾(𝛿𝜉),𝜆_𝛿^{−𝑚+1,𝛾}(𝜉)∶=𝜆_𝛿^{𝑚−1,𝛾}(𝜉)⁻¹=𝜆^−𝑚,𝛾(𝜉)𝜆^1,𝛾(𝛿𝜉), (3.15) for all real numbers 𝑚∈ℝ, 𝛾≥1, and 𝛿∈]0,1]. One has the obvious identities 𝜆₁^{𝑚−1,𝛾}(𝜉)≡𝜆^{𝑚−1,𝛾}(𝜉), 𝜆₁^{−𝑚+1,𝛾}(𝜉)≡𝜆₁^{−𝑚+1,𝛾}(𝜉)≡𝜆^{−𝑚+1,𝛾}(𝜉). However, to avoid confusion in the following, it is worthwhile to remark that functions 𝜆_𝛿^{−𝑚+1,𝛾}(𝜉) and 𝜆_𝛿^{−𝑚+1,𝛾}(𝜉) are no longer the same as soon as 𝛿 becomes strictly smaller than 1; indeed (3.15) gives 𝜆_𝛿^{−𝑚+1,𝛾}(𝜉)=𝜆^{−𝑚+2,𝛾}(𝜉)𝜆^−1,𝛾(𝛿𝜉).

A straightforward application of Leibniz's rule leads to the following result.

Lemma 3.4. For every 𝑚∈ℝ and all 𝛼∈ℕ^𝑛, there exists a positive constant 𝐶_𝑚,𝛼 such that 𝜕^𝛼_𝜉𝜆_𝛿^{𝑚−1,𝛾}(𝜉)≤𝐶_𝑚,𝛼𝜆_𝛿^{𝑚−1−|𝛼|,𝛾}(𝜉),∀𝜉∈ℝ^𝑛,∀𝛾≥1,∀𝛿∈]0,1]. (3.16)

Because of estimates (3.16), 𝜆_𝛿^{𝑚−1,𝛾}(𝜉) can be regarded as a 𝛾-depending symbol, in two different ways. On one hand, combining estimates (3.16) with the trivial inequality𝜆^−1,𝛾(𝛿𝜉)≤1 (3.17) immediately gives that {𝜆_𝛿^{𝑚−1,𝛾}}_0<𝛿≤1 is a bounded subset of Γ^𝑚.

On the other hand, the left inequality in𝛿𝜆^1,𝛾(𝜉)≤𝜆^1,𝛾(𝛿𝜉)≤𝜆^1,𝛾(𝜉),∀𝜉∈ℝ^𝑛,∀𝛿∈]0,1], (3.18) together with (3.16), also gives𝜕^𝛼_𝜉𝜆_𝛿^{𝑚−1,𝛾}(𝜉)≤𝐶_𝑚,𝛼𝛿⁻¹𝜆^{𝑚−1−|𝛼|,𝛾}(𝜉),∀𝜉∈ℝ^𝑛,∀𝛾≥1. (3.19) According to Definition 3.3, (3.19) means that 𝜆_𝛿^{𝑚−1,𝛾} actually belongs, for each fixed 𝛿, to Γ^𝑚−1; nevertheless, the family {𝜆_𝛿^{𝑚−1,𝛾}}_0<𝛿≤1 is unbounded as a subset of Γ^𝑚−1.

For later use, we also need to study the behavior of functions 𝜆_𝛿^{−𝑚+1,𝛾} as 𝛾-depending symbols.

Analogously to Lemma 3.4, one can prove the following result.

Lemma 3.5. For all 𝑚∈ℝ and 𝛼∈ℕ^𝑛, there exists 𝐶_𝑚,𝛼>0 such that 𝜕^𝛼_𝜉𝜆_𝛿^{−𝑚+1,𝛾}(𝜉)≤𝐶_𝑚,𝛼𝜆_𝛿^{−𝑚+1−|𝛼|,𝛾}(𝜉),∀𝜉∈ℝ^𝑛,∀𝛾≥1,∀𝛿∈]0,1]. (3.20)

In particular, Lemma 3.5 implies that the family {𝜆_𝛿^{−𝑚+1,𝛾}}_0<𝛿≤1 is a bounded subset of Γ^−𝑚+1 (it suffices to combine (3.20) with the right inequality in (3.18)).

Any symbol 𝑎=𝑎(𝑥,𝜉,𝛾)∈Γ^𝑚 defines a pseudodifferential operator Op^𝛾(𝑎)=𝑎(𝑥,𝐷,𝛾) on the Schwartz space 𝒮(ℝ^𝑛), by the standard formula∀𝑢∈𝒮ℝ^𝑛,∀𝑥∈ℝ^𝑛,Op^𝛾(𝑎)𝑢(𝑥)=𝑎𝑥,𝐷,𝛾𝑢(𝑥)=∶(2𝜋)^−𝑛_ℝ𝑛𝑒^{𝑖𝑥⋅𝜉}𝑎𝑥,𝜉,𝛾𝑢(𝜉)𝑑𝜉, (3.21) where, of course, we denote 𝑥⋅𝜉∶=∑^𝑛_𝑗=1𝑥_𝑗𝜉_𝑗. 𝑎 is called the symbol of the operator (3.21), and 𝑚 is its order. It comes from the classical theory that Op^𝛾(𝑎) defines a linear-bounded operator Op^𝛾(𝑎)∶𝒮ℝ^𝑛⟶𝒮ℝ^𝑛; (3.22) moreover, the latter extends to a linear-bounded operator on the space 𝒮^′(ℝ^𝑛) of tempered distributions in ℝ^𝑛.

An exhaustive account of the symbolic calculus for pseudodifferential operators with symbols in Γ^𝑚 can be found in [11] (see also [12]). Here, we just recall the following result, concerning the product and the commutator of two pseudodifferential operators.

Proposition 3.6. Let 𝑎∈Γ^𝑚 and 𝑏∈Γ^𝑙, for 𝑙,𝑚∈ℝ. Then Op^𝛾(𝑎)Op^𝛾(𝑏) is a pseudodifferential operator with symbol in Γ^𝑚+𝑙; moreover, if one lets 𝑎#𝑏 denote the symbol of the product, one has for every integer 𝑁≥1𝑎#𝑏−_|𝛼|<𝑁(−𝑖)^|𝛼|𝜕𝛼!^𝛼𝜉𝑎𝜕^𝛼_𝑥𝑏∈Γ^{𝑚+𝑙−𝑁}. (3.23) Under the same assumptions, the commutator [Op^𝛾(𝑎),Op^𝛾(𝑏)]∶=Op^𝛾(𝑎)Op^𝛾(𝑏)−Op^𝛾(𝑏)Op^𝛾(𝑎) is again a pseudodifferential operator with symbol 𝑐∈Γ^𝑚+𝑙. If one further assumes that one of the two symbols 𝑎 or 𝑏 is scalar-valued (so that 𝑎 and 𝑏 commute in the pointwise product), then the symbol 𝑐 of [Op^𝛾(𝑎),Op^𝛾(𝑏)] has order 𝑚+𝑙−1.

We point out that when the symbol 𝑏∈Γ^𝑙 of the preceding statement does not depend on the 𝑥 variables (i.e., 𝑏=𝑏(𝜉,𝛾)), then the symbol 𝑎#𝑏 of the product Op^𝛾(𝑎)Op^𝛾(𝑏) reduces to the pointwise product of symbols 𝑎 and 𝑏; in this case, the asymptotic formula (3.23) is replaced by the exact formula(𝑎#𝑏)𝑥,𝜉,𝛾=𝑎𝑥,𝜉,𝛾𝑏𝜉,𝛾. (3.24) According to (3.15), (3.21), we write𝜆_𝛿^{𝑚−1,𝛾}(𝐷)∶=Op^𝛾𝜆_𝛿^{𝑚−1,𝛾},𝜆_𝛿^{−𝑚+1,𝛾}(𝐷)∶=Op^𝛾𝜆_𝛿^{−𝑚+1,𝛾}. (3.25) In view of (3.15) and (3.24), the operator 𝜆_𝛿^{𝑚−1,𝛾}(𝐷) is invertible, and its two-sided inverse is given by 𝜆_𝛿^{−𝑚+1,𝛾}(𝐷).

Starting from the symbolic classes Γ^𝑚, 𝑚∈ℝ, we introduce now the class of conormal operators in ℝ^𝑛₊, to be used in the sequel.

Let 𝑎(𝑥,𝜉,𝛾) be a 𝛾-depending symbol in Γ^𝑚, 𝑚∈ℝ. The conormal operator with symbol 𝑎, denoted by Op^𝛾_♯(𝑎) (or equivalently 𝑎(𝑥,𝑍,𝛾)), is defined by setting∀𝑢∈𝐶^∞₍₀₎ℝ^𝑛₊,Op^𝛾_♯(𝑎)𝑢^♯=Op^𝛾(𝑎)𝑢^♯. (3.26) In other words, the operator Op^𝛾_♯(𝑎) is the composition of mappingsOp^𝛾_♯(𝑎)=♯⁻¹∘Op^𝛾(𝑎)∘♯. (3.27) As we already noted, 𝑢^♯∈𝒮(ℝ^𝑛) whenever 𝑢∈𝐶^∞₍₀₎(ℝ^𝑛₊); hence formula (3.26) makes sense and gives that Op^𝛾_♯(𝑎)𝑢 is a 𝐶^∞-function in ℝ^𝑛₊. Also Op^𝛾_♯(𝑎)∶𝐶^∞₍₀₎(ℝ^𝑛₊)→𝐶^∞(ℝ^𝑛₊) is a linear-bounded operator that extends to a linear-bounded operator from the space of distributions 𝑢∈𝒟^′(ℝ^𝑛₊) satisfying 𝑢^♯∈𝒮^′(ℝ^𝑛) into 𝒟^′(ℝ^𝑛₊) itself. (In principle, Op^𝛾_♯(𝑎) could be defined by (3.26) over all functions 𝑢∈𝐶^∞(ℝ^𝑛₊), such that 𝑢^♯∈𝒮(ℝ^𝑛). Then Op^𝛾_♯(𝑎) defines a linear-bounded operator on the latter function space, provided that it is equipped with the topology induced, via ♯, from the Fréchet topology of 𝒮(ℝ^𝑛).) Throughout the paper, we continue to denote this extension by Op^𝛾_♯(𝑎) (or 𝑎(𝑥,𝑍,𝛾) equivalently).

As an immediate consequence of (3.27), we have that for all symbols 𝑎∈Γ^𝑚, 𝑏∈Γ^𝑙, with 𝑚,𝑙∈ℝ, there holds∀𝑢∈𝐶^∞₍₀₎ℝ^𝑛₊,Op^𝛾_♯(𝑎)Op^𝛾_♯(𝑏)𝑢=Op^𝛾(𝑎)Op^𝛾(𝑏)𝑢^♯♯−1. (3.28) Then, it is clear that a functional calculus of conormal operators can be straightforwardly borrowed from the corresponding pseudodifferential calculus in ℝ^𝑛; in particular we find that products and commutators of conormal operators are still operators of the same type, and their symbols are computed according to the rules collected in Proposition 3.6.

Below, let us consider the main examples of conormal operators that will be met in Section 4.

As a first example, we quote the multiplication by a matrix-valued function 𝐵∈𝐶^∞₍₀₎(ℝ^𝑛₊). It is clear that this makes an operator of order zero according to (3.26); indeed (3.4) gives for any vector-valued 𝑢∈𝐶^∞₍₀₎(ℝ^𝑛₊)(𝐵𝑢)^♯(𝑥)=𝐵^♮(𝑥)𝑢^♯(𝑥), (3.29) and 𝐵^♮ is a 𝐶^∞-function in ℝ^𝑛, with bounded derivatives of any order, hence a symbol in Γ⁰.

We remark that, when computed for 𝐵^♮, the norm of order 𝑘∈ℕ, defined on symbols by (3.14), just reduces to𝐵^♮_0,𝑘=max_{|𝛼|≤𝑘}𝜕^𝛼𝐵^♮_{𝐿∞(ℝ𝑛)}=max_{|𝛼|≤𝑘}𝑍^𝛼𝐵_{𝐿∞(ℝ𝑛+)}, (3.30) where the second identity above exploits formulas (3.5) and that ♮ maps isometrically 𝐿^∞(ℝ^𝑛₊) onto 𝐿^∞(ℝ^𝑛).

Now, let ℒ∶=𝛾𝐼_𝑁+∑^𝑛_𝑗=1𝐴_𝑗(𝑥)𝑍_𝑗 be a first-order linear partial differential operator, with matrix-valued coefficients 𝐴_𝑗∈𝐶^∞₍₀₎(ℝ^𝑛₊) for 𝑗=1,…,𝑛 and 𝛾≥1. Since the leading part of ℒ only involves conormal derivatives, applying (3.4), (3.6), and (3.7) then gives 𝛾𝑢+^𝑛_𝑗=1𝐴_𝑗𝑍_𝑗𝑢^♯=1𝛾𝐼−2𝐴^♮₁𝑢^♯+^𝑛_𝑗=1𝐴^♮_𝑗𝜕_𝑗𝑢^♯=Op^𝛾(𝑎)𝑢^♯, (3.31) where 𝑎=𝑎(𝑥,𝜉,𝛾)∶=(𝛾𝐼_𝑁−(1/2)𝐴^♮₁(𝑥))+𝑖∑^𝑛_𝑗=1𝐴^♮_𝑗(𝑥)𝜉_𝑗 is a symbol in Γ¹. Then ℒ is a conormal operator of order 1, according to (3.26).

In Section 4, we will be mainly interested in the family of conormal operators𝜆_𝛿^{𝑚−1,𝛾}(𝑍)∶=Op^𝛾_♯𝜆_𝛿^{𝑚−1,𝛾},𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)∶=Op^𝛾_♯𝜆_𝛿^{−𝑚+1,𝛾}. (3.32) The operators 𝜆_𝛿^{𝑚−1,𝛾}(𝑍) are involved in the characterization of conormal regularity provided by Proposition 3.2 (remember that, after Lemma 3.4, 𝜆_𝛿^{𝑚−1,𝛾}∈Γ^𝑚−1). Indeed, from Plancherel's formula and the fact that the operator ♯ preserves the 𝐿²-norm, the following identities‖𝑢‖_ℝ^𝑛_{+,𝑚−1,tan,𝛾,𝛿}≡𝜆_𝛿^{𝑚−1,𝛾}(𝑍)𝑢_{𝐿2(ℝ𝑛+)} (3.33) can be straightforwardly established; hence, Proposition 3.2 can be restated in terms of the boundedness, with respect to 𝛿, of the 𝐿²-norms of functions 𝜆_𝛿^{𝑚−1,𝛾}(𝑍)𝑢. This observation is the key point that leads to the analysis performed in Section 4.

Another main feature of the conormal operators (3.32) is that 𝜆_𝛿^{−𝑚+1,𝛾}(𝑍) provides a two-sided inverse of 𝜆_𝛿^{𝑚−1,𝛾}(𝑍); this comes at once from the analogous property of the operators in (3.25) and formulas (3.26), (3.28).

3.3. Sobolev Continuity of Conormal Operators

Proposition 3.7. If 𝑠,𝑚∈ℝ then for all 𝑎∈Γ^𝑚 the pseudodifferential operator Op^𝛾(𝑎) extends as a linear-bounded operator from 𝐻_𝛾^𝑠+𝑚(ℝ^𝑛) into 𝐻^𝑠_𝛾(ℝ^𝑛), and the operator norm of such an extension is uniformly bounded with respect to 𝛾.

We refer the reader to [11] for a detailed proof of Proposition 3.7. A thorough analysis shows that the norm of Op^𝛾(𝑎), as a linear-bounded operator from 𝐻_𝛾^𝑠+𝑚(ℝ^𝑛) to 𝐻^𝑠_𝛾(ℝ^𝑛), actually depends only on a norm of type (3.14) of the symbol 𝑎, besides the Sobolev order 𝑠 and the symbolic order 𝑚 (cf. [11] for detailed calculations). This observation entails, in particular, that the operator norm is uniformly bounded with respect to 𝛾 and other additional parameters from which the symbol of the operator might possibly depend, as a bounded mapping.

From the Sobolev continuity of pseudodifferential operators quoted above, and using that the operator ♯ maps isomorphically conormal Sobolev spaces in ℝ^𝑛₊ onto ordinary Sobolev spaces in ℝ^𝑛, we easily derive the following result.

Proposition 3.8. If 𝑚∈ℤ and 𝑎∈Γ^𝑚, then the conormal operator Op^𝛾_♯(𝑎) extends to a linear-bounded operator from 𝐻^𝑠+𝑚_tan,𝛾(ℝ^𝑛₊) to 𝐻^𝑠_tan,𝛾(ℝ^𝑛₊), for every integer 𝑠≥0, such that 𝑠+𝑚≥0; moreover the operator norm of such an extension is uniformly bounded with respect to 𝛾.

Remark 3.9. We point out that the statement above only deals with integer orders of symbols and conormal Sobolev spaces. The reason is that, in Section 2, conormal Sobolev spaces were only defined for positive integer orders. In principle, this lack could be removed by extending the definition of conormal spaces 𝐻^𝑠_tan(ℝ^𝑛₊) to any real order 𝑠: this could be trivially done, just defining 𝐻^𝑠_tan(ℝ^𝑛₊) to be the pull-back, by the operator ♯, of functions in 𝐻^𝑠(ℝ^𝑛). However, this extension to fractional exponents seems to be useless for the subsequent developments.

As regards to the action of conormal operators on the mixed spaces ℋ^𝑠,𝑟_tan,𝛾(ℝ^𝑛₊), similar arguments to those used in the proof of Proposition 3.8 lead to the following.

Proposition 3.10. Let 𝑎=𝑎(𝑥,𝜉,𝛾) be a symbol in Γ^𝑚, for 𝑚∈ℤ. Then for all integers 𝑟,𝑠∈ℕ, such that 𝑠≥𝑟, 𝑠>0, and 𝑟+𝑚≥0, Op^𝛾_♯(𝑎)=𝑎(𝑥,𝑍,𝛾) extends by continuity to a linear-bounded operator Op^𝛾_♯(𝑎)∶ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾ℝ^𝑛₊⟶ℋ^𝑠,𝑟_tan,𝛾ℝ^𝑛₊. (3.34) Moreover, the operator norm of such an extension is uniformly bounded with respect to 𝛾.

4. Proof of Theorem 1.1

This section is entirely devoted to the proof of Theorem 1.1.

4.1. The Strategy of the Proof: Comparison with the Strongly Well-Posed Case

As it was already pointed out in the Introduction, in order to solve the BVP (1.2)-(1.3) in 𝐿², Theorem 1.1 requires an additional tangential/conormal regularity of the corresponding data. The precise increase of regularity needed for the data is prescribed by the energy inequality (1.7): to estimate the 𝐿²-norm of the solution, in the interior and on the boundary of the domain, we lose 𝑟 conormal derivatives and 𝑠−𝑟 tangential derivatives with respect to the interior source term 𝐹, and 𝑠 (tangential) derivatives with respect to the boundary datum 𝐺.

In [2], the conormal regularity of weak solutions to the BVP (1.2)-(1.3) was studied, under the assumption that no loss of derivatives occurs in the estimate of the solution by the data. To prove the result of [2, Theorem 15], the solution 𝑢 to (1.2)-(1.3) was regularized by a family of tangential mollifiers 𝐽_𝜀, 0<𝜀<1, defined by Nishitani and Takayama in [9] as a suitable combination of the operator ♯ and the standard Friedrichs'mollifiers. The essential point of the analysis performed in [2] was to notice that the mollified solution 𝐽_𝜀𝑢 solves the same problem (1.2)-(1.3), as the original solution 𝑢. The data of the problem for 𝐽_𝜀𝑢 come from the regularization, by 𝐽_𝜀, of the data involved in the original problem for 𝑢; furthermore, an additional term [𝐽_𝜀,𝐿]𝑢, where [𝐽_𝜀,𝐿] is the commutator between the differential operator 𝐿 and the tangential mollifier 𝐽_𝜀, appears into the equation satisfied by 𝐽_𝜀𝑢. Because the energy estimate associated to a strong 𝐿²-well-posed problem does not lose derivatives, actually this term can be incorporated into the source term of the equation satisfied by 𝐽_𝜀𝑢.

In the case of Theorem 1.1, where the a priori estimate (1.7) exhibits a finite loss of regularity with respect to the data, this technique seems to be unsuccessful, since [𝐽_𝜀,𝐿]𝑢 cannot be absorbed into the right-hand side without losing derivatives on the solution 𝑢; on the other hand, it seems that the same term cannot be merely reduced to a lower-order term involving the smoothed solution 𝐽_𝜀𝑢, as well (this should require a sharp control of the error term 𝑢−𝐽_𝜀𝑢).

These observations lead to develop another technique, where the tangential mollifier 𝐽_𝜀 is replaced by the family of operators (3.32), involved in the characterization of regularity given by Proposition 3.2. Instead of studying the problem satisfied by the smoothed solution 𝐽_𝜀𝑢, here we consider the problem satisfied by 𝜆_𝛿^{𝑚−1,𝛾}(𝑍)𝑢. As before, a new term [𝜆_𝛿^{𝑚−1,𝛾}(𝑍),𝐿]𝑢 appears which takes account of the commutator between the differential operator 𝐿 and the conormal operator 𝜆_𝛿^{𝑚−1,𝛾}(𝑍). Since we assume the weak well-posedness of the BVP (1.2)-(1.3) to be preserved under lower order terms, the approach consists of treating the commutator [𝜆_𝛿^{𝑚−1,𝛾}(𝑍),𝐿]𝑢 as a lower-order term within the interior equation for 𝜆_𝛿^{𝑚−1,𝛾}(𝑍)𝑢 (see (4.10)) (differently from the strongly 𝐿²-well-posed case studied in [2], the stability of problem (1.2)-(1.3) under lower-order perturbations is no longer a trivial consequence of the well-posedness itself. In Theorem 1.1, this stability is required as an additional hypothesis about the BVP); this is made possible by taking advantage from the invertibility of the operator 𝜆_𝛿^{𝑚−1,𝛾}(𝑍).

We argue by induction on the integer order 𝑚≥1. Let us take arbitrary data 𝐹∈ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾(ℝ^𝑛₊), 𝐺∈𝐻_𝛾^𝑠+𝑚(ℝ^𝑛−1), and fix an arbitrary matrix-valued function 𝐵∈𝐶^∞₍₀₎(ℝ^𝑛₊) (as the lower order term in the interior equation (1.2)).

Because of the inductive hypothesis, we already know that the unique 𝐿²-solution 𝑢 to (1.2)-(1.3) actually belongs to 𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊) and 𝑢^𝐼_∣𝑥1=0 belongs to 𝐻_𝛾^𝑚−1(ℝ^𝑛−1), provided that 𝛾 is taken to be larger than a certain constant 𝛾_𝑚−1≥1; moreover the solution 𝑢 obeys the estimate (1.8) of order 𝑚−1𝛾‖𝑢‖²_{𝐻𝑚−1tan,𝛾ℝ𝑛+}+𝑢^𝐼_∣𝑥1=0²_{𝐻𝛾𝑚−1ℝ𝑛−1}≤𝐶_𝑚−11𝛾^2𝑠+1‖𝐹‖²_{ℋ𝑠+𝑚−1,𝑟+𝑚−1tan,𝛾ℝ𝑛+}+1𝛾^2𝑠‖𝐺‖²_{𝐻𝛾𝑠+𝑚−1ℝ𝑛−1}, (4.1) where the positive constant 𝐶_𝑚−1, as well as 𝛾_𝑚−1, only depends on the smoothness order 𝑚 and the 𝐿^∞-norm of a finite number (depending on 𝑚 itself) of conormal derivatives of 𝐵 (cf. (3.30)), besides the coefficients 𝐴_𝑗  (1≤𝑗≤𝑛) of 𝐿 and the integer numbers 𝑟 and 𝑠.

In order to increase the conormal regularity of the solution 𝑢 by order one, we are going to act on 𝑢 by the conormal operator 𝜆_𝛿^{𝑚−1,𝛾}(𝑍); then we consider the analogue of the original problem (1.2)-(1.3) satisfied by 𝜆_𝛿^{𝑚−1,𝛾}(𝑍)𝑢.

4.2. A Modified Version of the Conormal Operator 𝜆_𝛿^{𝑚−1,𝛾}(𝑍)

Due to some technical reasons that will be clarified in Section 4.3, we need to slightly modify the conormal operator 𝜆_𝛿^{𝑚−1,𝛾}(𝑍) to be applied to the solution 𝑢 of the original BVP (1.2)-(1.3), as was described in the preceding section.

The first step is to decompose the weight function 𝜆_𝛿^{𝑚−1,𝛾} as the sum of two contributions. To do so, we proceed as follows. Firstly, we take an arbitrary positive, even function 𝜒∈𝐶^∞(ℝ^𝑛) with the following properties:0≤𝜒(𝑥)≤1,∀𝑥∈ℝ^𝑛(𝑥)|𝑥|≤1,𝜒≡1,for2,𝜒(𝑥)≡0,for|𝑥|>1. (4.2) Then, we set𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝜉)∶=𝜒(𝐷)𝜆_𝛿^{𝑚−1,𝛾}(𝜉)=ℱ⁻¹𝜒∗𝜆_𝛿^{𝑚−1,𝛾}(𝜉),𝑟_𝑚,𝛿𝜉,𝛾∶=𝜆_𝛿^{𝑚−1,𝛾}(𝜉)−𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝜉)=𝐼−𝜒(𝐷)𝜆_𝛿^{𝑚−1,𝛾}(𝜉). (4.3) The following result (the proof of which is postponed to Appendix A) shows that the function 𝜆^{𝑚−1,𝛾}_𝜒,𝛿 essentially behaves like 𝜆_𝛿^{𝑚−1,𝛾}.

Lemma 4.1. Let the function 𝜒∈𝐶^∞(ℝ^𝑛) satisfy the assumptions in (4.2). Then 𝜆^{𝑚−1,𝛾}_𝜒,𝛿 is a symbol in Γ^𝑚−1; moreover for every multi-index 𝛼∈ℕ^𝑛, there exists a positive constant 𝐶_𝑚,𝛼, independent of 𝛾 and 𝛿, such that 𝜕^𝛼_𝜉𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝜉)≤𝐶_𝑚,𝛼𝜆_𝛿^{𝑚−1−|𝛼|,𝛾}(𝜉),∀𝜉∈ℝ^𝑛. (4.4)

An immediate consequence of Lemma 4.1 and (4.3) is that 𝑟_𝑚,𝛿 is also a 𝛾-depending symbol in Γ^𝑚−1.

Let us define with the obvious meaning of the notations𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)∶=Op^𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿,𝑟_𝑚,𝛿𝐷,𝛾∶=Op^𝛾𝑟_𝑚,𝛿,𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)∶=Op^𝛾_♯𝜆^{𝑚−1,𝛾}_𝜒,𝛿,𝑟_𝑚,𝛿𝑍,𝛾∶=Op^𝛾_♯𝑟_𝑚,𝛿. (4.5) The second important result is concerned with the conormal operator 𝑟_𝑚,𝛿(𝑍,𝛾)=Op^𝛾_♯(𝑟_𝑚,𝛿) and tells that it essentially behaves as a regularizing operator on conormal Sobolev spaces.

Lemma 4.2. For every 𝑘∈ℕ, the conormal operator 𝑟_𝑚,𝛿(𝑍,𝛾) extends as a linear-bounded operator, still denoted by 𝑟_𝑚,𝛿(𝑍,𝛾), from 𝐿²(ℝ^𝑛₊) to 𝐻^𝑘_tan,𝛾(ℝ^𝑛₊). Moreover there exists a positive constant 𝐶_𝑚,𝑘, depending only on 𝑘 and 𝑚, such that for all 𝛾≥1 and 𝛿∈]0,1]𝑟_𝑚,𝛿𝑍,𝛾𝑢_{𝐻𝑘tan,𝛾(ℝ𝑛+)}≤𝐶_𝑚,𝑘𝛾^𝑘‖𝑢‖_𝐿²_(ℝ^𝑛₊₎,∀𝑢∈𝐿²ℝ^𝑛₊. (4.6)

Remark 4.3. In the framework of the general theory of pseudodifferential operators, the procedure adopted to define the symbol 𝜆^{𝑚−1,𝛾}_𝜒,𝛿 is standard and is used to modify an arbitrary symbol in such a way to make properly supported the corresponding pseudodifferential operator (see [13] for the definition of a properly supported operator and an extensive description of the method). As a general issue, one can prove that the resulting properly supported operator differs from the original one by a regularizing remainder. Essentially, an easy adaptation of the same arguments to the framework of conormal spaces in ℝ^𝑛₊ can be employed to prove the regularizing action of the conormal operator 𝑟_𝑚,𝛿(𝑍,𝛾) stated by Lemma 4.2.

According to (4.3), we decompose𝜆_𝛿^{𝑚−1,𝛾}(𝑍)=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)+𝑟_𝑚,𝛿𝑍,𝛾. (4.7) As a consequence of Lemmas 4.1 and 4.2, the role of the family of conormal operators {𝜆_𝛿^{𝑚−1,𝛾}(𝑍)}_0<𝛿≤1 in the characterization of the conormal regularity provided by Proposition 3.2 (cf. also (3.33)) can also be played by the family of “modified” operators {𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)}_0<𝛿≤1, namely, we have the following.

Corollary 4.4. For every positive integer 𝑚 and 𝛾≥1, 𝑢∈𝐻^𝑚_tan,𝛾(ℝ^𝑛₊) if and only if 𝑢∈𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊) and the set {‖𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢‖_{𝐿2(ℝ𝑛+)}}_0<𝛿≤1 is bounded.

In order to suitably handle the commutator between the differential operator 𝐿 and the conormal operator 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍), that comes from writing down the problem satisfied by 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢 (see Sections 4.3.1 and 4.3.2), it is useful to analyze the behavior of the pseudodifferential operators 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷), when interacting with another pseudodifferential operator by composition and commutation. The following lemma analyzes these situations; for later use, it is convenient to replace in our reasoning the function 𝜆^{𝑚−1,𝛾}_𝜒,𝛿 by a general 𝛾-depending symbol 𝑎_𝛿 preserving the same kind of decay properties as in (4.4).

Lemma 4.5. Let {𝑎_𝛿}_0<𝛿≤1 be a family of symbols 𝑎_𝛿=𝑎_𝛿(𝑥,𝜉,𝛾)∈Γ^𝑟−1, 𝑟∈ℝ, such that for all multi-indices 𝛼,𝛽∈ℕ^𝑛 there exists a positive constant 𝐶_{𝑟,𝛼,𝛽}, independent of 𝛾 and 𝛿, for which 𝜕^𝛼_𝜉𝜕^𝛽_𝑥𝑎_𝛿𝑥,𝜉,𝛾≤𝐶_{𝑟,𝛼,𝛽}𝜆_𝛿^{𝑟−1−|𝛼|,𝛾}(𝜉),∀𝑥,𝜉∈ℝ^𝑛. (4.8) Let 𝑏=𝑏(𝑥,𝜉,𝛾) be another symbol in Γ^𝑙, for 𝑙∈ℝ.
Then, for every 𝛿∈]0,1], the product Op^𝛾(𝑎_𝛿)Op^𝛾(𝑏) is a pseudodifferential operator with symbol 𝑎_𝛿#𝑏 in Γ^{𝑙+𝑟−1}. Moreover, for all multi-indices 𝛼,𝛽∈ℕ^𝑛 there exists a constant 𝐶_{𝑟,𝑙,𝛼,𝛽}, independent of 𝛾 and 𝛿, such that 𝜕^𝛼_𝜉𝜕^𝛽_𝑥𝑎_𝛿#𝑏𝑥,𝜉,𝛾≤𝐶_{𝑟,𝑙,𝛼,𝛽}𝜆_𝛿^{𝑙+𝑟−1−|𝛼|,𝛾}(𝜉),∀𝑥,𝜉∈ℝ^𝑛. (4.9) Under the same hypotheses, Op^𝛾(𝑎_𝛿)Op^𝛾(𝑏)𝜆_𝛿^{−𝑚+1,𝛾}(𝐷) is a pseudodifferential operator with symbol (𝑎_𝛿#𝑏)𝜆_𝛿^{−𝑚+1,𝛾} in Γ^{𝑙+𝑟−𝑚}; moreover, {(𝑎_𝛿#𝑏)𝜆_𝛿^{−𝑚+1,𝛾}}_0<𝛿≤1 is a bounded subset of Γ^{𝑙+𝑟−𝑚}. Eventually, if the symbol 𝑎_𝛿 is scalar-valued, [Op^𝛾(𝑎_𝛿),Op^𝛾(𝑏)]𝜆_𝛿^{−𝑚+1,𝛾}(𝐷) is a pseudodifferential operator with symbol 𝑐_𝛿∈Γ^{𝑙+𝑟−𝑚−1}, and {𝑐_𝛿}_0<𝛿≤1 is a bounded subset of Γ^{𝑙+𝑟−𝑚−1}.

The proof of Lemma 4.5 is postponed to Appendix A.

Remark 4.6. That Op^𝛾(𝑎_𝛿)Op^𝛾(𝑏), Op^𝛾(𝑎_𝛿)Op^𝛾(𝑏)𝜆_𝛿^{−𝑚+1,𝛾}(𝐷) and [Op^𝛾(𝑎_𝛿),Op^𝛾(𝑏)]𝜆_𝛿^{−𝑚+1,𝛾}(𝐷) have symbols belonging, respectively, to Γ^{𝑙+𝑟−1}, Γ^{𝑙+𝑟−𝑚} and Γ^{𝑙+𝑟−𝑚−1} (for scalar-valued 𝑎_𝛿) follows at once from the standard rules of symbolic calculus summarized in Proposition 3.6. The nontrivial part of the statement above (although deduced from the asymptotic formula (3.23) with a minor effort) is the one asserting that the symbol of Op^𝛾(𝑎_𝛿)Op^𝛾(𝑏) enjoys estimates (4.9); indeed, these estimates give the precise dependence on 𝛿 of the decay at infinity of this symbol. Then the remaining assertions in Lemma 4.5 easily follow from (4.9) itself.

Remark 4.7. In view of Proposition 3.8, the results on symbols collected in Lemma 4.5 can be used to study the conormal Sobolev continuity of the related conormal operators.
To be definite, for every nonnegative integer number 𝑠, such that 𝑠+𝑙+𝑟−𝑚 is also nonnegative, Proposition 3.8 and Lemma 4.5 imply that the conormal operator Op^𝛾_♯(𝑎_𝛿)Op^𝛾_♯(𝑏)𝜆_𝛿^{−𝑚+1,𝛾}(𝑍) extends as a linear-bounded mapping from 𝐻^{𝑠+𝑙+𝑟−𝑚}_tan,𝛾(ℝ^𝑛₊) into 𝐻^𝑠_tan,𝛾(ℝ^𝑛₊); moreover, its operator norm is uniformly bounded with respect to 𝛾 and 𝛿.
If in addition 𝑠+𝑙+𝑟−𝑚≥1 and 𝑎_𝛿 are scalar-valued, then [Op^𝛾_♯(𝑎_𝛿),Op^𝛾_♯(𝑏)]𝜆_𝛿^{−𝑚+1,𝛾}(𝑍) extends as a linear-bounded operator from 𝐻^{𝑠+𝑙+𝑟−𝑚−1}_tan,𝛾(ℝ^𝑛₊) into 𝐻^𝑠_tan,𝛾(ℝ^𝑛₊), and again its operator norm is uniformly bounded with respect to 𝛾 and 𝛿.
These mapping properties will be usefully applied in Sections 4.3 and 4.5.

4.3. The Interior Equation

We follow the strategy already explained in Section 4.1, where now the role of the operator 𝜆_𝛿^{𝑚−1,𝛾}(𝑍) is replaced by 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍). Since 𝜆^{𝑚−1,𝛾}_𝜒,𝛿∈Γ^𝑚−1 (because of Lemma 4.1) and, for 𝛾≥𝛾_𝑚−1, 𝑢∈𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊) (from the inductive hypothesis), after Proposition 3.8 we know that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢∈𝐿²(ℝ^𝑛₊).

Applying 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍) to (1.2) (where ℬ is just the multiplication by 𝐵), we find that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢 must solve𝛾+𝐿+𝐵𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢+𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿+𝐵𝑢=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝐹,inℝ^𝑛₊. (4.10) We are going now to show that the commutator term [𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿+𝐵]𝑢 in the above equation can be actually considered as a lower-order term with respect to 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢.

To this end, we may decompose this term as the sum of two contributions corresponding, respectively, to the tangential and normal components of 𝐿.

First, in view of (1.5), (1.6), we may write the coefficient 𝐴₁ of the normal derivative 𝜕₁ in the expression (1.4) of 𝐿 as 𝐴₁=𝐴¹₁+𝐴²₁,𝐴¹₁𝐴∶=^1𝐼,𝐼000,𝐴²_1∣𝑥1=0=0, (4.11) hence 𝐴²₁𝜕₁=𝐻₁𝑍₁, (4.12) where 𝐻₁(𝑥)=𝑥₁⁻¹𝐴²₁(𝑥)∈𝐶^∞₍₀₎(ℝ^𝑛₊). Accordingly, we split 𝐿 as 𝐿=𝐴¹₁𝜕₁+𝐿_tan,𝐿_tan∶=𝐻₁𝑍₁+^𝑛_𝑗=2𝐴_𝑗𝑍_𝑗. (4.13) Consequently, we have𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿+𝐵𝑢=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴¹₁𝜕₁𝑢+𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿_tan+𝐵𝑢. (4.14) Note that 𝐿_tan+𝐵 is just a conormal operator of order 1, according to the terminology introduced in Section 3.2.

4.3.1. The Tangential Commutator

Firstly, we study the tangential commutator [𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿_tan+𝐵]𝑢. Using the identity 𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝜆_𝛿^{𝑚−1,𝛾}(𝑍)=𝐼 and (4.7), the latter can be written in terms of 𝜆^{𝛾,𝑚−1}_𝜒,𝛿(𝑍)𝑢, modulo some smoothing reminder. Indeed we compute𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿_tan+𝐵𝑢=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿_tan+𝐵𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢+𝑟_𝑚,𝛿𝑢=𝜆𝑍,𝛾_{𝑚−1,𝛾𝜒,𝛿}(𝑍),𝐿_tan+𝐵𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢+𝑠_𝑚,𝛿𝑥,𝑍,𝛾𝑢, (4.15) where we have set for short𝑠_𝑚,𝛿𝑥,𝑍,𝛾∶=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿_tan+𝐵𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝑟_𝑚,𝛿𝑍,𝛾. (4.16)

4.3.2. The Normal Commutator

First of all, we notice that, due to the structure of the matrix 𝐴¹₁, the commutator [𝜆^{𝛾,𝑚−1}_𝜒,𝛿(𝑍),𝐴¹₁𝜕₁] acts nontrivially only on the noncharacteristic component of the vector function 𝑢; namely, we have: 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴¹₁𝜕₁𝜆𝑢=^{𝑚−1,𝛾𝜒,𝛿}(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑢^𝐼0. (4.17) Therefore, we focus on the study of the first nontrivial component of the commutator term. Note that the commutator [𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁] cannot be merely treated by the tools of the conormal calculus developed in Section 3.2, because of the presence of the effective normal derivative 𝜕₁ (recall that 𝐴₁^𝐼,𝐼 is invertible). This section is devoted to the study of the normal commutator [𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁]𝑢^𝐼. The following result is of fundamental importance.

Proposition 4.8. For all 𝛿∈]0,1], 𝛾≥1 and 𝑚∈ℕ, there exists a symbol 𝑞_𝑚,𝛿(𝑥,𝜉,𝛾)∈Γ^𝑚−2 such that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑤=𝑞_𝑚,𝛿𝑥,𝑍,𝛾𝜕₁𝑤,∀𝑤∈𝐶^∞₍₀₎ℝ^𝑛₊. (4.18) Moreover, the symbol 𝑞_𝑚,𝛿(𝑥,𝜉,𝛾) obeys the following estimates. For all 𝛼,𝛽∈ℕ^𝑛, there exists a positive constant 𝐶_{𝑚,𝛼,𝛽}, independent of 𝛾 and 𝛿, such that 𝜕^𝛼_𝜉𝜕^𝛽_𝑥𝑞_𝑚,𝛿𝑥,𝜉,𝛾≤𝐶_{𝑚,𝛼,𝛽}𝜆_𝛿^{𝑚−2−|𝛼|,𝛾}(𝜉),∀𝑥,𝜉∈ℝ^𝑛. (4.19)

Proof. That 𝑞_𝑚,𝛿(𝑥,𝜉,𝛾), satisfying estimates (4.19), is a symbol in Γ^𝑚−2 actually follows arguing from (4.19) and inequalities (3.18) as was already done for 𝜆_𝛿^{𝑚−1,𝛾}(𝜉) and 𝜆_𝛿^{−𝑚+1,𝛾}(𝜉) (see Section 3.2).
For given 𝑤∈𝐶^∞₍₀₎(ℝ^𝑛₊), let us explicitly compute ([𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁]𝑤)^♯; using the identity (𝜕₁𝑤)^♯=𝑒^−𝑥1(𝑍₁𝑤)^♯ and that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍) and 𝑍₁ commute, we find for every 𝑥∈ℝ^𝑛𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑤^♯(𝑥)=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)𝐴₁^{𝐼,𝐼,♮}𝑒^−(⋅)1𝑍₁𝑤^♯(𝑥)−𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−𝑥1𝑍₁𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑤^♯(𝑥)=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)𝐴₁^{𝐼,𝐼,♮}𝑒^−(⋅)1𝑍₁𝑤^♯(𝑥)−𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−𝑥1𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑍₁𝑤^♯(𝑥)=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)𝐴₁^{𝐼,𝐼,♮}𝑒^−(⋅)1𝑍₁𝑤^♯(𝑥)−𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−𝑥1𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)𝑍₁𝑤^♯(𝑥). (4.20) Observing that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷) acts on the space 𝒮(ℝ^𝑛) as the convolution by the inverse Fourier transform of 𝜆^{𝑚−1,𝛾}_𝜒,𝛿, the preceding expression can be equivalently restated as follows: 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑤^♯(𝑥)=ℱ⁻¹𝜆^{𝑚−1,𝛾}_𝜒,𝛿∗𝐴₁^{𝐼,𝐼,♮}𝑒^−(⋅)1𝑍₁𝑤^♯(𝑥)−𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−𝑥1ℱ⁻¹𝜆^{𝑚−1,𝛾}_𝜒,𝛿∗𝑍₁𝑤^♯=ℱ⁻¹𝜆^{𝑚−1,𝛾}_𝜒,𝛿,𝐴₁^{𝐼,𝐼,♮}(𝑥−⋅)𝑒^{−(𝑥1−(⋅)1)}𝑍₁𝑤^♯(𝑥−⋅)−𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−𝑥1⟨ℱ⁻¹𝜆^{𝑚−1,𝛾}_𝜒,𝛿,𝑍₁𝑤^♯()⟩=𝜂𝑥−⋅_{𝛿𝑚−1,𝛾},𝜒(⋅)𝐴₁^{𝐼,𝐼,♮}(𝑥−⋅)𝑒^{−(𝑥1−(⋅)1)}𝑍₁𝑤^♯(−𝜂𝑥−⋅)_{𝛿𝑚−1,𝛾},𝜒(⋅)𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−𝑥1𝑍₁𝑤^♯(=𝜂𝑥−⋅)_{𝛿𝑚−1,𝛾},𝜒(⋅)𝐴₁^{𝐼,𝐼,♮}(𝑥−⋅)𝜕₁𝑤^♯(−𝜂𝑥−⋅)_{𝛿𝑚−1,𝛾},𝜒(⋅)𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−(⋅)1𝜕₁𝑤^♯(=𝜂𝑥−⋅)_{𝛿𝑚−1,𝛾},𝜒(⋅)𝐴₁^{𝐼,𝐼,♮}(𝑥−⋅)−𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−(⋅)1𝜕₁𝑤^♯(𝑥−⋅), (4.21) where 𝜂_𝛿^{𝑚−1,𝛾}∶=ℱ⁻¹(𝜆_𝛿^{𝑚−1,𝛾}) and the identity ℱ⁻¹(𝜆^{𝑚−1,𝛾}_𝜒,𝛿)=𝜒𝜂_𝛿^{𝑚−1,𝛾} (following at once from (4.3)) has been used. Just for brevity, let us further set 𝒦𝑥,𝑦∶=𝐴₁^{𝐼,𝐼,♮}𝑥−𝑦−𝐴₁^{𝐼,𝐼,♮}(𝑥)𝑒^−𝑦1𝜒𝑦. (4.22) Thus the identity above reads as 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑤^♯(𝑥)=𝜂_𝛿^{𝑚−1,𝛾},𝒦(𝑥,⋅)𝜕₁𝑤^♯(𝑥−⋅), (4.23) where the “kernel” 𝒦(𝑥,𝑦) is a bounded function in 𝐶^∞(ℝ^𝑛×ℝ^𝑛), with bounded derivatives of all orders. This regularity of 𝒦 is due to the presence of the function 𝜒 in formula (4.22); actually the vanishing of 𝜒 at infinity prevents the blow-up of the exponential factor 𝑒^−𝑦1, as 𝑦₁→−∞. We point out that this is just the step of our analysis of the normal commutator, where this function 𝜒 is needed.
After (4.22), we also have that 𝒦(𝑥,0)=0; then, by a Taylor expansion with respect to 𝑦, we can represent the kernel 𝒦(𝑥,𝑦) as follows: 𝒦𝑥,𝑦=^𝑛_𝑘=1𝑏_𝑘𝑥,𝑦𝑦_𝑘, (4.24) where 𝑏_𝑘(𝑥,𝑦) are given bounded functions in 𝐶^∞(ℝ^𝑛×ℝ^𝑛), with bounded derivatives; it comes from (4.22) and (4.2) that 𝑏_𝑘 can be defined in such a way that for some 𝜀>1 and all 𝑥∈ℝ^𝑛 there holds supp𝑏_𝑘(𝑥,⋅)⊆𝑦≤𝜀. (4.25) Inserting (4.24) in (4.23) and using standard properties of the Fourier transform we get 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑤^♯(𝑥)=𝜂_𝛿^{𝑚−1,𝛾},^𝑛_𝑘=1𝑏_𝑘(𝑥,⋅)(⋅)_𝑘𝜕₁𝑤^♯(=𝑥−⋅)_𝑛𝑘=1(⋅)_𝑘ℱ⁻¹𝜆_𝛿^{𝑚−1,𝛾},𝑏_𝑘(𝑥,⋅)𝜕₁𝑤^♯(𝑥−⋅)=−^𝑛_𝑘=1ℱ⁻¹𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾},𝑏_𝑘(𝑥,⋅)𝜕₁𝑤^♯(𝑥−⋅)=−^𝑛_𝑘=1𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾},ℱ⁻¹𝑏_𝑘(𝑥,⋅)𝜕₁𝑤^♯(𝑥−⋅)=−^𝑛_{𝑘=1ℝ𝑛}𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}(𝜉)ℱ⁻¹𝑏_𝑘(𝑥,⋅)𝜕₁𝑤^♯(𝑥−⋅)(𝜉)𝑑𝜉=−^𝑛_𝑘=1(2𝜋)^−𝑛_ℝ𝑛𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}(𝜉)_ℝ𝑛𝑒^{𝑖𝜉⋅𝑦}𝑏_𝑘𝑥,𝑦𝜕₁𝑤^♯𝑥−𝑦𝑑𝑦𝑑𝜉, (4.26) where we have set 𝐷_𝑘∶=−𝑖𝜕_𝜉𝑘 (for each 𝑘=1,…,𝑛);   note that for 𝑤∈𝐶^∞₍₀₎(ℝ^𝑛₊) and any 𝑥∈ℝ^𝑛, the function 𝑏_𝑘(𝑥,⋅)(𝜕₁𝑤)^♯(𝑥−⋅) belongs to 𝒮(ℝ^𝑛); hence the last expression in (4.26) makes sense. Henceforth, we replace (𝜕₁𝑤)^♯ by any function 𝑣∈𝒮(ℝ^𝑛). Our next goal is writing the integral operator (2𝜋)^−𝑛_ℝ𝑛𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}(𝜉)_ℝ𝑛𝑒^{𝑖𝜉⋅𝑦}𝑏_𝑘𝑥,𝑦𝑣𝑥−𝑦𝑑𝑦𝑑𝜉 (4.27) as a pseudodifferential operator.
Firstly, we make use of the inversion formula for the Fourier transformation and Fubini's theorem to recast (4.27) as follows: _ℝ𝑛𝑒^{𝑖𝜉⋅𝑦}𝑏_𝑘𝑣=()𝑥,𝑦𝑥−𝑦𝑑𝑦2𝜋^−𝑛_ℝ𝑛𝑒^{𝑖𝜉⋅𝑦}𝑏_𝑘𝑥,𝑦_ℝ𝑛𝑒^{𝑖(𝑥−𝑦)⋅𝜂}𝑣𝜂=()𝑑𝜂𝑑𝑦2𝜋^−𝑛_ℝ𝑛𝑒^{𝑖𝑥⋅𝜂}_ℝ𝑛𝑒^{−𝑖𝑦⋅(𝜂−𝜉)}𝑏_𝑘𝑣𝜂=()𝑥,𝑦𝑑𝑦𝑑𝜂2𝜋^−𝑛_ℝ𝑛𝑒^{𝑖𝑥⋅𝜂}𝑏̂_𝑘𝑥,𝜂−𝜉𝑣𝜂𝑑𝜂; (4.28) for every index 𝑘, 𝑏̂_𝑘(𝑥,𝜁) denotes the partial Fourier transform of 𝑏_𝑘(𝑥,𝑦) with respect to 𝑦. Then, inserting (4.28) into (4.27), we obtain (2𝜋)^−𝑛_ℝ𝑛𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}(𝜉)_ℝ𝑛𝑒^{𝑖𝜉⋅𝑦}𝑏_𝑘𝑣=()𝑥,𝑦𝑥−𝑦𝑑𝑦𝑑𝜉2𝜋^−2𝑛_ℝ𝑛𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}(𝜉)_ℝ𝑛𝑒^{𝑖𝑥⋅𝜂}𝑏̂_𝑘𝑥,𝜂−𝜉𝑣𝜂𝑑𝜂𝑑𝜉. (4.29) Recall that, for each 𝑥∈ℝ^𝑛, the function 𝑦↦𝑏_𝑘(𝑥,𝑦) belongs to 𝐶^∞₀(ℝ^𝑛) (and its compact support does not depend on 𝑥, see (4.25)); thus, for each 𝑥∈ℝ^𝑛, 𝑏̂_𝑘(𝑥,𝜁) is rapidly decreasing in 𝜁. Because of the estimates for derivatives of 𝜆_𝛿^{𝑚−1,𝛾} and since 𝑣(𝜂) is also rapidly decreasing, Fubini's theorem can be used to change the order of the integrations within (4.29). So we get (2𝜋)^−2𝑛_ℝ𝑛𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}(𝜉)_ℝ𝑛𝑒^{𝑖𝑥⋅𝜂}𝑏̂_𝑘𝑣𝜂=()𝑥,𝜂−𝜉𝑑𝜂𝑑𝜉2𝜋^−2𝑛_ℝ𝑛𝑒^{𝑖𝑥⋅𝜂}_ℝ𝑛𝑏̂_𝑘𝑥,𝜂−𝜉𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}(𝜉)𝑣𝜂=()𝑑𝜉𝑑𝜂2𝜋^−𝑛_ℝ𝑛𝑒^{𝑖𝑥⋅𝜂}𝑞_{𝑘,𝑚,𝛿}𝑥,𝜂,𝛾𝑣𝜂𝑑𝜂, (4.30) where we have set 𝑞_{𝑘,𝑚,𝛿}𝑥,𝜉,𝛾∶=(2𝜋)^−𝑛_ℝ𝑛𝑏̂_𝑘𝑥,𝜂𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}𝜉−𝜂𝑑𝜂. (4.31) Notice that formula (4.31) defines 𝑞_{𝑘,𝑚,𝛿} as the convolution of the functions 𝑏̂_𝑘(𝑥,⋅) and 𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}; hence 𝑞_{𝑘,𝑚,𝛿} is a well-defined 𝐶^∞-function in ℝ^𝑛×ℝ^𝑛.
The proof of Proposition 4.8 will be accomplished, once the following lemma is proved.
Lemma 4.9. For every 𝑚∈ℕ, 𝑘=1,…,𝑛 and all 𝛼,𝛽∈𝑁^𝑛, there exists a positive constant 𝐶_{𝑘,𝑚,𝛼,𝛽}, independent of 𝛾 and 𝛿, such that 𝜕^𝛼_𝜉𝜕^𝛽_𝑥𝑞_{𝑘,𝑚,𝛿}𝑥,𝜉,𝛾≤𝐶_{𝑘,𝑚,𝛼,𝛽}𝜆_𝛿^{𝑚−2−|𝛼|,𝛾}(𝜉),∀𝑥,𝜉∈ℝ^𝑛. (4.32)
It comes from Lemma 4.9 and the left inequality in (3.18) that, for each index 𝑘, the function 𝑞_{𝑘,𝑚,𝛿} is a symbol in Γ^𝑚−2; notice however that the set {𝑞_{𝑘,𝑚,𝛿}}_0<𝛿≤1 is bounded in Γ^𝑚−1 but not in Γ^𝑚−2. The proof of Lemma 4.9 is postponed to Appendix A.
Now, we continue the proof of Proposition 4.8.
End of the Proof of Proposition 4.8
The last row of (4.30) provides the desired representation of (4.27) as a pseudodifferential operator; actually it gives the identity (2𝜋)^−𝑛_ℝ𝑛𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}(𝜉)_ℝ𝑛𝑒^{𝑖𝜉⋅𝑦}𝑏_𝑘𝑥,𝑦𝑣𝑥−𝑦𝑑𝑦𝑑𝜉=Op^𝛾𝑞_{𝑘,𝑚,𝛿}𝑣(𝑥), (4.33) for every 𝑣∈𝒮(ℝ^𝑛).
Inserting the above formula (with 𝑣=(𝜕₁𝑤)^♯) into (4.26) finally gives 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑤^♯(𝑥)=Op^𝛾𝑞_𝑚,𝛿𝜕₁𝑤^♯(𝑥), (4.34) where 𝑞_𝑚,𝛿=𝑞_𝑚,𝛿(𝑥,𝜉,𝛾) is the symbol in Γ^𝑚−2 defined by 𝑞_𝑚,𝛿𝑥,𝜉,𝛾∶=−^𝑛_𝑘=1𝑞_{𝑘,𝑚,𝛿}𝑥,𝜉,𝛾. (4.35) Of course, formula (4.18) is equivalent to (4.34), in view of (3.26). Estimates (4.19) are satisfied by 𝑞_𝑚,𝛿 by summation over 𝑘 of the similar estimates satisfied by 𝑞_{𝑚,𝛿,𝑘} (cf. Lemma 4.9).
This ends the proof of Proposition 4.8.

Now, we are going to show how the representation in (4.18) can be exploited to treat the normal commutator as a lower-order term in (4.10) satified by 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢.

Firstly, we notice that formula (4.18) has been deduced for smooth functions 𝑤, while 𝑢 is just an 𝐿²-function (actually it belongs to 𝐻^𝑚−1_tan(ℝ^𝑛₊), by the inductive hypothesis). In order to use (4.18) for 𝑢, we need to approximate the latter by smooth functions. This can be done by the help of [14, Proposition 1, Theorem 1]; from there, we know that there exists a sequence {𝑢_𝜈}_𝜈 in 𝐶^∞₍₀₎(ℝ^𝑛₊) such that𝑢_𝜈⟶𝑢,in𝐿²ℝ^𝑛₊,(𝐿+𝐵)𝑢_𝜈⟶(𝐿+𝐵)𝑢,in𝐿²ℝ^𝑛₊,𝑢^𝐼_{𝜈∣𝑥1=0}⟶𝑢^𝐼_∣𝑥1=0,in𝐻^−1/2ℝ^𝑛−1,as𝜈⟶+∞. (4.36) For each index 𝜈, the regular function 𝑢_𝜈∈𝐶^∞₍₀₎(ℝ^𝑛₊) solves the same BVP as the function 𝑢, with new data 𝐹_𝜈, 𝐺_𝜈 defined by𝐹_𝜈∶=𝛾+𝐿+𝐵𝑢_𝜈,𝐺_𝜈∶=𝑀𝑢_{𝜈∣𝑥1=0}. (4.37) It immediately follows from (4.36) that the regular data 𝐹_𝜈,𝐺_𝜈 approximate the original data 𝐹,𝐺 by𝐹_𝜈⟶𝐹,in𝐿²ℝ^𝑛₊,𝐺_𝜈⟶𝐺,in𝐻^−1/2ℝ^𝑛−1,as𝜈⟶+∞. (4.38) The same analysis performed to the BVP (1.2)-(1.3) can be applied to the BVP solved by 𝑢_𝜈, for each 𝜈; in particular, we find that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈 satisfies the analogue to (4.10), where 𝐹 is replaced by 𝐹_𝜈. Because of the regularity of 𝑢_𝜈, formula (4.18) can be used to represent the normal commutator term [𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁]𝑢^𝐼_𝜈. Directly from system (𝛾+𝐿+𝐵)𝑢_𝜈=𝐹_𝜈, 𝜕₁𝑢^𝐼_𝜈 can be represented in terms of tangential derivatives of 𝑢_𝜈 only, as follows:𝜕₁𝑢^𝐼_𝜈=𝐴₁^{𝐼,𝐼−1}𝐹^𝐼_𝜈+ℒ^𝐼𝑢_𝜈, (4.39) where ℒ^𝐼=ℒ^𝐼(𝑥,𝑍,𝛾) denotes the tangential partial differential operatorℒ^𝐼𝑢_𝜈∶=−𝐴₁^{𝐼,𝐼−1}𝛾𝑢^𝐼_𝜈+𝐻₁𝑍₁𝑢_𝜈^𝐼𝐼+^𝑛_𝑗=2𝐴_𝑗𝑍_𝑗𝑢_𝜈+𝐵𝑢_𝜈^𝐼, (4.40) and we have set 𝐻₁∶=𝑥₁⁻¹𝐴₁^{𝐼,𝐼𝐼} (recall that 𝐻₁∈𝐶^∞₍₀₎(ℝ^𝑛₊) since 𝐴^{𝐼,𝐼𝐼}_1∣𝑥1=0=0). Inserting (4.39) into (4.18), written for 𝑤=𝑢^𝐼_𝜈, leads to𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑢^𝐼_𝜈=𝑞_𝑚,𝛿𝑥,𝑍,𝛾𝐴₁^{𝐼,𝐼−1}𝐹^𝐼_𝜈+ℒ^𝐼𝑢_𝜈. (4.41) On the other hand, plugging 𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)+𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝑟_𝑚,𝛿(𝑍,𝛾)=𝐼 into (4.41) gives𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑢^𝐼_𝜈=𝑞_𝑚,𝛿𝑥,𝑍,𝛾𝐴₁^{𝐼,𝐼−1}𝐹^𝐼_𝜈+𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈+𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝑟_𝑚,𝛿𝑍,𝛾𝑢_𝜈. (4.42) Now, we let 𝜈→+∞.

From (1.2) (written for 𝑢 and 𝑢_𝜈), and using that 𝑢^♯_𝜈→𝑢^♯ in 𝐿²(ℝ^𝑛), one finds 𝐴₁𝜕₁𝑢_𝜈^♯⟶𝐴₁𝜕₁𝑢^♯,in𝒮^′ℝ^𝑛, (4.43) hence (because ♯⁻¹ is a linear continuous operator from 𝒮^′(ℝ^𝑛)⊂𝒟^′(ℝ^𝑛) to 𝒟^′(ℝ^𝑛₊))𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝐴₁^𝐼,𝐼𝜕₁𝑢^𝐼_𝜈⟶𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝐴₁^𝐼,𝐼𝜕₁𝑢^𝐼,in𝒟^′ℝ^𝑛₊. (4.44) On the other hand, 𝑢^♯_𝜈→𝑢^♯ in 𝐿²(ℝ^𝑛) implies that𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈⟶𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)(𝑢),in𝒟^′ℝ^𝑛₊, (4.45) hence𝐴₁^𝐼,𝐼𝜕₁𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_𝜈⟶𝐴₁^𝐼,𝐼𝜕₁𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼,in𝒟^′ℝ^𝑛₊. (4.46) Adding (4.44), (4.46) then gives𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑢^𝐼_𝜈⟶𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑢^𝐼,in𝒟^′ℝ^𝑛₊. (4.47) As to the right-hand side of (4.42), all the involved operators, acting on 𝐹_𝜈 and 𝑢_𝜈, are conormal. Hence the 𝐿²-convergences 𝑢_𝜈→𝑢 and 𝐹_𝜈→𝐹 and the fact that conormal operators continuously extend to the space of distributions 𝑢∈𝒟^′(ℝ^𝑛₊), for which 𝑢^♯∈𝒮^′(ℝ^𝑛), give the convergences𝑞_𝑚,𝛿𝑥,𝑍,𝛾𝐴₁^{𝐼,𝐼−1}𝐹^𝐼_𝜈⟶𝑞_𝑚,𝛿𝑥,𝑍,𝛾𝐴₁^{𝐼,𝐼−1}𝐹^𝐼,𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈⟶𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢,𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝑟_𝑚,𝛿𝑍,𝛾𝑢_𝜈⟶𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝑟_𝑚,𝛿𝑍,𝛾𝑢,in𝒟^′ℝ^𝑛₊. (4.48) Therefore, the uniqueness of the limit in 𝒟^′(ℝ^𝑛₊) together with (4.47), (4.48) implies that (4.42) holds true for the 𝐿²-solution 𝑢 of (1.2)-(1.3), that is,𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐴₁^𝐼,𝐼𝜕₁𝑢^𝐼=𝑞_𝑚,𝛿𝑥,𝑍,𝛾𝐴₁^{𝐼,𝐼−1}𝐹^𝐼+𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢+𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝑟_𝑚,𝛿𝑍,𝛾𝑢. (4.49) Let us come back to the commutator term [𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿+𝐵]𝑢 appearing in the interior equation (4.10).

Substituting (4.15), (4.49) into (4.14) gives for this term the following representation:𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿+𝐵𝑢=𝜌_𝑚,𝛿𝑥,𝑍,𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢+𝜏_𝑚,𝛿𝑥,𝑍,𝛾𝑢+𝜂_𝑚,𝛿𝑥,𝑍,𝛾𝐹, (4.50) where we have set for short𝜌_𝑚,𝛿𝑥,𝑍,𝛾∶=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿_tan+𝐵𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)+𝑞_𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)0,𝜏_𝑚,𝛿𝑥,𝑍,𝛾∶=𝑠_𝑚,𝛿+𝑞𝑥,𝑍,𝛾^𝑚,𝛿𝑥,𝑍,𝛾ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍)𝑟_𝑚,𝛿0,𝜂𝑍,𝛾_𝑚,𝛿𝑞𝑥,𝑍,𝛾∶=^𝑚,𝛿𝑥,𝑍,𝛾𝐴₁^{𝐼,𝐼−1}0.00 (4.51) Consequently, the interior equation (4.10) can be restated as𝛾+𝐿+𝐵+𝜌_𝑚,𝛿𝑥,𝑍,𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢+𝜏_𝑚,𝛿𝑥,𝑍,𝛾𝑢+𝜂_𝑚,𝛿𝑥,𝑍,𝛾𝐹=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝐹,inℝ^𝑛₊. (4.52) Since 𝐿_tan+𝐵 and ℒ^𝐼 are conormal operators with symbol in Γ¹, Lemma 4.5 and Proposition 4.8 imply that 𝜌_𝑚,𝛿(𝑥,𝑍,𝛾) is a conormal operator with symbol in Γ⁰, for each 0<𝛿≤1; moreover, it amounts that the family of symbols {𝜌_𝑚,𝛿}_0<𝛿≤1 is a bounded subset of Γ⁰. Therefore, 𝜌_𝑚,𝛿(𝑥,𝑍,𝛾) can be regarded, jointly with 𝐵, as a lower-order term in (4.52), as considered in the statement of Theorem 1.1.

Concerning the terms 𝜏_𝑚,𝛿(𝑥,𝑍,𝛾)𝑢, 𝜂_𝑚,𝛿(𝑥,𝑍,𝛾)𝐹, they can be both moved into the right-hand side of (4.52), to be treated as a part of the interior source term, as will be detailed in Section 4.5.

4.4. The Boundary Condition

Now we are going to seek for an appropriate boundary condition to be coupled with the interior equation (4.10), in order to state a BVP solved by 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢.

To this end, it is worthwhile to make an additional hypothesis about the smooth function 𝜒 involved in the definition of 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍). We assume that 𝜒 has the form∀𝑥=𝑥₁,𝑥^′∈ℝ^𝑛,𝜒(𝑥)=𝜒₁𝑥₁𝜒𝑥^′, (4.53) where 𝜒₁∈𝐶^∞(ℝ) and 𝜒∈𝐶^∞(ℝ^𝑛−1) are given positive even functions, to be chosen in such a way that conditions (4.2) are made satisfied.

As it was done for the analysis of the normal commutator (cf. Proposition 4.8), we start our reasoning by dealing with smooth functions. In this case, following closely the arguments employed to prove Proposition 4.8 and Lemma 4.9, we are able to get the following.

Proposition 4.10. Assume that 𝜒 obeys the assumptions (4.2), (4.53). Then, for all 𝛿∈]0,1], 𝛾≥1, and 𝑚∈ℕ, the function 𝑏^′_𝑚,𝛿(𝜉^′,𝛾) defined by 𝑏^′_𝑚,𝛿𝜉^′,𝛾∶=(2𝜋)^−𝑛_ℝ𝑛𝜆_𝛿^{𝑚−1,𝛾}𝜂₁,𝜂^′+𝜉^′𝑒^(⋅)1/2𝜒₁^∧1𝜂₁𝜒𝜂^′𝑑𝜂,∀𝜉^′∈ℝ^𝑛−1, (4.54) (where 𝜉^′∶=(𝜉₂,…,𝜉_𝑛) are the Fourier dual variables of the tangential variables 𝑥^′=(𝑥₂,…,𝑥_𝑛)) is a 𝛾-depending symbol in ℝ^𝑛−1 belonging to Γ^𝑚−1. Moreover, for all functions 𝑤∈𝐶^∞₍₀₎(ℝ^𝑛₊) there holds ∀𝑥^′∈ℝ^𝑛−1,𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑤_∣𝑥1=0𝑥^′=𝑏^′_𝑚,𝛿𝐷^′,𝛾𝑤_∣𝑥1=0𝑥^′, (4.55) where one has set 𝐷^′=(𝐷₂,…,𝐷_𝑛), 𝐷_𝑗=−𝑖𝜕_𝑗 (for 𝑗=2,…,𝑛) and 𝑏^′_𝑚,𝛿(𝐷^′,𝛾) stands for the ordinary pseudodifferential operator in ℝ^𝑛−1 with symbol 𝑏^′_𝑚,𝛿.
Eventually, the following estimates are satisfied by the symbol 𝑏^′_𝑚,𝛿(𝜉^′,𝛾): for all 𝛼^′=(𝛼₂,…,𝛼_𝑛)∈ℕ^𝑛−1 there exists a positive constant 𝐶_𝑚,𝛼′, independent of 𝛾 and 𝛿, such that 𝜕^𝛼′_𝜉′𝑏^′_𝑚,𝛿𝜉^′,𝛾≤𝐶_𝑚,𝛼′𝜆^{𝑚−1−|𝛼′}_𝛿^|,𝛾𝜉^′,∀𝜉^′∈ℝ^𝑛−1. (4.56)

Proof. That 𝑏^′_𝑚,𝛿 belongs to Γ^𝑚−1 immediately follows from estimates (4.56), using the (𝑛−1)-dimensional counterpart of (3.18).
Let 𝑤∈𝐶^∞₍₀₎(ℝ^𝑛₊); to find a symbol 𝑏^′_𝑚,𝛿 satisfying (4.55), from (4.3) we firstly compute 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑤^♯(𝑥)=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)𝑤^♯(𝑥)=ℱ⁻¹𝜆^{𝑚−1,𝛾}_𝜒,𝛿∗𝑤^♯(𝑥)=⟨ℱ⁻¹𝜆^{𝑚−1,𝛾}_𝜒,𝛿,𝑤^♯()⟩=ℱ𝑥−⋅₋₁𝜆_𝛿^{𝑚−1,𝛾},𝜒(⋅)𝑒^{(𝑥1−(⋅)1)/2}𝑤𝑒^{𝑥1−(⋅)1},𝑥^′−(⋅)^′,∀𝑥₁,𝑥^′∈ℝ^𝑛, (4.57) hence 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑤(𝑥)=ℱ⁻¹𝜆_𝛿^{𝑚−1,𝛾},𝜒(⋅)𝑒^−(⋅)1/2𝑤𝑥₁𝑒^−(⋅)1,𝑥^′−(⋅)^′=𝜆_𝛿^{𝑚−1,𝛾},ℱ⁻¹𝜒(⋅)𝑒^−(⋅)1/2𝑤𝑥₁𝑒^−(⋅)1,𝑥^′−(⋅)^′=(2𝜋)^−𝑛𝜆_𝛿^{𝑚−1,𝛾}(𝜉)𝑒^{𝑖𝜉⋅𝑦}𝜒𝑦𝑒^−𝑦1/2𝑤𝑥₁𝑒^−𝑦1,𝑥^′−𝑦^′𝑑𝑦𝑑𝜉,∀𝑥₁>0,∀𝑥^′∈ℝ^𝑛−1. (4.58) The regularity of 𝑤 legitimates all the above calculations. Setting 𝑥₁=0 in the last expression above, we deduce the corresponding expression for the trace on the boundary of 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑤𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑤_∣𝑥1=0𝑥^′=(2𝜋)^−𝑛𝜆_𝛿^{𝑚−1,𝛾}(𝜉)𝑒^{𝑖𝜉⋅𝑦}𝜒𝑦e^−𝑦1/2𝑤_∣𝑥1=0𝑥^′−𝑦^′𝑑𝑦𝑑𝜉. (4.59) Now we substitute (4.53) into the 𝑦-integral appearing in the last expression above; then Fubini's theorem gives 𝑒^{𝑖𝜉⋅𝑦}𝜒₁𝑦₁𝜒𝑦^′𝑒^−𝑦1/2𝑤_∣𝑥1=0𝑥^′−𝑦^′=𝑒𝑑𝑦_{𝑖𝜉′⋅𝑦′}𝑒^{𝑖𝜉1𝑦1}𝑒^−𝑦1/2𝜒₁𝑦₁𝑑𝑦₁𝜒𝑦^′𝑤_∣𝑥1=0𝑥^′−𝑦^′𝑑𝑦^′=𝑒^(⋅)1/2𝜒₁^∧1𝜉₁𝑒^{𝑖𝜉′⋅𝑦′}𝜒𝑦^′𝑤_∣𝑥1=0𝑥^′−𝑦^′𝑑𝑦^′, (4.60) where ∧₁ is used to denote the one-dimensional Fourier transformation with respect to 𝑦₁. Writing, by the inversion formula, (𝑤_∣𝑥1=0)(𝑥^′−𝑦^′)=(2𝜋)^−𝑛+1∫𝑒^{𝑖(𝑥′−𝑦′)⋅𝜂′}𝑤_∣𝑥1=0(𝜂^′)𝑑𝜂^′ and using once more Fubini's theorem, we further obtain 𝑒^{𝑖𝜉′⋅𝑦′}𝜒𝑦^′𝑤_∣𝑥1=0𝑥^′−𝑦^′𝑑𝑦^′=(2𝜋)^−𝑛+1𝑒^{𝑖𝜉′⋅𝑦′}𝜒𝑦^′𝑒^{𝑖(𝑥′−𝑦′)⋅𝜂′}𝑤_∣𝑥1=0𝜂^′𝑑𝜂^′𝑑𝑦^′=𝑒^{𝑖𝑥′⋅𝜂′}(2𝜋)^−𝑛+1𝑒^{𝑖(𝜉′−𝜂′)⋅𝑦′}𝜒𝑦^′𝑑𝑦^′𝑤_∣𝑥1=0𝜂^′𝑑𝜂^′=(2𝜋)^−𝑛+1𝑒^{𝑖𝑥′⋅𝜂′}𝜒𝜉^′−𝜂^′𝑤_∣𝑥1=0𝜂^′𝑑𝜂^′; (4.61)∧ is used here to denote the (𝑛−1)-dimensional Fourier transformation with respect to 𝑥^′. Inserting (4.60), (4.61) into (4.59) then leads to 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑤_∣𝑥1=0𝑥^′=(2𝜋)^−𝑛𝜆_𝛿^{𝑚−1,𝛾}(𝜉)𝑒^(⋅)1/2𝜒₁^∧1𝜉₁(2𝜋)^−𝑛+1𝑒^{𝑖𝑥′⋅𝜂′}𝜒𝜉^′−𝜂^′𝑤_∣𝑥1=0𝜂^′𝑑𝜂^′𝑑𝜉. (4.62) Because (𝑒^(⋅)1/2𝜒₁)^∧1∈𝒮(ℝ), 𝜒∈𝒮(ℝ^𝑛−1), and 𝑤_∣𝑥1=0∈𝒮(ℝ^𝑛−1), the double integral 𝑒^{𝑖𝑥′⋅𝜂′}𝜆_𝛿^{𝑚−1,𝛾}(𝜉)𝑒^(⋅)1/2𝜒₁^∧1𝜉₁𝜒𝜉^′−𝜂^′𝑤_∣𝑥1=0𝜂^′𝑑𝜂^′𝑑𝜉 (4.63) converges absolutely; hence Fubini's theorem allows to exchange the order of the integrations in (4.62) and find 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑤_∣𝑥1=0𝑥^′=(2𝜋)^−𝑛+1𝑒^{𝑖𝑥′⋅𝜂′}𝑏^′_𝑚,𝛿𝜂^′,𝛾𝑤_∣𝑥1=0𝜂^′𝑑𝜂^′, (4.64) where 𝑏^′_𝑚,𝛿(𝜂^′,𝛾) is defined by (4.54). This shows the identity (4.55).
The proof of estimates (4.56) is similar to that of estimates (4.32) in Lemma 4.9 (see Appendix A); so we will omit it.

Let us now illustrate how formula (4.55) can be used to derive the desired boundary condition satisfied by 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢.

Again, let 𝑢 be the 𝐿²-solution to the original BVP (1.2)-(1.3) and {𝑢_𝜈}^+∞_𝜈=1 the corresponding sequence in 𝐶^∞₍₀₎(ℝ^𝑛₊), approximating 𝑢 in the sense of (4.36).

The last convergence in (4.36) and the Sobolev continuity of standard pseudodifferential operators gives in particular that𝑏^′_𝑚,𝛿𝐷^′,𝛾𝑢^𝐼_{𝜈∣𝑥1=0}⟶𝑏^′_𝑚,𝛿𝐷^′,𝛾𝑢^𝐼_∣𝑥1=0,in𝐻^−𝑚+1/2ℝ^𝑛−1. (4.65) On the other hand, (4.36) and (4.52) (written for 𝑢 and 𝑢_𝜈) can be used to prove that (𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼)_∣𝑥1=0 and (𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_𝜈)_∣𝑥1=0, for each 𝜈, are traces well defined in 𝐻^−1/2(ℝ^𝑛−1) and the convergence𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_{𝜈∣𝑥1=0}⟶𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_∣𝑥1=0 (4.66) holds true, at least in 𝒟^′(ℝ^𝑛−1). The proof of this assertion is postponed to Appendix A (see Lemma A.2).

Since 𝑢_𝜈 are smooth functions, from Proposition 4.10, it comes that for each 𝜈:𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_{𝜈∣𝑥1=0}=𝑏^′_𝑚,𝛿𝐷^′,𝛾𝑢^𝐼_{𝜈∣𝑥1=0}. (4.67) Then, letting 𝜈→+∞, (4.65) and (4.66) yield𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_∣𝑥1=0=𝑏^′_𝑚,𝛿𝐷^′,𝛾𝑢^𝐼_∣𝑥1=0. (4.68) Recalling that 𝑀=(𝐼_𝑑,0), from the boundary condition (1.3) and (4.68), we immediately find𝑀𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_∣𝑥1=0=𝑀𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_∣𝑥1=0=𝑀𝑏^′_𝑚,𝛿𝐷^′,𝛾𝑢^𝐼_∣𝑥1=0=𝑏^′_𝑚,𝛿𝐷^′,𝛾𝑀𝑢_∣𝑥1=0=𝑏^′_𝑚,𝛿𝐷^′,𝛾𝐺. (4.69)

4.5. Derivation of the Conormal Regularity at the Order 𝑚

We are now in the position to get the desired conormal regularity of the solution 𝑢 to (1.2)-(1.3), under the assumptions that 𝐹∈ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾(ℝ^𝑛₊), 𝐺∈𝐻_𝛾^𝑠+𝑚(ℝ^𝑛−1). From now on, assume that 𝛾≥𝛾_𝑚−1; so far, from the inductive hypothesis we know that, for such a 𝛾, 𝑢 belongs to 𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊), 𝑢^𝐼_∣𝑥1=0∈𝐻_𝛾^𝑚−1(ℝ^𝑛−1) and the estimate (4.1) is satisfied (see the end of Section 4.1). Because of the calculations made in Sections 4.3 and 4.4, it follows that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢 is an 𝐿² solution of the problem𝛾+𝐿+𝐵+𝜌_𝑚,𝛿𝑥,𝑍,𝛾𝜆^{𝛾,𝑚−1}_𝜒,𝛿(𝑍)𝑢=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)−𝜂_𝑚,𝛿𝑥,𝑍,𝛾𝐹−𝜏_𝑚,𝛿𝑥,𝑍,𝛾𝑢,inℝ^𝑛₊,𝑀𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢=𝑏^′_𝑚,𝛿𝐷^′,𝛾𝐺,onℝ^𝑛−1. (4.70)(4.71) The previous one reads as the original BVP (1.2)-(1.3) solved by 𝑢, where the role of the lower-order term ℬ is played here by the conormal operator 𝐵+𝜌_𝑚,𝛿(𝑥,𝑍,𝛾). As already discussed in the end of Section 4.3, in view of Proposition 3.8 and Lemma 4.5, the symbol of 𝐵+𝜌_𝑚,𝛿(𝑥,𝑍,𝛾) belongs to Γ⁰, and {𝜌_𝑚,𝛿}_0<𝛿≤1 is a bounded subset of Γ⁰.

As regards to the terms 𝜏_𝑚,𝛿(𝑥,𝑍,𝛾)𝑢 and 𝜂_𝑚,𝛿(𝑥,𝑍,𝛾)𝐹 appearing into the right-hand side of (4.70), they can be regarded as a part of the source term in the interior equation (4.70) (this is the reason why they have been moved in the right-hand side of (4.70)).

Let us firstly focus on 𝜏_𝑚,𝛿(𝑥,𝑍,𝛾)𝑢. After Lemma 4.5 and Proposition 3.8 (see also Proposition 4.8, Remark 4.7, and formulas (4.16), (4.51)), we know that for any 𝑘∈ℕ the operators [𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍),𝐿_tan+𝐵]𝜆_𝛿^{−𝑚+1,𝛾}(𝑍) and 𝑞_𝑚,𝛿(𝑥,𝑍,𝛾)ℒ^𝐼𝜆_𝛿^{−𝑚+1,𝛾}(𝑍) extend as linear-bounded mappings from 𝐻^𝑘_tan,𝛾(ℝ^𝑛₊) into itself, and their operator norms are uniformly bounded with respect to 𝛾 and 𝛿. Combining with the result of Lemma 4.2, it follows that a positive constant 𝐶_𝑠>0, independent of 𝛾 and 𝛿, can be found in such a way that𝜏_𝑚,𝛿𝑥,𝑍,𝛾𝑢_{ℋ𝑠,𝑟tan,𝛾(ℝ𝑛+)}≤𝜏_𝑚,𝛿𝑥,𝑍,𝛾𝑢_{𝐻𝑠tan,𝛾(ℝ𝑛+)}≤𝐶_𝑠𝛾^𝑠‖𝑢‖_𝐿²_(ℝ^𝑛₊₎. (4.72) Concerning now the term (𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)−𝜂_𝑚,𝛿(𝑥,𝑍,𝛾))𝐹, after Lemma 4.5 and Proposition 4.8 we already know that the symbol 𝜂_𝑚,𝛿(𝑥,𝜉,𝛾) has order 𝑚−2 and obeys the same decay estimates as in (4.19). From (3.17) it then follows that {𝜂_𝑚,𝛿}_0<𝛿≤1 is a bounded subset of Γ^𝑚−1; because {𝜆^{𝑚−1,𝛾}_𝜒,𝛿}_0<𝛿≤1 is also a bounded subset of Γ^𝑚 (as a consequence of (4.4) and (3.17) again), after Proposition 3.10 we conclude that there exists a positive constant 𝐶_{𝑚,𝑠,𝑟} such that𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)−𝜂_𝑚,𝛿𝑥,𝑍,𝛾𝐹_{ℋ𝑠,𝑟tan,𝛾(ℝ𝑛+)}≤𝐶_{𝑚,𝑠,𝑟}‖𝐹‖_ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾(ℝ^𝑛₊₎. (4.73) As regards to the boundary datum 𝑏^′_𝑚,𝛿(𝐷^′,𝛾)𝐺 in (4.71), the family of symbols {𝑏^′_𝑚,𝛿}_0<𝛿≤1 in ℝ^𝑛−1 defines a bounded subset of Γ^𝑚; this follows from estimates (4.56) and the inequality (3.17) (in dimension 𝑛−1). Therefore, the Sobolev continuity of ordinary pseudodifferential operators in ℝ^𝑛−1 implies the existence of a positive constant 𝐶_𝑚,𝑠, independent of 𝛾 and 𝛿, such that:𝑏^′_𝑚,𝛿𝐷^′,𝛾𝐺_{𝐻𝑠𝛾(ℝ𝑛−1)}≤C_𝑚,𝑠‖𝐺‖_𝐻𝛾^𝑠+𝑚_(ℝ^𝑛−1₎. (4.74) From the assumptions made about the BVP (1.2)-(1.3) in Theorem 1.1, we find some constants 𝛾_𝑚≥1,  𝐶_𝑚>0 such that for all 𝛾≥𝛾_𝑚∶=max{𝛾_𝑚−1,𝛾_𝑚} and every 𝛿∈]0,1], 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢 is the only 𝐿²-solution of (4.70)-(4.71) and obeys the estimate𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢²_{𝐿2ℝ𝑛+}+𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_∣𝑥1=0²_{𝐿2ℝ𝑛−1}≤𝐶_𝑚1𝛾^2𝑠+1𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)−𝜂_𝑚,𝛿𝑥,𝑍,𝛾𝐹−𝜏_𝑚,𝛿𝑥,𝑍,𝛾𝑢²_{ℋ𝑠,𝑟tan,𝛾ℝ𝑛+}+1𝛾^2𝑠𝑏^′_𝑚,𝛿𝐷^′,𝛾𝐺²_{𝐻s𝛾ℝ𝑛−1}. (4.75) We remark that, according to the statement of Theorem 1.1, the constants 𝛾_𝑚, 𝐶_𝑚 are only dependent on a 𝛿,𝛾-uniform upper bound of the 𝑘th-order norm (3.14), computed on the symbol of 𝐵+𝜌_𝑚,𝛿(𝑥,𝑍,𝛾), besides the coefficients A_𝑗, 1≤𝑗≤𝑛, and the integer numbers 𝑟,𝑠.

Then, using (4.72)–(4.74) and (1.7) to estimate the right-hand side of (4.75), we get𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢²_{𝐿2ℝ𝑛+}+𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_∣𝑥1=0²_{𝐿2ℝ𝑛−1}≤𝐶_{𝑚,𝑠,𝑟}1𝛾^2𝑠+1‖𝐹‖²_ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾ℝ^𝑛₊+1𝛾^2𝑠‖𝐺‖²_{𝐻𝛾𝑠+𝑚ℝ𝑛−1}, (4.76) where 𝐶_{𝑚,𝑠,𝑟} is a suitable positive constant independent of 𝛾≥𝛾_𝑚 and 𝛿. Since the 𝐿²-norms ‖𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢‖_{𝐿2(ℝ𝑛+)} are bounded by (4.76), uniformly with respect to 𝛿∈]0,1], Corollary 4.4 gives 𝑢∈𝐻^𝑚_tan,𝛾(ℝ^𝑛₊).

As to the Sobolev regularity of the trace on the boundary of the noncharacteristic component 𝑢^𝐼 of the solution, the estimate (4.76) gives a bound of ‖(𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼)_∣𝑥1=0‖_{𝐿2(ℝ𝑛−1)}=‖𝑏^′_𝑚,𝛿(𝐷^′,𝛾)(𝑢^𝐼_∣𝑥1=0)‖_{𝐿2(ℝ𝑛−1)} uniform with respect to 𝛿∈]0,1] (cf. (4.68)). Then 𝑢^𝐼_∣𝑥1=0∈𝐻^𝑚_𝛾(ℝ^𝑛−1) can be derived from the next result, the proof of which will be given in Appendix A.

Lemma 4.11. For 𝑚∈ℕ and 𝛿∈]0,1], let 𝑏^′_𝑚,𝛿(𝜉^′,𝛾) be defined by (4.54). Then there exists a symbol 𝛽_𝑚,𝛿(𝜉^′,𝛾)∈Γ^𝑚−2 such that 𝑏^′_𝑚,𝛿𝜉^′,𝛾=𝜆_𝛿^{𝑚−1,𝛾}𝜉^′+𝛽_𝑚,𝛿𝜉^′,𝛾,∀𝜉^′∈ℝ^𝑛−1. (4.77) In addition, the symbol 𝛽_𝑚,𝛿 satisfies the following estimates: for every 𝛼^′∈ℕⁿ⁻¹, there exists a positive constant 𝐶_𝑚,𝛼′, independent of 𝛾 and 𝛿, such that 𝜕^𝛼′_𝜉′𝛽_𝑚,𝛿𝜉^′,𝛾≤𝐶_𝑚,𝛼′𝜆^{𝑚−2−|𝛼′}_𝛿^|,𝛾𝜉^′,∀𝜉^′∈ℝ^𝑛−1. (4.78)

Arguing as was done to derive Corollary 4.4 from Lemma 4.2, from Lemma 4.11 we deduce the following.

Corollary 4.12. For every positive integer 𝑚 and 𝛾≥1, 𝑣∈𝐻^𝑚_𝛾(ℝ^𝑛−1) if and only if 𝑣∈𝐻_𝛾^𝑚−1(ℝ^𝑛−1) and the set {‖𝑏^′_𝑚,𝛿(𝐷^′,𝛾)𝑣‖_{𝐿2(ℝ𝑛−1)}}_0<𝛿≤1 is bounded.

After the result of Corollary 4.12, we conclude that 𝑢^𝐼_∣𝑥1=0∈𝐻^𝑚_𝛾(ℝ^𝑛−1).

It remains to prove that the solution of (1.2)-(1.3) satisfies the a priori estimate (1.8) of order 𝑚. From estimate (4.76) and the use of the identities (4.7), (4.68), (4.77), we also deduce 𝛾𝜆_𝛿^{𝑚−1,𝛾}(𝑍)𝑢²_{𝐿2ℝ𝑛+}+𝜆_𝛿^{𝑚−1,𝛾}𝐷^′𝑢^𝐼_∣𝑥1=0²_{𝐿2ℝ𝑛−1}≤𝐶^′_{𝑚,𝑠,𝑟}1𝛾^2𝑠+1‖𝐹‖²_ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾ℝ^𝑛₊+1𝛾^2𝑠‖𝐺‖²_{𝐻𝛾𝑠+𝑚ℝ𝑛−1}+2𝛾𝑟_𝑚,𝛿𝑍,𝛾𝑢²_{𝐿2ℝ𝑛+}+2𝛽_𝑚,𝛿𝐷^′,𝛾𝑢^𝐼_∣𝑥1=0²_{𝐿2ℝ𝑛−1}, (4.79) where the positive constant 𝐶^′_{𝑚,𝑠,𝑟} is again independent of 𝛾≥𝛾_𝑚 and 𝛿. On the other hand, using Lemma 4.2 and that {𝛽_𝑚,𝛿}_0<𝛿≤1 is a bounded subset of Γ^𝑚−1 (that follows at once from Lemma 4.11 and inequality (3.17)), one can estimate𝛾𝑟_𝑚,𝛿𝑍,𝛾𝑢²_{𝐿2ℝ𝑛+}+𝛽_𝑚,𝛿𝐷^′,𝛾𝑢^𝐼_∣𝑥1=0²_{𝐿2ℝ𝑛−1}≤𝐶_𝑚𝛾‖𝑢‖²_{𝐿2ℝ𝑛+}+𝑢^𝐼_∣𝑥1=0²_{𝐻𝛾𝑚−1ℝ𝑛−1} (4.80) with positive constant 𝐶_𝑚 independent, once again, of 𝛾 and 𝛿.

In the end, combining (4.79), (4.80) and using the a priori estimate (4.1) of order 𝑚−1 on 𝑢, which holds true by the inductive assumption, we conclude that there exists a constant 𝐶^′′_{𝑚,𝑠,𝑟}>0 such that𝛾𝜆_𝛿^{𝑚−1,𝛾}(𝑍)𝑢²_{𝐿2ℝ𝑛+}+𝜆_𝛿^{𝑚−1,𝛾}𝐷^′𝑢^𝐼_∣𝑥1=0²_{𝐿2ℝ𝑛−1}≤𝐶^′′_{𝑚,𝑠,𝑟}1𝛾^2𝑠+1‖𝐹‖²_ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾ℝ^𝑛₊+1𝛾^2𝑠‖𝐺‖²_{𝐻𝛾𝑠+𝑚ℝ𝑛−1} (4.81) for all 𝛾≥𝛾_𝑚 and 𝛿∈]0,1].

The energy estimate (1.8) of order 𝑚 hence follows by letting 𝛿→0 into the left-hand side of (4.81) (for an arbitrarily fixed 𝛾≥𝛾_𝑚) and exploiting the results of Propositions 3.1 and 3.2.

Appendices

A. Proof of Some Technical Lemmata

A.1. Proof of Lemma 4.1

The proof that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿 obeys estimates (4.4) relies on the following 𝛾-weighted version of Peetre's inequality.

For all 𝑠∈ℝ, 𝛾≥1, and 𝜉,𝜂∈ℝ^𝑛𝜆^𝑠,𝛾(𝜉)≤2^|𝑠|𝜆^𝑠,𝛾𝜉−𝜂𝜆^|𝑠|𝜂. (A.1) The proof of (A.1) follows by an easy account of the parameter 𝛾 into the arguments used to prove the classical Peetre's inequality (cf., e.g., [11], [15, Lemma 1.18]). As an easy consequence of (A.1) and (3.15), it can be also proved that the following holds:𝜆_𝛿^{𝑠−1,𝛾}(𝜉)≤2^|𝑠|+1𝜆_𝛿^{𝑠−1,𝛾}𝜉−𝜂𝜆^|𝑠|𝜂𝜆¹𝛿𝜂,∀𝜉,𝜂∈ℝ^𝑛, (A.2) for an arbitrary 𝛿∈]0,1].

For an arbitrary 𝛼∈ℕ^𝑛, we use (A.2) with 𝑠=𝑚−|𝛼| and (3.16) to find𝜕^𝛼𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝜉)=ℱ⁻¹𝜒∗𝜕^𝛼𝜆_𝛿^{𝑚−1,𝛾}(𝜉)≤ℱ⁻¹𝜒𝜂𝜕^𝛼𝜆_𝛿^{𝑚−1,𝛾}𝜉−𝜂𝑑𝜂≤𝐶_𝑚,𝛼2^{𝑚+1+|𝛼|}𝜆_𝛿^{𝑚−1−|𝛼|,𝛾}(𝜉)ℱ⁻¹𝜒𝜂𝜆^{𝑚+1+|𝛼|}𝜂𝑑𝜂. (A.3) Since ℱ⁻¹𝜒𝜆^{𝑚+1+|𝛼|}∈𝒮(ℝ^𝑛), the integral above is finite; moreover it does not depend on 𝛾 and 𝛿. Therefore (A.3) is precisely an estimate of type (4.4). That 𝜆^{𝑚−1,𝛾}_𝜒,𝛿∈Γ^𝑚−1 follows at once from the same arguments applied to 𝜆_𝛿^{𝑚−1,𝛾} (see (3.19)).

A.2. Proof of Lemma 4.5

The symbol of Op^𝛾(𝑎_𝛿)Op^𝛾(𝑏) is 𝑎_𝛿#𝑏. Because of (3.17), we already know that {𝑎_𝛿}_0<𝛿≤1 is a bounded subset of Γ^𝑟. Then the rules of symbolic calculus give that 𝑎_𝛿#𝑏 is a symbol in Γ^𝑟+𝑙, for every 𝛿; moreover, from (3.23) one has𝑎_𝛿#𝑏𝑥,𝜉,𝛾=_|𝛼|<𝑁(−𝑖)^|𝛼|𝜕𝛼!^𝛼𝜉𝑎_𝛿𝑥,𝜉,𝛾𝜕^𝛼_𝑥𝑏𝑥,𝜉,𝛾+ℛ_𝑁,𝛿𝑥,𝜉,𝛾, (A.4) for every integer 𝑁≥1, and {ℛ_𝑁,𝛿}_0<𝛿≤1 is a bounded subset of Γ^{𝑟+𝑙−𝑁}.

In particular, setting 𝑁=1 in (A.4) gives𝑎_𝛿#𝑏𝑥,𝜉,𝛾=𝑎_𝛿𝑥,𝜉,𝛾𝑏𝑥,𝜉,𝛾+ℛ_1,𝛿𝑥,𝜉,𝛾, (A.5) where ℛ_1,𝛿∈Γ^{𝑟+𝑙−1} and for all 𝛼,𝛽∈ℕ^𝑛 there exists 𝐶_{𝑟,𝑙,𝛼,𝛽}>0, independent of 𝛾 and 𝛿, such that𝜕^𝛼_𝜉𝜕^𝛽_𝑥ℛ_1,𝛿𝑥,𝜉,𝛾≤𝐶_{𝑟,𝑙,𝛼,𝛽}𝜆^{𝑟+𝑙−1−|𝛼|,𝛾}(𝜉),∀𝑥,𝜉∈ℝ^𝑛. (A.6) Then, combining (A.6) with the right inequality in (3.18), we easily derive that ℛ_1,𝛿 satisfies the estimates (4.9). By Leibniz's rule and the use of (4.8), one can trivially check that estimates (4.9) are satisfied by the product of symbols 𝑎_𝛿(𝑥,𝜉,𝛾)𝑏(𝑥,𝜉,𝛾) as well. That 𝑎_𝛿#𝑏∈Γ^{𝑟+𝑙−1} follows again from estimates (4.9) themselves and the left inequality in (3.18).

As regards to the remaining assertions about the symbols of the operators Op^𝛾(𝑎_𝛿)Op^𝛾(𝑏)𝜆_𝛿^{−𝑚+1,𝛾}(𝐷) and [Op^𝛾(𝑎_𝛿), Op^𝛾(𝑏)]𝜆_𝛿^{−𝑚+1,𝛾}(𝐷) (in the case of scalar-valued 𝑎_𝛿), they follow at once from Leibniz's rule and Proposition 3.6, combined with the estimates (4.9) and (3.20).

A.3. Proof of Lemma 4.9

Recall that we have defined for each 𝑘=1,…,𝑛𝑞_{𝑘,𝑚,𝛿}𝑥,𝜉,𝛾∶=(2𝜋)^−𝑛_ℝ𝑛𝑏̂_𝑘𝑥,𝜂𝐷_𝑘𝜆_𝛿^{𝑚−1,𝛾}𝜉−𝜂𝑑𝜂, (A.7) where the functions 𝑏_𝑘=𝑏_𝑘(𝑥,𝑦) (cf. (4.24)) are given in 𝐶^∞(ℝ^𝑛×ℝ^𝑛), have bounded derivatives in ℝ^𝑛×ℝ^𝑛, and satisfy for all 𝑥∈ℝ^𝑛supp𝑏_𝑘(𝑥,⋅)⊆𝑦≤𝜀, (A.8) with some constant 𝜀>1. Recall also that 𝑏̂_𝑘(𝑥,𝜁) denotes the partial Fourier transform of 𝑏_𝑘(𝑥,𝑦) with respect to 𝑦.

In the sequel, we remove the subscript 𝑘 for simplicity.

The following lemma is concerned with the behavior at infinity of 𝑏̂(𝑥,𝜁).

Lemma A.1. Let the function 𝑏=𝑏(𝑥,𝑦)∈𝐶^∞(ℝ^𝑛×ℝ^𝑛) obey all of the preceding assumptions. Then, for every positive integer 𝑁 and all multi-indices 𝛼∈ℕ^𝑛, there exists a positive constant 𝐶_𝑁,𝛼 such that 1+𝜁^2𝑁𝜕^𝛼_𝑥𝑏̂(𝑥,𝜁)≤𝐶_𝑁,𝛼,∀𝑥,𝜁∈ℝ^𝑛. (A.9)

Proof. Since for each 𝑥∈ℝ^𝑛, the function 𝑏(𝑥,⋅) has compact support (independent of 𝑥), integrating by parts we get for an arbitrary integer 𝑁>01+𝜁^2𝑁𝑏̂(𝑥,𝜁)=_{|𝛼|≤𝑁}𝑁!𝛼!(𝑁−|𝛼|)!_{{|𝑦|≤𝜀}}𝜁^2𝛼𝑒^{−𝑖𝜁⋅𝑦}𝑏𝑥,𝑦𝑑𝑦=_{|𝛼|≤𝑁}𝑁!(|𝛼!()𝛼!𝑁−|)−1^|𝛼|_{{|𝑦|≤𝜀}}𝑒^{−𝑖𝜁⋅𝑦}𝜕_𝑦^2𝛼𝑏𝑥,𝑦𝑑𝑦, (A.10) from which (A.9) trivially follows, using that 𝑦-derivatives of 𝑏(𝑥,𝑦) are bounded in ℝ^𝑛×ℝ^𝑛 by a positive constant independent of 𝑥.

We are going now to analyze the behavior at infinity of the derivatives of the symbol 𝑞_𝑚,𝛿(𝑥,𝜉,𝛾) defined as in (A.7), where 𝑏_𝑘 is replaced by 𝑏. For all multi-indices 𝛼,𝛽∈ℕ^𝑛, differentiation under the integral sign in (A.7) gives𝜕^𝛼_𝜉𝜕^𝛽_𝑥𝑞_𝑚,𝛿𝑥,𝜉,𝛾=−𝑖(2𝜋)^−𝑛𝜕^𝛽_𝑥𝑏̂𝑥,𝜂𝜕^{𝛼+𝑒𝑘}𝜆_𝛿^{𝑚−1,𝛾}𝜉−𝜂𝑑𝜂, (A.11) where 𝑒^𝑘∶=(0,…,1_}𝑘,…,0). Then using (3.16) and (A.9) and combining with (A.2), for 𝑠=𝑚−1−|𝛼|, we obtain𝜕^𝛼_𝜉𝜕^𝛽_𝑥𝑞_𝑚,𝛿𝑥,𝜉,𝛾≤𝐶_𝑁,𝛽𝐶_𝑚,𝛼𝜆^−2𝑁𝜂𝜆_𝛿^{𝑚−2−|𝛼|,𝛾}𝜉−𝜂𝑑𝜂≤𝐶_{𝑁,𝑚,𝛼,𝛽}𝜆_𝛿^{𝑚−2−|𝛼|,𝛾}(𝜉)𝜆^{𝑚+2+|𝛼|−2𝑁}𝜂𝑑𝜂, (A.12) where the integral in the last line is finite, provided that the integer 𝑁 is taken to be sufficiently large. This provides the estimate (4.32), with constant 𝐶_{𝑁,𝑚,𝛼,𝛽}∫𝜆^{𝑚+2+|𝛼|−2𝑁}(𝜂)𝑑𝜂 independent of 𝛾 and 𝛿.

A.4. Proof of Lemma 4.11

Setting for short𝜙(𝑥)∶=𝑒^𝑥1/2𝜒₁𝑥₁𝜒𝑥^′, (A.13) the symbol (4.54) can be rewritten as𝑏^′_𝑚,𝛿𝜉^′,𝛾=(2𝜋)^−𝑛𝜆_𝛿^{𝑚−1,𝛾}𝜂₁,𝜂^′+𝜉^′𝜙𝜂𝑑𝜂. (A.14) By a first-order Taylor expansion of 𝜂↦𝜆_𝛿^{𝑚−1,𝛾}(𝜂₁,𝜂^′+𝜉^′) about 𝜂=0, we further obtain𝑏^′_𝑚,𝛿𝜉^′=(),𝛾2𝜋^−𝑛𝜆_𝛿^{𝑚−1,𝛾}𝜉^′𝜙𝜂𝑑𝜂−𝑖(2𝜋)^𝑛−𝑛_𝑗=1¹₀𝜕_𝑗𝜆_𝛿^{𝑚−1,𝛾}𝑡𝜂₁,𝑡𝜂^′+𝜉^′𝑑𝑡𝜕_𝑗𝜙𝜂𝑑𝜂. (A.15) Then, using (2𝜋)^−𝑛∫𝜙(𝜂)𝑑𝜂=𝜙(0)=1 and the definition of 𝜙 (see (4.2)), (A.15) yields (4.77), where𝛽_𝑚,𝛿𝜉^′,𝛾∶=−𝑖(2𝜋)^𝑛−𝑛_𝑗=1¹₀𝜕_𝑗𝜆_𝛿^{𝑚−1,𝛾}𝑡𝜂₁,𝑡𝜂^′+𝜉^′𝑑𝑡𝜕_𝑗𝜙𝜂𝑑𝜂. (A.16) To prove (4.78), differentiation under the integral sign of (A.16) gives for an arbitrary 𝛼^′∈ℕ^𝑛−1𝜕^𝛼′_𝜉′𝛽_𝑚,𝛿𝜉^′,𝛾=−𝑖(2𝜋)^𝑛−𝑛_𝑗=1¹₀𝜕^{𝑒𝑗+(0,𝛼′)}𝜆_𝛿^{𝑚−1,𝛾}𝑡𝜂₁,𝑡𝜂^′+𝜉^′𝑑𝑡𝜕_𝑗𝜙𝜂𝑑𝜂, (A.17) hence from (3.16) we get𝜕^𝛼′_𝜉′𝛽_𝑚,𝛿𝜉^′,𝛾≤𝐶_𝑚,𝛼′^𝑛_𝑗=1¹₀𝜆^{𝑚−2−|𝛼′}_𝛿^|,𝛾𝑡𝜂₁,𝑡𝜂^′+𝜉^′𝑑𝑡𝜕_𝑗𝜙𝜂𝑑𝜂. (A.18) Then, applying (A.2) (for 𝑠=𝑚−1−|𝛼^′|) to estimate the right-hand side of (A.18) and using that 𝜕_𝑗𝜙∈𝒮(ℝ^𝑛) for each 1≤𝑗≤𝑛, we get𝜕^𝛼′_𝜉′𝛽_𝑚,𝛿𝜉^′,𝛾≤𝐶^′_{𝑚,𝑁,𝛼′}𝜆^{𝑚−2−|𝛼′|,𝛾}𝜉^′𝜆^{𝑚+2+|𝛼′|−𝑁}𝜂𝑑𝜂, (A.19) for an arbitrary integer 𝑁>0 and 𝐶^′_{𝑚,𝑁,𝛼′}>0 independent of 𝛾 and 𝛿. This provides the estimate of type (4.78), once 𝑁 is chosen large enough to ensure the convergence of the last integral.

A.5. A Further Technical Result

We conclude this appendix with the following result, that was involved in the arguments given in Section 4.4.

Lemma A.2. Let 𝑢∈𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊) be a solution to (1.2)-(1.3), with data 𝐹∈ℋ^{𝑠+𝑚,𝑟+𝑚}_tan,𝛾(ℝ^𝑛₊), 𝐺∈𝐻_𝛾^𝑠+𝑚(ℝ^𝑛−1), such that 𝑢^𝐼_∣𝑥1=0∈𝐻_𝛾^𝑚−1(ℝ^𝑛−1), for a given integer 𝑚≥1. Let {𝑢_𝜈} be a sequence in 𝐶^∞₍₀₎(ℝ^𝑛₊) approximating the solution 𝑢 in the sense of (4.36). Then the trace (𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼)_∣𝑥1=0 is well defined in 𝐻^−1/2(ℝ^𝑛−1) and one has 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_{𝜈∣𝑥1=0}⟶𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_∣𝑥1=0,in𝒟^′ℝ^𝑛−1. (A.20)

Proof. Since 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍) is of order 𝑚−1, in view of Proposition 3.8 it follows from 𝑢∈𝐻^𝑚−1_tan,𝛾(ℝ^𝑛₊) that 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢∈𝐿²(ℝ^𝑛₊). We use (4.52) to find (𝐿+𝐵)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)−𝜂_𝑚,𝛿𝑥,𝑍,𝛾𝐹−𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢−𝜌_𝑚,𝛿𝑥,𝑍,𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢−𝜏_𝑚,𝛿𝑥,𝑍,𝛾𝑢, (A.21) where the operators involved in the right-hand side are defined by (4.51). Because of (4.73), (4.74) and since 𝜌_𝑚,𝛿(𝑥,𝑍,𝛾) is 𝐿²-bounded (by Proposition 3.8), from (A.21) we derive that (𝐿+𝐵)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢∈𝐿²(ℝ^𝑛₊). Then, it is known from [14] that the trace on the boundary of (𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢)^𝐼=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼 exists in 𝐻^−1/2(ℝ^𝑛−1); moreover the Green formula _ℝ𝑛+(𝐿+𝐵)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢𝑣=_ℝ𝑛+𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢(𝐿+𝐵)^∗𝑣+_ℝ𝑛−1𝐴₁𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_∣𝑥1=0𝑣_∣𝑥1=0𝑑𝑥^′ (A.22) holds true for all functions 𝑣∈𝐶^∞₍₀₎(ℝ^𝑛₊).
Notice that 𝑢_𝜈∈𝐶^∞₍₀₎(ℝ^𝑛₊) implies that (𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈)^♯=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)(𝑢^♯_𝜈)∈𝐿²(ℝ^𝑛), and then 𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈∈𝐿²(ℝ^𝑛₊). Therefore, starting from the same equation as (4.52), where 𝑢 and 𝐹 are replaced by 𝑢_𝜈 and 𝐹_𝜈∶=(𝛾+𝐿+𝐵)𝑢_𝜈, and arguing as before, one also gets (𝐿+𝐵)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈∈𝐿²(ℝ^𝑛₊); then (𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢^𝐼_𝜈)_∣𝑥1=0∈𝐻^−1/2(ℝ^𝑛−1), for each 𝜈, and (A.22) is fulfilled, where 𝑢 is replaced by 𝑢_𝜈.
Because the Green formulas hold for 𝑢 and 𝑢_𝜈, (A.20) is true, granted that the convergences _ℝ𝑛+(𝐿+𝐵)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈𝑣⟶_ℝ𝑛+(𝐿+𝐵)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢𝑣,_ℝ𝑛+𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈(𝐿+𝐵)^∗𝑣⟶_ℝ𝑛+𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢(𝐿+𝐵)^∗𝑣 (A.23) have been proved, whenever 𝑣∈𝐶^∞₍₀₎(ℝ^𝑛₊).
For each 𝜈, we use (A.21) (and a change of variables) to get _ℝ𝑛+(𝐿+𝐵)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢_𝜈𝑣=_ℝ𝑛𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)−𝜂_𝑚,𝛿𝑥,𝐷,𝛾𝐹^♯_𝜈−𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)𝑢^♯_𝜈−𝜌_𝑚,𝛿𝑥,𝐷,𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)𝑢^♯_𝜈−𝜏_𝑚,𝛿𝑥,𝐷,𝛾𝑢^♯_𝜈𝑣^♯=_ℝ𝑛𝐹^♯_𝜈𝑎𝑥,𝐷,𝛾^∗𝑣^♯+𝑢^♯_𝜈𝑏𝑥,𝐷,𝛾^∗𝑣^♯, (A.24) where we have set 𝑎𝑥,𝐷,𝛾∶=𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)−𝜂_𝑚,𝛿,𝑏𝑥,𝐷,𝛾𝑥,𝐷,𝛾∶=−𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)−𝜌_𝑚,𝛿𝑥,𝐷,𝛾𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝐷)−𝜏_𝑚,𝛿𝑥,𝐷,𝛾. (A.25) Repeating the same calculations on ∫(𝐿+𝐵)(𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢)𝑣 also gives _ℝ𝑛+(𝐿+𝐵)𝜆^{𝑚−1,𝛾}_𝜒,𝛿(𝑍)𝑢−𝑢_𝜈𝑣=_ℝ𝑛𝐹^♯_𝜈−𝐹^♯𝑎𝑥,𝐷,𝛾^∗𝑣^♯+𝑢^♯_𝜈−𝑢^♯𝑏𝑥,𝐷,𝛾^∗𝑣^♯. (A.26) Then the first convergence in (A.23) is proven, as a consequence of the convergences 𝐹^♯_𝜈→𝐹^♯, 𝑢^♯_𝜈→𝑢^♯ in 𝐿²(ℝ^𝑛) and Cauchy-Schwarz's inequality.
In a completely similar way, one can check the validity of the second convergence in (A.23).

Acknowledgment

The paper was supported by the national research project PRIN 2007 “Equations of Fluid Dynamics of Hyperbolic Type and Conservation Laws.”