Introduction

In this paper, we consider the double layer potential associated with the fundamental solution of a second order differential operator with constant coefficients. Unless otherwise specified, we assume that

$$n\in {\mathbb {N}}{\setminus }\{0,1\}\,,$$

where \({\mathbb {N}}\) denotes the set of natural numbers including 0. Let \(\alpha \in ]0,1]\), \(m\in {\mathbb {N}}\setminus \{0\}\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{m,\alpha }\). For the notation and standard properties of the Hölder and of the Schauder spaces \(C^{m,\alpha }\), we refer to [8, §2], [6, §2.6, 2.11]. Let \(\nu \equiv (\nu _{l})_{l=1,\dots ,n}\) denote the external unit normal to \(\partial \Omega\). Let \(N_{2}\) denote the number of multi-indexes \(\gamma \in {\mathbb {N}}^{n}\) with \(|\gamma |\le 2\). For each

$$\begin{aligned} {\textbf{a}}\equiv (a_{\gamma })_{|\gamma |\le 2}\in {\mathbb {C}}^{N_{2}}\,, \end{aligned}$$
(1.1)

we set

$$a^{(2)}\equiv (a_{lj} )_{l,j=1,\dots ,n}\qquad a^{(1)}\equiv (a_{j})_{j=1,\dots ,n}\qquad a\equiv a_{0}\,.$$

with \(a_{lj} \equiv 2^{-1}a_{e_{l}+e_{j}}\) for \(j\ne l\), \(a_{jj} \equiv a_{e_{j}+e_{j}}\), and \(a_{j}\equiv a_{e_{j}}\), where \(\{e_{j}:\,j=1,\dots ,n\}\) is the canonical basis of \({\mathbb {R}}^{n}\). We note that the matrix \(a^{(2)}\) is symmetric. Then, we assume that \({\textbf{a}}\in {\mathbb {C}}^{N_{2}}\) satisfies the following ellipticity assumption

$$\begin{aligned} \inf _{ \xi \in {\mathbb {R}}^{n}, |\xi |=1 }{\textrm{Re}}\,\left\{ \sum _{|\gamma |=2}a_{\gamma }\xi ^{\gamma }\right\} >0\,, \end{aligned}$$
(1.2)

and we consider the case in which

$$\begin{aligned} a_{lj} \in {\mathbb {R}}\qquad \forall l,j=1,\dots ,n\,. \end{aligned}$$
(1.3)

Then, we introduce the operators

$$\begin{aligned} P[{\textbf{a}},D]u\equiv & {} \sum _{l,j=1}^{n}\partial _{x_{l}}(a_{lj}\partial _{x_{j}}u) + \sum _{l=1}^{n}a_{l}\partial _{x_{l}}u+au,\\ B_{\Omega }^{*}v\equiv & {} \sum _{l,j=1}^{n} \overline{a}_{jl}\nu _{l}\partial _{x_{j}}v -\sum _{l=1}^{n}\nu _{l}\overline{a}_{l}v, \end{aligned}$$

for all \(u,v\in C^{2}(\overline{\Omega })\), and a fundamental solution \(S_{{\textbf{a}} }\) of \(P[{\textbf{a}},D]\), and the boundary integral operator corresponding to the double layer potential

$$\begin{aligned} W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\mu ](x) \equiv \int _{\partial \Omega }\mu (y)\overline{B^{*}_{\Omega ,y}}\left( S_{{\textbf{a}}}(x-y)\right) \,d\sigma _{y} \\ \begin{array}{lll} &{}\qquad \qquad \qquad \qquad =- \int _{\partial \Omega }\mu (y)\sum _{l,j=1}^{n} a_{jl}\nu _{l}(y)\frac{\partial S_{ {\textbf{a}} } }{\partial x_{j}}(x-y)\,d\sigma _{y}\\ &{}\qquad \qquad \qquad \qquad -\int _{\partial \Omega }\mu (y)\sum _{l=1}^{n}\nu _{l}(y)a_{l} S_{ {\textbf{a}} }(x-y)\,d\sigma _{y} \end{array}\nonumber \end{aligned}$$
(1.4)

for all \(x\in \partial \Omega\), where the density or moment \(\mu\) is a function from \(\partial \Omega\) to \({\mathbb {C}}\) and \(d\sigma _{y}\) is the ordinary \((n-1)\)-dimensional measure. Here, the subscript y of \(\overline{B^{*}_{\Omega ,y}}\) means that we are taking y as variable of the differential operator \(\overline{B^{*}_{\Omega ,y}}\). If \(\Omega\) is at least of class \(C^{1,\alpha }\) for some \(\alpha \in ]0,1]\), the kernel \(\overline{B^{*}_{\Omega ,y}}\left( S_{{\textbf{a}}}(x-y)\right)\) is well-known to have a weak singularity (cf., e.g., [8, Lem. 5.1 (i)]), and accordingly the integral in (1.4) exists in the sense of Lebesgue as long as \(\mu\) is essentially bounded. The role of the double layer potential in the solution of boundary value problems for the operator \(P[{\textbf{a}},D]\) is well known (cf., e.g., Günter [11], Kupradze, Gegelia, Basheleishvili and Burchuladze [17], Mikhlin [26], Mikhlin and Prössdorf [27], Buchukuri, Chkadua, Duduchava, and Natroshvili [1].)

For an account of known results on the boundary behavior of the double layer potential in Schauder spaces with \(m=1\), \(\alpha \in ]0,1]\) or in case \(\Omega\) is a bounded open Lipschitz set, we refer to the survey paper [22].

We now briefly summarize some known results in the classical case of the boundary behavior of the double layer potential in Schauder spaces with \(m\ge 2\). Instead for the regularity properties of the double layer potential in Schauder spaces with \(m\ge 2\) outside of the boundary, we refer to Günter [11], Kupradze, Gegelia, Basheleishvili and Burchuladze [17], Mikhlin [26], Mikhlin and Prössdorf [27], Miranda [28, 29], Wiegner [34], Dalla Riva [5], Dalla Riva, Morais and Musolino [7], Mitrea, Mitrea and Verdera [32] and references therein.

In case \(n=3\), \(m\ge 2\), \(\alpha \in ]0,1]\) and \(\Omega\) is of class \(C^{m,\alpha }\) and if \(P[{\textbf{a}},D]\) is the Laplace operator, Günter [11, Appendix, § IV, Thm. 3] has proved that \(W[\partial \Omega ,{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is bounded from \(C^{m-2,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha '}(\partial \Omega )\) for \(\alpha '\in ]0,\alpha [\).

In case \(n\ge 2\), \(m\ge 2\), \(\alpha \in ]0,1]\), O. Chkadua [3] has pointed out that one could exploit Kupradze, Gegelia, Basheleishvili and Burchuladze [17, Chap. IV, Sect. 2, Thm 2.9, Chap. IV, Sect. 3, Theorems 3.26 and 3.28] and prove that if \(\Omega\) is of class \(C^{m,\alpha }\), then \(W[\partial \Omega ,{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is bounded from \(C^{m-1,\alpha '}(\partial \Omega )\) to \(C^{m,\alpha '}(\partial \Omega )\) for \(\alpha '\in ]0,\alpha [\).

In case \(n=3\) and \(\Omega\) is of class \(C^{2}\), \(\alpha \in ]0,1[\) and if \(P[{\textbf{a}},D]\) is the Helmholtz operator, Colton and Kress [4] have developed the previous work of Günter [11] and Mikhlin [26] and proved that the operator \(W[\partial \Omega ,{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is bounded from \(C^{0,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\).

In case \(n\ge 2\), \(\alpha \in ]0,1[\) and \(\Omega\) is of class \(C^{2}\) and if \(P[{\textbf{a}},D]\) is the Laplace operator, Hsiao and Wendland [13, Remark 1.2.1] deduce that the operator \(W[\partial \Omega ,{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is bounded from \(C^{0,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) by the work of Mikhlin and Prössdorf [27].

In case \(n=3\), \(m\ge 2\), \(\alpha \in ]0,1[\) and \(\Omega\) is of class \(C^{m,\alpha }\) and if \(P[{\textbf{a}},D]\) is the Helmholtz operator, Kirsch [15, Thm. 3.3 (a)] has developed the previous work of Günter [11], Mikhlin [26] and Colton and Kress [4] and has proved that the operator \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is bounded from \(C^{m-1,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\).

Von Wahl [33] has considered the case of Sobolev spaces and has proved that if \(\Omega\) is of class \(C^{\infty }\) and if \(S_{{\textbf{a}}}\) is the fundamental solution of the Laplace operator, then the double layer improves the regularity of one unit on the boundary. Then, Heinemann [12] developed the ideas of von Wahl in the frame of Schauder spaces and proved that if \(\Omega\) is of class \(C^{m+5}\) and if \(S_{{\textbf{a}}}\) is the fundamental solution of the Laplace operator, then the double layer improves the regularity of one unit on the boundary, i.e., \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is linear and continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m+1,\alpha }(\partial \Omega )\).

Maz’ya and Shaposhnikova [25] have proved that \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous in fractional Sobolev spaces under sharp regularity assumptions on the boundary and if \(P[{\textbf{a}},D]\) is the Laplace operator.

Dondi and the author [8] have proved that if \(m\ge 2\) and \(\Omega\) is of class \(C^{m,\alpha }\) with \(\alpha \in ]0,1[\), then the double layer potential \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) associated with the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients is bounded from \(C^{m,\beta }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) for all \(\beta \in ]0,\alpha ]\).

For the corresponding results for the fundamental solution of the heat equation, we refer to the author and Luzzini [23, 24] and references therein.

In this paper, we plan to prove that if \(m\ge 2\) and \(\Omega\) is of class \(C^{m,\alpha }\) with \(\alpha \in ]0,1]\), then the double layer potential \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) associated with the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients is bounded from \(C^{m-1,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case \(\alpha <1\) and to the generalized Schauder space \(C^{m,\omega _{1}(\cdot )}(\partial \Omega )\) of functions with m-th order derivatives which satisfy a generalized \(\omega _{1}(\cdot )\)-Hölder condition with \(\omega _{1}(\cdot )\) as in (3.11) and thus with

$$\omega _{1}(r)\sim r|\ln r| \qquad {\textrm{as}}\ r\rightarrow 0,$$

in case \(\alpha =1\) (cf. Theorem 7.1). For the classical definition of the generalized Hölder or Schauder spaces on the boundary, we refer the reader to the author and Dondi [8, §2] and to Dalla Riva, the author and Musolino [6, §2.6, 2.20].

Hence, we sharpen the work of the above-mentioned authors in the sense that if \(\Omega\) is of class \(C^{m,\alpha }\) with \(m\ge 2\), then the class of regularity of the target space of \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is precisely \(C^{m,\alpha }\) if \(\alpha <1\) and is the generalized Schauder space \(C^{m,\omega _1(\cdot )}\) if \(\alpha =1\).

Moreover, we extend the above mentioned result of Kirsch [15] in the sense that Kirsch [15] considered the Helmholtz operator in case \(n=3\), \(\alpha <1\) and we consider a general fundamental solution \(S_{{\textbf{a}}}\) with \({\textbf{a}}\) as in (1.1), (1.2), (1.3), \(\alpha \le 1\) and \(n\ge 2\).

Instead, in the present paper, we do not consider case \(m=1\), a case in which Theorem 7.1 does have a different form and for which we refer the reader to [20, Thms. 5.1, 5.5], [21, Thm. 1.1], [22].

Notation

Let \(M_n({\mathbb {R}})\) denote the set of \(n\times n\) matrices with real entries. |A| denotes the operator norm of a matrix A, \(A^{t}\) denotes the transpose matrix of A. Let \(O_{n}({\mathbb {R}})\) denote the set of \(n\times n\) orthogonal matrices with real entries. We set

$$\begin{aligned} {\mathbb {B}}_n(\xi ,r)\equiv \left\{ \eta \in {\mathbb {R}}^n:\, |\xi -\eta |<r\right\} \qquad \forall (\xi ,r)\in {\mathbb {R}}^n\times ]0,+\infty [ \,. \end{aligned}$$
(2.1)

If \({\mathbb {D}}\) is a subset of \({\mathbb {R}}^n\), then we set

$$B({\mathbb {D}})\equiv \left\{ f\in {\mathbb {C}}^{\mathbb {D}}:\,f\ \text {is\ bounded} \right\} \,,\quad \Vert f\Vert _{B({\mathbb {D}})}\equiv \sup _{\mathbb {D}}|f|\qquad \forall f\in B({\mathbb {D}})\,.$$

Then, \(C^0({\mathbb {D}})\) denotes the set of continuous functions from \({\mathbb {D}}\) to \({\mathbb {C}}\) and we introduce the subspace \(C^0_b({\mathbb {D}})\equiv C^0({\mathbb {D}})\cap B({\mathbb {D}})\) of \(B({\mathbb {D}})\). Let \(\omega\) be a function from \([0,+\infty [\) to itself such that

$$\begin{aligned}{} & {} \qquad \qquad \omega (0)=0,\qquad \omega (r)>0\qquad \forall r\in ]0,+\infty [,\\{} & {} \qquad \qquad \omega \ {\text {is\ increasing,}}\ \lim _{r\rightarrow 0^{+}}\omega (r)=0,\nonumber \\{} & {} \qquad \qquad {\text {and}}\ \sup _{(a,t)\in [1,+\infty [\times ]0,+\infty [} \frac{\omega (at)}{a\omega (t)}<+\infty .\nonumber \end{aligned}$$
(2.2)

Here ‘\(\omega\) is increasing’ means that \(\omega (r_1)\le \omega (r_2)\) whenever \(r_1\), \(r_2\in [0,+\infty [\) and \(r_1<r_2\). If f is a function from a subset \({\mathbb {D}}\) of \({\mathbb {R}}^n\) to \({\mathbb {C}}\), then we denote by \(|f:{\mathbb {D}}|_{\omega (\cdot )}\) the \(\omega (\cdot )\)-Hölder constant of f, which is delivered by the formula

$$|f:{\mathbb {D}}|_{\omega (\cdot ) } \equiv \sup \left\{ \frac{|f( x )-f( y)|}{\omega (| x- y|) }: x, y\in {\mathbb {D}}, x\ne y\right\} \,.$$

If \(|f:{\mathbb {D}}|_{\omega (\cdot )}<+\infty\), we say that f is \(\omega (\cdot )\)-Hölder continuous. Sometimes, we simply write \(|f|_{\omega (\cdot )}\) instead of \(|f:{\mathbb {D}}|_{\omega (\cdot )}\). The subset of \(C^{0}({\mathbb {D}} )\) whose functions are \(\omega (\cdot )\)-Hölder continuous is denoted by \(C^{0,\omega (\cdot )} ({\mathbb {D}})\) and \(|f:{\mathbb {D}}|_{\omega (\cdot )}\) is a semi-norm on \(C^{0,\omega (\cdot )} ({\mathbb {D}})\). Then, we consider the space \(C^{0,\omega (\cdot )}_{b}({\mathbb {D}} ) \equiv C^{0,\omega (\cdot )} ({\mathbb {D}} )\cap B({\mathbb {D}} )\) with the norm

$$\Vert f\Vert _{ C^{0,\omega (\cdot )}_{b}({\mathbb {D}} ) }\equiv \sup _{x\in {\mathbb {D}} }|f(x)|+|f|_{\omega (\cdot )}\qquad \forall f\in C^{0,\omega (\cdot )}_{b}({\mathbb {D}} )\,.$$

Remark 2.3

Let \(\omega\) be as in (2.2). Let \({\mathbb {D}}\) be a subset of \({\mathbb {R}}^{n}\). Let f be a bounded function from \({\mathbb {D}}\) to \({\mathbb {C}}\), \(a\in ]0,+\infty [\). Then,

$$\sup _{x,y\in {\mathbb {D}},\ |x-y|\ge a}\frac{|f(x)-f(y)|}{\omega (|x-y|)} \le \frac{2}{\omega (a)} \sup _{{\mathbb {D}}}|f|\,.$$

In the case in which \(\omega (\cdot )\) is the function \(r^{\alpha }\) for some fixed \(\alpha \in ]0,1]\), a so-called Hölder exponent, we simply write \(|\cdot :{\mathbb {D}}|_{\alpha }\) instead of \(|\cdot :{\mathbb {D}}|_{r^{\alpha }}\), \(C^{0,\alpha } ({\mathbb {D}})\) instead of \(C^{0,r^{\alpha }} ({\mathbb {D}})\), \(C^{0,\alpha }_{b}({\mathbb {D}})\) instead of \(C^{0,r^{\alpha }}_{b} ({\mathbb {D}})\), and we say that f is \(\alpha\)-Hölder continuous provided that \(|f:{\mathbb {D}}|_{\alpha }<+\infty\). For the standard properties of the spaces of Hölder or Lipschitz continuous functions, we refer to [8, §2], [6, §2.6]. Let \(\Omega\) be an open subset of \({\mathbb {R}}^n\). Let \(s\in {\mathbb {N}}\setminus \{0\}\), \(f\in \left( C^{1}(\Omega )\right) ^{s}\). Then, Df denotes the Jacobian matrix of f.

Special classes of potential type kernels in \({\mathbb {R}}^n\)

In this section, we collect some basic properties of the classes of kernels that we need. For the proofs, we refer to [19, §3]. If X and Y are subsets of \({\mathbb {R}}^n\), then we denote by \({\mathbb {D}}_{X\times Y}\) the diagonal of \(X\times Y\), i.e., we set

$$\begin{aligned} {\mathbb {D}}_{X\times Y}\equiv \left\{ (x,y)\in X\times Y:\,x=y \right\} \end{aligned}$$
(3.1)

and if \(X=Y\), then we denote by \({\mathbb {D}}_{X}\) the diagonal of \(X\times X\), i.e., we set

$${\mathbb {D}}_X\equiv {\mathbb {D}}_{X\times X}\,.$$

An off-diagonal function in \(X\times Y\) is a function from \((X\times Y)\setminus {\mathbb {D}}_{X\times Y}\) to \({\mathbb {C}}\). We plan to consider the well-known class of potential type off-diagonal kernels as in the following definition.

Definition 3.2

Let X and Y be subsets of \({\mathbb {R}}^n\). Let \(s\in {\mathbb {R}}\). We denote by \({\mathcal {K}}_{s,X\times Y}\) (or more simply by \({\mathcal {K}}_s\)), the set of continuous functions K from \((X\times Y)\setminus {\mathbb {D}}_{ X\times Y }\) to \({\mathbb {C}}\) such that

$$\Vert K\Vert _{ {\mathcal {K}}_{s,X\times Y} }\equiv \sup _{(x,y)\in (X\times Y){\setminus } {\mathbb {D}}_{ X\times Y } }|K(x,y)|\,|x-y|^s<+\infty \,.$$

The elements of \({\mathcal {K}}_{s,X\times Y}\) are said to be kernels of potential type s in \(X\times Y\).

We plan to consider specific classes of “potential type” kernels that are suitable to prove continuity theorems for integral operators in Hölder spaces as in the following definition, which is a generalization of related classes as in Gegelia, Basheleishvili, and Burchuladze [17] (see also Dondi and the author [8], where such classes have been introduced in a form that generalizes those of Giraud [10], Gegelia [9] and Gegelia, Basheleishvili and Burchuladze [17, Chap. IV]).

Definition 3.3

Let X, \(Y\subseteq {\mathbb {R}}^n\). Let \(s_1\), \(s_2\), \(s_3\in {\mathbb {R}}\). We denote by \({\mathcal {K}}_{s_1, s_2, s_3} (X\times Y)\) the set of continuous functions K from \((X\times Y)\setminus {\mathbb {D}}_{X\times Y}\) to \({\mathbb {C}}\) such that

$$\begin{aligned} \begin{matrix} \Vert K\Vert _{ {\mathcal {K}}_{ s_1, s_2, s_3 }(X\times Y) } \equiv \sup \biggl \{\biggr . |x-y|^{ s_{1} }|K(x,y)|:\,(x,y)\in X\times Y, x\ne y \biggl .\biggr \}\\ \qquad +\sup \biggl \{\biggr . \frac{|x'-y|^{s_{2}}}{|x'-x''|^{s_{3}}} | K(x',y)- K(x'',y) |:\,\\ \qquad \qquad \qquad \qquad \qquad \qquad x',x''\in X, x'\ne x'', y\in Y\setminus {\mathbb {B}}_{n}(x',2|x'-x''|) \biggl .\biggr \}<+\infty . \end{matrix} \end{aligned}$$

One can easily verify that \(({\mathcal {K}}_{ s_{1},s_{2},s_{3} }(X\times Y),\Vert \cdot \Vert _{ {\mathcal {K}}_{s_{1},s_{2},s_{3} }(X\times Y) })\) is a normed space. By our definition, if \(s_1\), \(s_2\), \(s_3\in {\mathbb {R}}\), we have

$${\mathcal {K}}_{s_{1},s_{2},s_{3} }(X\times Y) \subseteq {\mathcal {K}}_{s_{1}, X\times Y}$$

and

$$\Vert K\Vert _{{\mathcal {K}}_{s_{1}, X\times Y} }\le \Vert K\Vert _{ {\mathcal {K}}_{s_{1},s_{2},s_{3} }(X\times Y) } \qquad \forall K\in {\mathcal {K}}_{s_{1},s_{2},s_{3} }(X\times Y) \,.$$

We note that if we choose \(s_2=s_1+s_3\), we have a so-called class of standard kernels. However, we note that if X and Y are bounded, known kernels as \(\ln |x-y|\) in \((X\times Y)\setminus D_{X\times Y}\) belong to \({\mathcal {K}}_{\epsilon ,1,1 }(X\times Y)\) for all \(\epsilon \in ]0,1[\) and that \(1\ne \epsilon +1\) (cf. [8, Lem. 3.2 (v)])). Moreover, logarithmic terms normally appear in convolution kernels as those associated with a general fundamental solution \(S_{{\textbf{a}}}\) with \({\textbf{a}}\) as in (1.1), (1.2), (1.3) and \(n\ge 2\) or in its partial derivatives (see formula (4.4) below).

Then, we have the following elementary known embedding lemma (cf., e.g., [19, Lem. 3.1]).

Lemma 3.4

Let X, \(Y\subseteq {\mathbb {R}}^n\). Let \(s_1\), \(s_2\), \(s_3\in {\mathbb {R}}\). If \(a\in ]0,+\infty [\), then \({\mathcal {K}}_{s_{1},s_{2},s_{3} }(X\times Y)\) is continuously embedded into \({\mathcal {K}}_{s_{1},s_{2}-a,s_{3}-a }(X\times Y)\).

Next, we state the following two product rule statements (cf. [19, Thm. 3.1, Prop. 3.1]).

Theorem 3.5

Let X, \(Y\subseteq {\mathbb {R}}^n\). Let \(s_1\), \(s_2\), \(s_3\), \(t_1\), \(t_2\), \(t_3\in {\mathbb {R}}\).

  1. (i)

    If \(K_1\in {\mathcal {K}}_{s_{1},s_{2},s_{3} }(X\times Y)\) and \(K_2\in {\mathcal {K}}_{t_{1},t_{2},t_{3} }(X\times Y)\), then the following inequality holds

    $$\begin{array}{c}\vert K_1(x',y)K_2(x',y)-K_1(x'',y)K_2(x'',y)\vert\\ \;\leq\Arrowvert K_1\Arrowvert_{{\mathcal K}_{s_1,s_2,s_3}(X\times Y)}\Arrowvert K_2\Arrowvert_{{\mathcal K}_{t_1,t_2,t_3}(X\times Y)}\\ \times\left(\frac{\vert x'-x''\vert^{s_3}}{\vert x'-y\vert^{s_2+t_1}}+\frac{2^{\vert s_1\vert}\vert x'-x''\vert^{t_3}}{\vert x'-y\vert^{t_2+s_1}}\right)\end{array}$$

    for all \(x',x''\in X\), \(x'\ne x''\), \(y\in Y\setminus {\mathbb {B}}_{n}(x',2|x'-x''|)\).

  2. (ii)

    The pointwise product is bilinear and continuous from

    $${\mathcal {K}}_{s_{1},s_1+s_3,s_{3} }(X\times Y)\times {\mathcal {K}}_{t_{1},t_{1}+s_3,s_{3} }(X\times Y) \quad \text {to}\quad {\mathcal {K}}_{s_1+t_{1},s_{1}+s_3+t_1,s_{3} }(X\times Y)\,.$$

Proposition 3.6

Let X, \(Y\subseteq {\mathbb {R}}^n\). Let \(s_1\), \(s_2\), \(s_3 \in {\mathbb {R}}\), \(\alpha \in ]0,1]\). Then, the following statements hold.

  1. (i)

    If \(K\in {\mathcal {K}}_{s_{1},s_2,s_{3} }(X\times Y)\) and \(f\in C^{0,\alpha }_b(X)\), then

    $$|K(x,y)f(x)|\,|x-y|^{s_1}\le \Vert K\Vert _{ {\mathcal {K}}_{s_{1},X\times Y } }\sup _X|f| \quad \forall (x,y)\in X\times Y{\setminus }{\mathbb {D}}_{X\times Y}\,.$$

    and

    $$\begin{aligned}&|K(x',y)f(x')-K(x'',y)f(x'')|\\&\qquad \qquad \le \Vert K\Vert _{ {\mathcal {K}}_{s_{1},s_2,s_{3} }(X\times Y) }\Vert f\Vert _{ C^{0,\alpha }_b(X) } \left\{ \frac{|x'-x''|^{s_3}}{|x'-y|^{s_2}}+2^{|s_1|}\frac{|x'-x''|^{\alpha }}{|x'-y|^{s_1}} \right\} \end{aligned}$$

    for all \(x',x''\in X\), \(x'\ne x''\), \(y\in Y\setminus {\mathbb {B}}_{n}(x',2|x'-x''|)\).

  2. (ii)

    If \(s_2\ge s_1\) and X and Y are both bounded, then the map from

    $${\mathcal {K}}_{s_{1},s_2,s_{3} }(X\times Y)\times C^{0,s_3}_b(X)\quad \text {to}\quad {\mathcal {K}}_{s_{1},s_2,s_{3} }(X\times Y)$$

    that takes the pair (K, f) to the kernel K(x, y)f(x) of the variable \((x,y)\in (X\times Y)\setminus {\mathbb {D}}_{X\times Y}\) is bilinear and continuous.

  3. (iii)

    The map from

    $${\mathcal {K}}_{s_{1},s_2,s_{3} }(X\times Y)\times C^{0}_b(Y)\quad \text {to}\quad {\mathcal {K}}_{s_{1},s_2,s_{3} }(X\times Y)$$

    that takes the pair (K, f) to the kernel K(x, y)f(y) of the variable \((x,y)\in (X\times Y)\setminus {\mathbb {D}}_{X\times Y}\) is bilinear and continuous.

Next, we have the following embedding statement that holds for bounded sets (cf. [19, Prop. 3.2]).

Proposition 3.7

Let X, Y be bounded subsets of \({\mathbb {R}}^n\). Let \(s_1\), \(s_2\), \(s_3\), \(t_1\), \(t_2\), \(t_3\in {\mathbb {R}}\). Then, the following statements hold.

  1. (i)

    If \(t_1\ge s_1\), then \({\mathcal {K}}_{s_1,X\times Y}\) is continuously embedded into \({\mathcal {K}}_{t_1,X\times Y}\).

  2. (ii)

    If \(t_1\ge s_1\), \(t_3\le s_3\) and \((t_2-t_3)\ge (s_2-s_3)\), then \({\mathcal {K}}_{s_{1},s_{2},s_{3} }(X\times Y)\) is continuously embedded into \({\mathcal {K}}_{t_{1},t_{2},t_{3} }(X\times Y)\).

  3. (iii)

    If \(t_1\ge s_1\), \(t_3\le s_3\), then \({\mathcal {K}}_{s_{1},s_{1}+s_3,s_{3} }(X\times Y)\) is continuously embedded into the space \({\mathcal {K}}_{t_{1},t_{1}+t_3,t_{3} }(X\times Y)\).

We now show that we can associate a potential type kernel to all Hölder continuous functions (cf. [19, Lem. 3.3]).

Lemma 3.8

Let X, Y be subsets of \({\mathbb {R}}^n\). Let \(\alpha \in ]0,1]\). Let \(C^{0,\alpha } (X\cup Y)\) be endowed with the Hölder seminorm \(|\cdot :X\cup Y|_\alpha\). Then, the following statements hold.

  1. (i)

    If \(\mu \in C^{0,\alpha }(X\cup Y)\), then the map \(\Xi [\mu ]\) defined by

    $$\begin{aligned} \Xi [\mu ](x,y)\equiv \mu (x)-\mu (y)\qquad \forall (x,y)\in (X\times Y)\setminus {\mathbb {D}}_{X\times Y} \end{aligned}$$
    (3.9)

    belongs to \({\mathcal {K}}_{-\alpha ,0,\alpha }(X\times Y)\).

  2. (ii)

    The operator \(\Xi\) from \(C^{0,\alpha }(X\cup Y)\) to \({\mathcal {K}}_{-\alpha ,0,\alpha }(X\times Y)\) that takes \(\mu\) to \(\Xi [\mu ]\) is linear and continuous.

In order to introduce a result of [19, Thm. 6.3], we need to introduce a further norm for kernels in the case in which Y is a compact manifold of class \(C^1\) that is embedded in \(M={\mathbb {R}}^n\) and \(X=Y\).

Definition 3.10

Let Y be a compact manifold of class \(C^1\) that is embedded in \({\mathbb {R}}^n\). Let \(s_1\), \(s_2\), \(s_3\in {\mathbb {R}}\). We set

$$\begin{aligned}&{\mathcal {K}}_{ s_1, s_2, s_3 }^\sharp (Y\times Y) \equiv \biggl \{\biggr . K\in {\mathcal {K}}_{ s_1, s_2, s_3 }(Y\times Y):\,\\&\qquad \qquad \sup _{x\in Y}\sup _{r\in ]0,+\infty [} \left| \int _{Y\setminus {\mathbb {B}}_n(x,r)}K(x,y)\, d\sigma _y \right| <+\infty \biggl .\biggr \} \end{aligned}$$

and

$$\begin{aligned}&\Vert K\Vert _{{\mathcal {K}}_{ s_1, s_2, s_3 }^\sharp (Y\times Y)} \equiv \Vert K\Vert _{{\mathcal {K}}_{ s_1, s_2, s_3 }(Y\times Y)}\\&\qquad \qquad \qquad + \sup _{x\in Y}\sup _{r\in ]0,+\infty [} \left| \int _{Y\setminus {\mathbb {B}}_n(x,r)}K(x,y)\,d\sigma _y \right| \quad \forall K\in {\mathcal {K}}_{ s_1, s_2, s_3 }^\sharp (Y\times Y)\,. \end{aligned}$$

Clearly, \(({\mathcal {K}}^\sharp _{ s_{1},s_{2},s_{3} }(Y\times Y),\Vert \cdot \Vert _{ {\mathcal {K}}^\sharp _{s_{1},s_{2},s_{3} }(Y\times Y) })\) is a normed space. By definition, \({\mathcal {K}}^\sharp _{ s_{1},s_{2},s_{3} }(Y\times Y)\) is continuously embedded into \({\mathcal {K}}_{ s_{1},s_{2},s_{3} }(Y\times Y)\). Next, we introduce a function that we need for a generalized Hölder norm. For each \(\theta \in ]0,1]\), we define the function \(\omega _{\theta }(\cdot )\) from \([0,+\infty [\) to itself by setting

$$\begin{aligned} \omega _{\theta }(r)\equiv \left\{ \begin{array}{ll} 0 &{}r=0\,, \\ r^{\theta }|\ln r | &{}r\in ]0,r_{\theta }]\,, \\ r_{\theta }^{\theta }|\ln r_{\theta } | &{} r\in ]r_{\theta },+\infty [\,, \end{array} \right. \end{aligned}$$
(3.11)

where \(r_{\theta }\equiv e^{-1/\theta }\) for all \(\theta \in ]0,1]\). Obviously, \(\omega _{\theta }(\cdot )\) is concave and satisfies condition (2.2). We also note that if \({\mathbb {D}}\subseteq {\mathbb {R}}^n\), then the continuous embedding

$$C^{0, \theta }_b({\mathbb {D}})\subseteq C^{0,\omega _\theta (\cdot )}_b({\mathbb {D}})\subseteq C^{0,\theta '}_b({\mathbb {D}})$$

holds for all \(\theta '\in ]0,\theta [\). We also need to consider convolution kernels, thus we introduce the following notation. If \(n\in {\mathbb {N}}\setminus \{0\}\), \(m\in {\mathbb {N}}\), \(h\in {\mathbb {R}}\), \(\alpha \in ]0,1]\), then we set

$$\begin{aligned} {\mathcal {K}}^{m,\alpha }_h \equiv \biggl \{ k\in C^{m,\alpha }_{ {\textrm{loc}}}({\mathbb {R}}^n\setminus \{0\}):\, k\ {\text {is\ positively\ homogeneous\ of \ degree}}\ h \biggr \}\,, \end{aligned}$$
(3.12)

where \(C^{m,\alpha }_{ {\textrm{loc}}}({\mathbb {R}}^n\setminus \{0\})\) denotes the set of functions of \(C^{m}({\mathbb {R}}^n\setminus \{0\})\) whose restriction to \(\overline{\Omega }\) is of class \(C^{m,\alpha }(\overline{\Omega })\) for all bounded open subsets \(\Omega\) of \({\mathbb {R}}^n\) such that \(\overline{\Omega }\subseteq {\mathbb {R}}^n\setminus \{0\}\) and we set

$$\Vert k\Vert _{ {\mathcal {K}}^{m,\alpha }_h}\equiv \Vert k\Vert _{C^{m,\alpha }(\partial {\mathbb {B}}_n(0,1))}\qquad \forall k\in {\mathcal {K}}^{m,\alpha }_h\,.$$

We can easily verify that \(\left( {\mathcal {K}}^{m,\alpha }_h, \Vert \cdot \Vert _{ {\mathcal {K}}^{m,\alpha }_h}\right)\) is a Banach space. We also mention the following variant of a well-known statement (cf., e.g., [20, Lem. 3.11]).

Lemma 3.13

Let \(n\in {\mathbb {N}}\setminus \{0\}\), \(h\in [0,+\infty [\). If \(k\in C^{0,1}_{ {\textrm{loc}} }({\mathbb {R}}^n\setminus \{0\})\) is positively homogeneous of degree \(-h\), then \(k(x-y)\in {\mathcal {K}}_{h,h+1,1}({\mathbb {R}}^n\times {\mathbb {R}}^n)\). Moreover, the map from \({\mathcal {K}}^{0,1}_{-h}\) to \({\mathcal {K}}_{h,h+1,1}({\mathbb {R}}^n\times {\mathbb {R}}^n)\) which takes k to \(k(x-y)\) is linear and continuous (see (3.12) for the definition of \({\mathcal {K}}^{0,1}_{-h}\)).

If X and Y are subsets of \({\mathbb {R}}^n\), then the restriction operator

$$\text {from}\ {\mathcal {K}}_{h,h+1,1}({\mathbb {R}}^n\times {\mathbb {R}}^n)\ \text {to}\ {\mathcal {K}}_{h,h+1,1}(X\times Y)$$

is linear and continuous. Thus, Lemma 3.13 implies that if \(h\in [0,+\infty [\), then the map

$$\text {from}\ {\mathcal {K}}^{0,1}_{-h} \ \ \text { to}\ {\mathcal {K}}_{h,h+1,1}(X\times Y)\,,$$

which takes k to \(k(x-y)\), is linear and continuous.

Remark 3.14

As Lemma 3.13 shows the convolution kernels associated with positively homogeneous functions of negative degree are standard kernels. We note, however, that there exist potential type kernels that belong to a class \({\mathcal {K}}_{s_1,s_2,s_3} (X\times Y)\) with \(s_2\ne s_1+s_3\).

Technical preliminaries on the differential operator \(P[{\textbf{a}},D]\).

Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{1}\). The kernel of the boundary integral operator corresponding to the double layer potential is the following

$$\begin{aligned} \overline{B^{*}_{\Omega ,y}}\left( S_{{\textbf{a}}}(x-y)\right)\equiv & {} - \sum _{l,j=1}^{n} a_{jl}\nu _{l}(y)\frac{\partial S_{ {\textbf{a}} } }{\partial x_{j}}(x-y)\\- & {} \sum _{l=1}^{n}\nu _{l}(y)a_{l} S_{ {\textbf{a}} }(x-y)\qquad \forall (x,y)\in (\partial \Omega )^2\setminus {\mathbb {D}}_{\partial \Omega } \nonumber \end{aligned}$$
(4.1)

(cf. (1.4)). In order to analyze the kernel of the double layer potential, we need some more information on the fundamental solution \(S_{ {\textbf{a}} }\) of \(P[{\textbf{a}},D]\). To do so, we introduce the fundamental solution \(S_{n}\) of the Laplace operator. Namely, we set

$$S_{n}(x)\equiv \left\{ \begin{array}{lll} \frac{1}{s_{n}}\ln |x| \qquad &{} \forall x\in {\mathbb {R}}^{n}\setminus \{0\},\quad &{} {\textrm{if}}\ n=2\,,\\ \frac{1}{(2-n)s_{n}}|x|^{2-n}\qquad &{} \forall x\in {\mathbb {R}}^{n}\setminus \{0\},\quad &{} {\textrm{if}}\ n>2\,, \end{array} \right.$$

where \(s_{n}\) denotes the \((n-1)\) dimensional measure of \(\partial {\mathbb {B}}_{n}(0,1)\) and we follow a formulation of Dalla Riva [5, Thm. 5.2, 5.3] and Dalla Riva, Morais and Musolino [7, Thm. 5.5], that we state as in Dondi and the author [8, Cor. 4.2] (see also John [14], and Miranda [28] for homogeneous operators, and Mitrea and Mitrea [30, p. 203]).

Proposition 4.2

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Then, there exist an invertible matrix \(T\in M_{n}({\mathbb {R}})\) such that

$$\begin{aligned} a^{(2)}=TT^{t}\,, \end{aligned}$$
(4.3)

a real analytic function \(A_{1}\) from \(\partial {\mathbb {B}}_{n}(0,1)\times {\mathbb {R}}\) to \({\mathbb {C}}\) such that \(A_{1}(\cdot ,0)\) is odd, \(b_{0}\in {\mathbb {C}}\), a real analytic function \(B_{1}\) from \({\mathbb {R}}^{n}\) to \({\mathbb {C}}\) such that \(B_{1}(0)=0\), and a real analytic function C from \({\mathbb {R}}^{n}\) to \({\mathbb {C}}\) such that

$$\begin{aligned} S_{ {\textbf{a}} }(x)= & {} \frac{1}{\sqrt{\det a^{(2)} }}S_{n}(T^{-1}x)\nonumber \\+ & {} |x|^{3-n}A_{1}(\frac{x}{|x|},|x|) +(B_{1}(x)+b_{0}(1-\delta _{2,n}))\ln |x|+C(x), \end{aligned}$$
(4.4)

for all \(x\in {\mathbb {R}}^{n}\setminus \{0\}\), and such that both \(b_{0}\) and \(B_{1}\) equal zero if n is odd. Moreover,

$$\frac{1}{\sqrt{\det a^{(2)} }}S_{n}(T^{-1}x)$$

is a fundamental solution for the principal part of \(P[{\textbf{a}},D]\).

In particular for the statement that \(A_{1}(\cdot ,0)\) is odd, we refer to Dalla Riva, Morais and Musolino [7, Thm. 5.5, (32)], where \(A_{1}(\cdot ,0)\) coincides with \({\textbf{f}}_1({\textbf{a}},\cdot )\) in that paper. Here, we note that a function A from \((\partial {\mathbb {B}}_{n}(0,1))\times {\mathbb {R}}\) to \({\mathbb {C}}\) is said to be real analytic provided that it has a real analytic extension to an open neighborhood of \((\partial {\mathbb {B}}_{n}(0,1))\times {\mathbb {R}}\) in \({\mathbb {R}}^{n+1}\). Then, we have the following elementary lemma (cf., e.g., [20, Lem. 4.2]).

Lemma 4.5

Let \(n\in {\mathbb {N}}\setminus \{0,1\}\). A function A from \((\partial {\mathbb {B}}_{n}(0,1))\times {\mathbb {R}}\) to \({\mathbb {C}}\) is real analytic if and only if the function \(\tilde{A}\) from \(({\mathbb {R}}^n\setminus \{0\}) \times {\mathbb {R}}\) defined by

$$\begin{aligned} \tilde{A}(x,r)\equiv A(\frac{x}{|x|},r)\qquad \forall (x,r)\in ({\mathbb {R}}^n\setminus \{0\}) \times {\mathbb {R}} \end{aligned}$$
(4.6)

is real analytic.

Then, one can prove the following formula for the gradient of the fundamental solution (see Dondi and the author [8, Lem. 4.3, (4.8) and the following 2 lines]. Here one should remember that \(A_1(\cdot ,0)\) is odd and that \(b_0=0\) if n is odd).

Proposition 4.7

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(T\in M_{n}({\mathbb {R}})\) be as in (4.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(B_{1}\), C be as in Proposition 4.2. Then, there exists a real analytic function \(A_{2}\) from \(\partial {\mathbb {B}}_{n}(0,1)\times {\mathbb {R}}\) to \({\mathbb {C}}^{n}\) such that

$$\begin{aligned}&DS_{ {\textbf{a}} }(x)=\frac{1}{ s_{n}\sqrt{\det a^{(2)} } } |T^{-1}x|^{-n}x^{t}(a^{(2)})^{-1}\nonumber \\&\qquad +|x|^{2-n}A_{2}(\frac{x}{|x|},|x|)+DB_{1}(x)\ln |x|+DC(x) \quad \forall x\in {\mathbb {R}}^{n}\setminus \{0\}\,. \end{aligned}$$
(4.8)

Moreover, \(A_2(\cdot ,0)\) is even.

Next, we introduce the following technical lemma (see Dondi and the author [8, Lem. 3.2 (v), 3.3]). See also [20, Lem. 4.5].

Lemma 4.9

Let Y be a nonempty bounded subset of \({\mathbb {R}}^{n}\). Then, the following statements hold.

  1. (i)

    Let \({\textrm{diam}}\,(Y)\) be the diameter of Y, \(F\in {\textrm{Lip}}(\partial {\mathbb {B}}_{n}(0,1)\times [0,{\textrm{diam}}\,(Y)])\) with

    $$\begin{array}{l}{\textrm{Lip}}(F) \equiv \biggl \{\biggr . \frac{|F(\theta ',r')-F(\theta '',r'')|}{ |\theta '-\theta ''|+|r'-r''| }:\\(\theta ',r'),(\theta '',r'')\in \partial {\mathbb {B}}_{n}(0,1)\times [0,{\textrm{diam}}\,(Y)],\ (\theta ',r')\ne (\theta '',r'') \biggl .\biggr \}.\end{array}$$

    Then,

    $$\begin{array}{l}\left| F\left( \frac{x'-y}{|x'-y|},|x'-y| \right) - F\left( \frac{x''-y}{|x''-y|},|x''-y| \right) \right| \\ \le {\textrm{Lip}}(F) (2+ {\textrm{diam}}\,(Y)) \frac{|x'-x''|}{|x'-y|}\, \quad \forall y\in Y \setminus {\mathbb {B}}_{n}(x',2|x'-x''|),\end{array}$$
    (4.10)

    for all \(x',x''\in Y\), \(x'\ne x''\). In particular, if \(f\in C^{1}(\partial {\mathbb {B}}_{n}(0,1)\times {\mathbb {R}},{\mathbb {C}})\), then

    $$\begin{array}{c}M_{f,Y}\equiv \sup \biggl \{\biggr . \left| f\left( \frac{x'-y}{|x'-y|},|x'-y| \right) - f\left( \frac{x''-y}{|x''-y|},|x''-y| \right) \right|\\\times \frac{|x'-y|}{|x'-x''|}:\,x',x''\in Y, x'\ne x'', y\in Y \setminus {\mathbb {B}}_{n}(x',2|x'-x''|) \biggl .\biggr \} \end{array}$$

    is finite and thus the kernel \(f\left( \frac{x-y}{|x-y|},|x-y| \right)\) belongs to \({\mathcal {K}}_{0,1,1}(Y\times Y)\).

  2. (ii)

    Let W be an open neighborhood of \(\overline{Y-Y}\). Let \(f\in C^{1}(W,{\mathbb {C}})\). Then,

    $$\begin{array}{c}\tilde{M}_{f,Y}\equiv \sup \biggl \{\biggr . | f(x'-y)-f(x''-y)|\,|x'-x''|^{-1}\\:\,x',x''\in Y, x'\ne x'', y\in Y \biggl .\biggr \}<+\infty .\end{array}$$

    Here \(Y-Y\equiv \{y_{1}-y_{2}:\ y_{1}, y_{2}\in Y\}\). In particular, the kernel \(f(x-y)\) belongs to the class \({\mathcal {K}}_{0,0,1}(Y\times Y)\), which is continuously embedded into \({\mathcal {K}}_{0,1,1}(Y\times Y)\).

  3. (iii)

    The kernel \(\ln |x-y|\) belongs to \({\mathcal {K}}_{\epsilon ,1,1}(Y\times Y)\) for all \(\epsilon \in ]0,1[\).

In order to prove regularity results for the double layer potential, we need the definition of tangential derivative and some auxiliary operators that we now introduce. Let \(\Omega\) be an open subset of \({\mathbb {R}}^n\) of class \(C^1\). If \(l,r\in \{1,\dots ,n\}\), then \(M_{lr}\) denotes the tangential derivative operator from \(C^{1}(\partial \Omega )\) to \(C^{0}(\partial \Omega )\) that takes f to

$$\begin{aligned} M_{lr}[f]\equiv \nu _{l}\frac{\partial \tilde{f}}{\partial x_{r}}- \nu _{r}\frac{\partial \tilde{f}}{\partial x_{l}}\qquad {\text {on}}\ \partial \Omega \,, \end{aligned}$$
(4.11)

where \(\tilde{f}\) is any continuously differentiable extension of f to an open neighborhood of \(\partial \Omega\). We note that \(M_{lr}[f]\) is independent of the specific choice of \(\tilde{f}\) (cf., e.g., Dalla Riva, the author and Musolino [6, §2.21]). The tangential gradient \({\textrm{grad}}_{\partial \Omega } f\) of \(f\in C^{1}(\partial \Omega )\) is defined as

$$({\textrm{grad}}_{\partial \Omega } f)_h\equiv \frac{\partial \tilde{f}}{\partial x_h}-( (D\tilde{f})\,\nu )\nu _h \qquad {\text {on}}\ \partial \Omega \,,$$

for all \(h\in \{1,\dots ,n\}\) where \(\tilde{f}\) is an extension of f of class \(C^{1}\) in an open neighborhood of \(\partial \Omega\). We note that \({\textrm{grad}}_{\partial \Omega } f\) is independent of the specific choice of \(\tilde{f}\) (cf., e.g., Dalla Riva, the author and Musolino [6, §2.21]). See also Kirsch and Hettlich [16, A.5], Chavel [2, Chap. 1]. Then, we set

$$\begin{aligned} Q_j[g,\mu ](x) =\int _{\partial \Omega }(g(x)-g(y))\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\mu (y)\,d\sigma _{y}\quad \forall x\in \partial \Omega \,, \end{aligned}$$
(4.12)

for all \((g,\mu )\in C^{0,1}(\partial \Omega )\times L^{\infty }(\partial \Omega )\) for all \(j\in \{1,\dots ,n\}\). As a first step, we prove the following technical statement that determines the second order partial derivatives of the kernel \(S_{ {\textbf{a}} }(x-y)\), the class membership of the corresponding kernels and the class of the tangential gradient of the kernel \(\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\) with respect to its first variable on the boundary of an open set of class \(C^{1,\alpha }\)for all\(j\in \{1,\dots ,n\}\).

Lemma 4.13

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(B_{1}\), C be as in Proposition 4.2. Let \(A_2\) be as in Proposition 4.7. Let \(j,h\in \{1,\dots ,n\}\). Then, the following statements hold.

  1. (i)
    $$\begin{aligned}{} & {} \frac{\partial }{\partial x_h}\frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y) \nonumber \\{} & {} \qquad \quad =\frac{-n |T^{-1}(x-y)|^{-n-1} }{s_{n}\sqrt{\det a^{(2)} }} \frac{ \sum _{s,t=1}^n(T^{-1})_{st}(x_t-y_t)(T^{-1})_{sh} }{|T^{-1}(x-y)|} \nonumber \\{} & {} \qquad \quad \times \sum _{s=1}^n(x_s-y_s)((a^{(2)})^{-1})_{sj} + \frac{|T^{-1}(x-y)|^{-n} }{s_{n}\sqrt{\det a^{(2)} }} ((a^{(2)})^{-1})_{hj} \nonumber \\{} & {} \qquad \quad +(2-n)|x-y|^{1-n}\frac{x_h-y_h}{|x-y|}A_{2,j}\left( \frac{x-y}{|x-y|},|x-y|\right) \nonumber \\{} & {} \qquad \quad +|x-y|^{2-n}\biggl \{\sum _{s=1}^n\frac{\partial A_{2,j}}{\partial x_s}\left( \frac{x-y}{|x-y|},|x-y|\right) \nonumber \\{} & {} \qquad \quad \times \biggl ( \delta _{sh}|x-y|-\frac{(x_s-y_s)(x_h-y_h)}{|x-y|} \biggr )|x-y|^{-2} \nonumber \\{} & {} \qquad \quad +\frac{\partial A_{2,j}}{\partial r} \left( \frac{x-y}{|x-y|},|x-y|\right) \frac{x_h-y_h}{|x-y|}\biggr \} \nonumber \\{} & {} \qquad \quad +\frac{\partial ^2B_1}{\partial x_h\partial x_j}(x-y)\ln |x-y| +\frac{\partial B_1}{\partial x_j}(x-y)\frac{x_h-y_h}{|x-y|^2} \nonumber \\{} & {} \qquad \quad +\frac{\partial ^2C}{\partial x_h\partial x_j}(x-y) \end{aligned}$$
    (4.14)

    for all x, \(y\in {\mathbb {R}}^n\), \(x\ne y\).

  2. (ii)

    If G is a nonempty bounded subset of \({\mathbb {R}}^{n}\), then the kernel \(\frac{\partial }{\partial x_h}\frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y)\) belongs to \({\mathcal {K}}_{n,n+1,1}(G\times G)\).

  3. (iii)

    Let \(\alpha \in ]0,1]\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{1,\alpha }\). Then,

    $$\begin{aligned}{} & {} \left( {\textrm{grad}}_{\partial \Omega ,x} \left( \frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y)\right) \right) _h\nonumber \\{} & {} \qquad \qquad =\frac{\partial }{\partial x_h}\frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y) -\sum _{l=1}^{n}\frac{\partial }{\partial x_l}\frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y)\nu _l(x)\nu _h(x), \end{aligned}$$
    (4.15)

    for all \((x,y)\in (\partial \Omega )^2\setminus {\mathbb {D}}_{\partial \Omega }\) and the kernel \(\left( {\textrm{grad}}_{\partial \Omega ,x} \left( \frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y) \right) \right) _h\) belongs to \({\mathcal {K}}_{n,n+ \alpha ,\alpha }((\partial \Omega )\times (\partial \Omega ))\). Here \({\textrm{grad}}_{\partial \Omega ,x}\) denotes the tangential gradient with respect to the x variable.

Proof

Statement (i) holds by formula (4.8) and by standard differentiation rules. (ii) By Lemma 3.13, the kernel

$$\frac{-n |T^{-1}(x-y)|^{-n-1} }{s_{n}\sqrt{\det a^{(2)} }} \frac{ \sum _{s,t=1}^n(T^{-1})_{st}(x_t-y_t)(T^{-1})_{sh} }{|T^{-1}(x-y)|} \sum _{s=1}^n(x_s-y_s)((a^{(2)})^{-1})_{sj}$$

and the kernel

$$\frac{|T^{-1}(x-y)|^{-n} }{s_{n}\sqrt{\det a^{(2)} }} ((a^{(2)})^{-1})_{hj}$$

belong to \({\mathcal {K}}_{n,n+1,1}(G\times G)\). Since \(A_{2,j}\) is real analytic in \(\partial {\mathbb {B}}_n(0,1)\times {\mathbb {R}}\), Lemma 4.9 (i) implies that \(A_{2,j}\left( \frac{x-y}{|x-y|},|x-y|\right)\) belongs to \({\mathcal {K}}_{0,1,1}(G\times G)\). By Lemma 3.13, the kernel \(|x-y|^{1-n}\frac{x_h-y_h}{|x-y|}\) belongs to \({\mathcal {K}}_{n-1,n,1}(G\times G)\). Then, the product Theorem 3.5 (ii) implies that the product

$$(2-n)|x-y|^{1-n}\frac{x_h-y_h}{|x-y|}A_{2,j}(\frac{x-y}{|x-y|},|x-y|)$$

belongs to \({\mathcal {K}}_{n-1,n,1}(G\times G)\).

Since \(\frac{\partial A_{2,j}}{\partial x_s}\) is real analytic in \(\partial {\mathbb {B}}_n(0,1)\times {\mathbb {R}}\), Lemma 4.9 (i) implies that \(\frac{\partial A_{2,j}}{\partial x_s}\left( \frac{x-y}{|x-y|},|x-y|\right)\) belongs to \({\mathcal {K}}_{0,1,1}(G\times G)\). By Lemma 3.13, the kernel

$$|x-y|^{2-n}\biggl ( \delta _{sh}|x-y|-\frac{(x_s-y_s)(x_h-y_h)}{|x-y|} \biggr )|x-y|^{-2}$$

belongs to \({\mathcal {K}}_{n-1,n,1}(G\times G)\). Then, the product Theorem 3.5 (ii) implies that the product

$$|x-y|^{-n} \sum _{s=1}^n\frac{\partial A_{2,j}}{\partial x_s}\left( \frac{x-y}{|x-y|},|x-y|\right) \biggl ( \delta _{sh}|x-y|-\frac{(x_s-y_s)(x_h-y_h)}{|x-y|} \biggr )$$

belongs to \({\mathcal {K}}_{n-1,n,1}(G\times G)\).

Since \(\frac{\partial A_{2,j}}{\partial r}\) is real analytic in \(\partial {\mathbb {B}}_n(0,1)\times {\mathbb {R}}\), Lemma 4.9 (i) implies that \(\frac{\partial A_{2,j}}{\partial r}\left( \frac{x-y}{|x-y|},|x-y|\right)\) belongs to \({\mathcal {K}}_{0,1,1}(G\times G)\). By Lemma 3.13, the kernel

$$|x-y|^{2-n} \frac{x_h-y_h}{|x-y|}$$

belongs to \({\mathcal {K}}_{n-2,n-1,1}(G\times G)\). Then, the product Theorem 3.5 (ii) implies that the product

$$|x-y|^{2-n}\frac{\partial A_2}{\partial r} \left( \frac{x-y}{|x-y|},|x-y|\right) \frac{x_h-y_h}{|x-y|}$$

belongs to \({\mathcal {K}}_{n-2,n-1,1}(G\times G)\).

Since \(B_1\) is real analytic, Lemma 4.9 (ii) implies that \(\frac{\partial ^2B_1}{\partial x_h\partial x_j}(x-y)\) belongs to \({\mathcal {K}}_{0,1,1}(G\times G)\). By Lemma 4.9 (iii), the kernel \(\ln |x-y|\) belongs to \({\mathcal {K}}_{\epsilon ,1,1}(G\times G)\) for all \(\epsilon \in ]0,1[\). By the embedding Proposition 3.7 (ii), \({\mathcal {K}}_{\epsilon ,1,1}(G\times G)\) is contained in \({\mathcal {K}}_{\epsilon ,\epsilon +1,1}(G\times G)\) for all \(\epsilon \in ]0,1[\).

Then, the product Theorem 3.5 (ii) implies that the product

$$\frac{\partial ^2B_1}{\partial x_h\partial x_j}(x-y)\ln |x-y|$$

belongs to \({\mathcal {K}}_{\epsilon ,\epsilon +1,1}(G\times G)\) for all \(\epsilon \in ]0,1[\).

Since \(B_1\) is real analytic, Lemma 4.9 (ii) implies that the kernel \(\frac{\partial B_1}{\partial x_j}(x-y)\) belongs to \({\mathcal {K}}_{0,1,1}(G\times G)\). By Lemma 3.13, the kernel \(\frac{x_h-y_h}{|x-y|^2}\) belongs to \({\mathcal {K}}_{1,2,1}(G\times G)\). Then, the product Theorem 3.5 (ii) implies that the product

$$\frac{\partial B_1}{\partial x_j}(x-y)\frac{x_h-y_h}{|x-y|^2}$$

belongs to \({\mathcal {K}}_{1,2,1}(G\times G)\).

Since C is real analytic, Lemma 4.9 (ii) implies that \(\frac{\partial ^2C}{\partial x_h\partial x_j}(x-y)\) belongs to \({\mathcal {K}}_{0,1,1}(G\times G)\). Thus, we have proved that each addendum in the right-hand side of the formula (4.15) is contained in one of the following classes

$$\begin{aligned}{} & {} {\mathcal {K}}_{n,n+1,1}(G\times G),\quad {\mathcal {K}}_{n-1,n,1}(G\times G),\quad {\mathcal {K}}_{n-2,n-1,1}(G\times G),\quad \\{} & {} {\mathcal {K}}_{\epsilon ,\epsilon +1,1}(G\times G)\quad \forall \epsilon \in ]0,1[,\quad {\mathcal {K}}_{1,2,1}(G\times G),\qquad {\mathcal {K}}_{0,1,1}(G\times G) . \end{aligned}$$

Now, the embedding Proposition 3.7 (iii) implies that each of such classes is contained in \({\mathcal {K}}_{n,n+1,1}(G\times G)\), and thus the proof of statement (ii) is complete.

(iii) Formula (4.15) holds by formula (4.14) and by the definition of tangential gradient. By the elementary Lemma 3.4, we have

$$\begin{aligned}{} & {} {\mathcal {K}}_{n,n+1,1}((\partial \Omega )\times (\partial \Omega ))\\{} & {} \qquad \subseteq {\mathcal {K}}_{n,n+1-(1-\alpha ),1-(1-\alpha )}((\partial \Omega )\times (\partial \Omega )) ={\mathcal {K}}_{n,n+\alpha ,\alpha }((\partial \Omega )\times (\partial \Omega )). \end{aligned}$$

Then, the membership of the components of \(\nu\) in \(C^{0,\alpha }(\partial \Omega )\), statement (i) with \(G=\partial \Omega\) condition \(n\le n+\alpha\) and the product Proposition 3.6 (ii) imply that the right-hand side of formula (4.15) defines a kernel of class \({\mathcal {K}}_{n,n+\alpha ,\alpha }((\partial \Omega )\times (\partial \Omega ))\) and thus the proof is complete. \(\square\)

Then, we introduce the following technical statement (cf. [21, Thm. 3.2]).

Theorem 4.16

Let \(n\in {\mathbb {N}}\), \(n\ge 2\), \(\tilde{\alpha }\in ]0,1]\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^n\) of class \(C^{1,\tilde{\alpha }}\). Then, there exists \(c^*_{\partial \Omega ,\tilde{\alpha }}\in ]0,+\infty [\) such that

$$\begin{aligned} \sup _{x\in \partial \Omega } \sup _{\epsilon \in ]0,+\infty [} \left| \int _{(\partial \Omega )\setminus {\mathbb {B}}_{n}(x,\epsilon )} k(x-y)\,d\sigma _y \right| \le c^*_{\partial \Omega ,\tilde{\alpha }}\left\| k \right\| _{{\mathcal {K}}^{0,1}_{-(n-1) }} \forall k\in {\mathcal {K}}^{0,1}_{-(n-1);o }\,, \end{aligned}$$
(4.17)

where \({\mathcal {K}}^{0,1}_{-(n-1);o } \equiv \{k\in {\mathcal {K}}^{0,1}_{-(n-1)}:\, k\ \text {is\ odd}\}\) (cf. (3.12)).

Next, we prove the following technical lemma.

Lemma 4.18

Let \(n\in {\mathbb {N}}\setminus \{0,1\}\). Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(\alpha \in ]0,1]\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^n\) of class \(C^{1, \alpha }\). Let \(j,h,z\in \{1,\dots ,n\}\). Then,

$$\begin{aligned} \sup _{x\in \partial \Omega } \sup _{r\in ]0,+\infty [} \left| \, \int _{(\partial \Omega )\setminus {\mathbb {B}}_{n}(x,r)} (x_z-y_z)\frac{\partial }{\partial x_h}\frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y) \,d\sigma _y\right| <+\infty \,. \end{aligned}$$
(4.19)

Proof

By formula (4.14) and by the known inequalities

$$\begin{aligned} \sup _{x\in \partial \Omega }\int _{\partial \Omega }|x-y|^{-\gamma }\,d\sigma _y<+\infty \,, \quad \sup _{x\in \partial \Omega }\int _{\partial \Omega } |\ln |x-y|| \,d\sigma _y<+\infty \end{aligned}$$
(4.20)

for \(\gamma \in ]-\infty ,(n-1)[\) (cf., e.g., [8, Lem. 3.5]), we have

$$\begin{aligned}{} & {} \sup _{x\in \partial \Omega } \sup _{r\in ]0,+\infty [} \biggl | \, \int _{(\partial \Omega )\setminus {\mathbb {B}}_{n}(x,r)} (x_z-y_z)\frac{\partial }{\partial x_h}\frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y) \nonumber \\{} & {} \qquad \quad -(x_z-y_z) \frac{-n |T^{-1}(x-y)|^{-n-1} }{s_{n}\sqrt{\det a^{(2)} }} \frac{ \sum _{s,t=1}^n(T^{-1})_{st}(x_t-y_t)(T^{-1})_{sh} }{|T^{-1}(x-y)|} \nonumber \\{} & {} \qquad \quad \times \sum _{s=1}^n(x_s-y_s)((a^{(2)})^{-1})_{sj} -(x_z-y_z) \frac{|T^{-1}(x-y)|^{-n} }{s_{n}\sqrt{\det a^{(2)} }} ((a^{(2)})^{-1})_{hj} \,d\sigma _y \biggr | \nonumber \\{} & {} \qquad \le \sup _{x\in \partial \Omega } \int _{\partial \Omega }|2-n||x-y|^{2-n} \left| A_{2,j}\left( \frac{x-y}{|x-y|},|x-y|\right) \right| \nonumber \\{} & {} \qquad \quad +|x-y|^{3-n}\biggl \{\sum _{s=1}^n\left| \frac{\partial A_{2,j}}{\partial x_s}\left( \frac{x-y}{|x-y|},|x-y|\right) \right| 2|x-y|^{-1}\nonumber \\{} & {} \qquad \quad +\left| \frac{\partial A_{2,j}}{\partial r} \left( \frac{x-y}{|x-y|},|x-y|\right) \right| \biggr \} +\left| \frac{\partial ^2B_1}{\partial x_h\partial x_j}(x-y)\right| |x-y|\ln |x-y| \nonumber \\{} & {} \qquad \quad +\left| \frac{\partial B_1}{\partial x_j}(x-y)\right| +\left| \frac{\partial ^2C}{\partial x_h\partial x_j}(x-y)\right| |x-y|\,d\sigma _y<+\infty . \end{aligned}$$
(4.21)

Since the function

$$\begin{aligned}{} & {} \xi _z \frac{-n |T^{-1}\xi |^{-n-1} }{s_{n}\sqrt{\det a^{(2)} }} \frac{ \sum _{s,t=1}^n(T^{-1})_{st}\xi _t(T^{-1})_{sh} }{|T^{-1}\xi |} \sum _{s=1}^n\xi _s((a^{(2)})^{-1})_{sj}\\{} & {} \qquad \qquad \qquad \quad + \xi _z \frac{|T^{-1}\xi |^{-n} }{s_{n}\sqrt{\det a^{(2)} }} ((a^{(2)})^{-1})_{hj} \qquad \forall \xi \in {\mathbb {R}}^n\setminus \{0\}\end{aligned}$$

is positively homogeneous of degree \(-(n-1)\), Theorem 4.16 implies that

$$\begin{aligned}{} & {} \sup _{x\in \partial \Omega } \sup _{r\in ]0,+\infty [} \biggl |\, \int _{(\partial \Omega )\setminus {\mathbb {B}}_{n}(x,r)} (x_z-y_z) \frac{-n |T^{-1}(x-y)|^{-n-1} }{s_{n}\sqrt{\det a^{(2)} }} \\{} & {} \qquad \quad \times \frac{ \sum _{s,t=1}^n(T^{-1})_{st}(x_t-y_t)(T^{-1})_{sh} }{|T^{-1}(x-y)|} \sum _{s=1}^n(x_s-y_s)((a^{(2)})^{-1})_{sj} \\{} & {} \qquad \quad +(x_z-y_z) \frac{|T^{-1}(x-y)|^{-n} }{s_{n}\sqrt{\det a^{(2)} }} ((a^{(2)})^{-1})_{hj} \,d\sigma _y \biggr | \end{aligned}$$

is finite. Then, the above inequality implies the validity of the statement. \(\square\)

An extension of a classical theorem for the single layer potential

We plan to prove the following extension of a known classical result for the single layer potential

$$\begin{aligned} v_\Omega [S_{ {\textbf{a}} },\mu ](x)\equiv \int _{\partial \Omega }S_{ {\textbf{a}} }(x-y)\mu (y)\,d\sigma _{y} \qquad \forall x\in {\mathbb {R}}^{n}\,, \end{aligned}$$
(5.1)

for all \(\mu \in C^{0,\alpha }(\partial \Omega )\) (cf. Miranda [28], Kirsch [15, Thm. 3.3 (a)], Wiegner [34], Dalla Riva [5], Dalla Riva, Morais and Musolino [7] and references therein.)

Theorem 5.2

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(m\in {\mathbb {N}}\setminus \{0\}\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{m,1}\). If \(\mu \in C^{m-1,1}(\partial \Omega )\), then the restriction

$$\begin{aligned} V_\Omega [S_{{\textbf{a}}} ,\mu ] \equiv v_\Omega [S_{ {\textbf{a}} },\mu ]_{|\partial \Omega } \end{aligned}$$
(5.3)

belongs to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\). Moreover, the map from the space \(C^{m-1,1}(\partial \Omega )\) to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) that takes \(\mu\) to \(V_\Omega [S_{{\textbf{a}}},\mu ]\) is continuous.

Proof

We proceed by induction on m. Let \(m=1\). By the definition of norm in \(C^{1,\omega _1(\cdot )}(\partial \Omega )\), it suffices to show that

  1. (j)

    \(V_\Omega [S_{{\textbf{a}}},\cdot ]\) is linear and continuous from \(C^{0,1}(\partial \Omega )\) to \(C^{0}(\partial \Omega )\).

  2. (jj)

    \(V_\Omega [S_{{\textbf{a}}},\mu ]\) is continuously differentiable on \(\partial \Omega\) for all \(\mu\) in \(C^{0,1}(\partial \Omega )\).

  3. (jjj)

    \(M_{jl}[V_\Omega [S_{{\textbf{a}}},\cdot ]]\) is linear and continuous from \(C^{0,1}(\partial \Omega )\) to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\) for all \(j,l\in \{1,\dots ,n\}\),

(cf., e.g., [8, Lem. 2.3]). Since \(C^{1,1}(\partial \Omega )\) is continuously embedded into \(C^{1,\alpha }(\partial \Omega )\) for all \(\alpha \in ]0,1[\), statements (j), (jj) hold by [8, Thm. 7.1 (i)].

We now consider statement (jjj). To do so, we plan to write a formula for the tangential derivatives of the single layer in terms of the tangential derivatives of the density on \(\partial \Omega\). Let \(v^+_\Omega [S_{ {\textbf{a}} },\mu ]\) denote the restriction of \(v_\Omega [S_{ {\textbf{a}} },\mu ]\) to \(\overline{\Omega }\). Since \(\Omega\) is of class \(C^{1,1}\) and accordingly of class \(C^{1,\alpha }\) for all \(\alpha \in ]0,1[\) and \(\mu \in C^{0,\alpha }(\partial \Omega )\), we know that \(v^+_\Omega [S_{ {\textbf{a}} },\mu ]\in C^{1,\alpha }(\overline{\Omega })\) for all \(\alpha \in ]0,1[\) (cf. [8, Thm. 7.1 (i)]). Let \(j,l\in \{1,\dots ,n\}\). Since \(\Omega\) is of class \(C^{1,1}\) and \(\nu\) is of class \(C^{0,1}\), there exists \(\tilde{\nu }\in C^{0,1}({\mathbb {R}}^n)\) with compact support such that \(\tilde{\nu }_{|\partial \Omega }=\nu\) (cf., e.g., [6, Thm. 2.85]). Next, we find convenient to introduce the notation

$$M^{\sharp }_{jl}[f](x)\equiv \tilde{\nu }_{j}(x)\frac{\partial f}{\partial x_{l}}(x) - \tilde{\nu }_{l}(x)\frac{\partial f}{\partial x_{j}}(x)\qquad \forall x\in \overline{\Omega }\,,$$

for all \(f\in C^{1}(\overline{\Omega })\). If necessary, we write \(M^{\sharp }_{jl,x}\) to emphasize that we are taking x as variable of the differential operator \(M^{\sharp }_{jl}\). Next, we fix \(x\in \Omega\) and we note that

$$\begin{aligned}{} & {} M^{\sharp }_{jl}\left[ v^+_\Omega [S_{ {\textbf{a}} },\mu ]\right] (x)\nonumber \\{} & {} \qquad \qquad \quad =\tilde{\nu }_j(x)\frac{\partial }{\partial x_l}v^+_\Omega [S_{ {\textbf{a}} },\mu ](x) -\tilde{\nu }_l(x)\frac{\partial }{\partial x_j}v^+_\Omega [S_{ {\textbf{a}} },\mu ](x) \nonumber \\{} & {} \qquad \qquad \quad = \int _{\partial \Omega }(\tilde{\nu }_j(x)-\nu _j(y))\frac{\partial }{\partial x_l}S_{{\textbf{a}}}(x-y) \mu (y) \nonumber \\{} & {} \qquad \qquad \qquad -(\tilde{\nu }_l(x)-\nu _l(y))\frac{\partial }{\partial x_j}S_{{\textbf{a}}}(x-y) \mu (y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \qquad + \int _{\partial \Omega }\left[ \nu _j(y)\frac{\partial }{\partial x_l}\left( S_{{\textbf{a}}}(x-y)\right) -\nu _l(y)\frac{\partial }{\partial x_j}\left( S_{{\textbf{a}}}(x-y)\right) \right] \mu (y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \quad = \int _{\partial \Omega }(\tilde{\nu }_j(x)-\tilde{\nu }_j(y))\frac{\partial }{\partial x_l}S_{{\textbf{a}}}(x-y) \mu (y) \nonumber \\{} & {} \qquad \qquad \qquad -(\tilde{\nu }_l(x)-\tilde{\nu }_l(y))\frac{\partial }{\partial x_j}S_{{\textbf{a}}}(x-y) \mu (y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \qquad -\int _{\partial \Omega } \left[ \tilde{\nu }_j(y)\frac{\partial }{\partial y_l}S_{{\textbf{a}}}(x-y) - \tilde{\nu }_l(y) \frac{\partial }{\partial y_j}S_{{\textbf{a}}}(x-y) \right] \mu (y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \quad = \int _{\partial \Omega }(\tilde{\nu }_j(x)-\tilde{\nu }_j(y))\frac{\partial }{\partial x_l}S_{{\textbf{a}}}(x-y) \mu (y) \nonumber \\{} & {} \qquad \qquad \qquad -(\tilde{\nu }_l(x)-\tilde{\nu }_l(y))\frac{\partial }{\partial x_j}S_{{\textbf{a}}}(x-y) \mu (y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \qquad -\int _{\partial \Omega } M_{jl,y} \left[ S_{{\textbf{a}}}(x-y) \right] \mu (y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \quad = \int _{\partial \Omega }(\tilde{\nu }_j(x)-\tilde{\nu }_j(y))\frac{\partial }{\partial x_l}S_{{\textbf{a}}}(x-y) \mu (y) \nonumber \\{} & {} \qquad \qquad \qquad -(\tilde{\nu }_l(x)-\tilde{\nu }_l(y))\frac{\partial }{\partial x_j}S_{{\textbf{a}}}(x-y) \mu (y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \qquad + \int _{\partial \Omega }S_{{\textbf{a}}}(x-y) M_{jl}[\mu ](y) \,d\sigma _y \end{aligned}$$
(5.4)

(cf. Lemma A.1 of the Appendix). Since \((\tilde{\nu },\mu )\in C^{0,1}(\overline{\Omega },{\mathbb {R}}^n)\times L^\infty (\partial \Omega )\) the first integral in the right-hand side of (5.4) defines a continuous function of \(x\in \overline{\Omega }\) (cf. [8, Thm. 8.1 (i)]). Since \(\Omega\) is of class \(C^{1,1}\) and \(M_{lj}[\mu ]\in L^\infty (\partial \Omega )\), the second integral in the right-hand side of (5.4) defines a continuous function of \(x\in \overline{\Omega }\) (cf. [8, Lem. 4.2 (i)] with \(G\equiv \partial \Omega\), \(\gamma \in ]0,1[\), together with [8, Lem. 6.2 ] with \(\gamma _1\equiv n-1-\gamma\)). Since \(v^+_\Omega [S_{ {\textbf{a}} },\mu ]\) is of class \(C^1(\overline{\Omega })\), then \(M^{\sharp }_{lj}\left[ v^+_\Omega [S_{ {\textbf{a}} },\mu ]\right]\) is continuous on \(\overline{\Omega }\). Hence, the left and right-hand sides of (5.4) must be equal for all \(x\in \overline{\Omega }\) and thus we have

$$\begin{aligned}{} & {} M_{jl}[V_\Omega [S_{{\textbf{a}}},\mu ]](x)=M^{\sharp }_{jl}\left[ v^+_\Omega [S_{ {\textbf{a}} },\mu ]\right] (x)\nonumber \\{} & {} \qquad \qquad \quad = \int _{\partial \Omega }(\nu _j(x)-\nu _j(y))\frac{\partial }{\partial x_l}S_{{\textbf{a}}}(x-y) \mu (y) \nonumber \\{} & {} \qquad \qquad \qquad -( \nu _l(x)-\nu _l(y))\frac{\partial }{\partial x_j}S_{{\textbf{a}}}(x-y) \mu (y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \qquad + \int _{\partial \Omega }S_{{\textbf{a}}}(x-y) M_{jl}[\mu ](y) \,d\sigma _y \nonumber \\{} & {} \qquad \qquad \quad =Q_{l}[\nu _j,\mu ](x)-Q_{j}[\nu _l,\mu ](x)+V_\Omega [S_{{\textbf{a}}},M_{jl}[\mu ]](x) \qquad \forall x\in \partial \Omega . \end{aligned}$$
(5.5)

Since the components of \(\nu\) are of class \(C^{0,1}\), the first two terms in the right-hand side of (5.5) define linear and continuous maps of the variable \(\mu\) from \(L^\infty (\partial \Omega )\) to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\) (cf. [8, Thm. 8.2 (i)]). Since \(M_{lj}\) is continuous from \(C^{0,1}(\partial \Omega )\) to \(L^\infty (\partial \Omega )\) and \(V_\Omega [S_{{\textbf{a}}},\cdot ]\) is linear and continuous from \(L^\infty (\partial \Omega )\) to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\) (cf. [8, Thm. 7.2], [20, Prop. 5.3]), we conclude that the right-hand side of (5.5) defines a linear and continuous map of the variable \(\mu\) from \(C^{0,1}(\partial \Omega )\) to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\). Hence, equality (5.5) implies the validity of statement (jjj) and the proof is complete.

Next, we assume that the statement holds for \(m\ge 1\), and we prove it for \(m+1\). By the definition of norm in \(C^{m+1,\omega _1(\cdot )}(\partial \Omega )\), it suffices to show that

  1. (l)

    \(V_\Omega [S_{{\textbf{a}}},\cdot ]\) is linear and continuous from \(C^{m,1}(\partial \Omega )\) to \(C^{0}(\partial \Omega )\).

  2. (ll)

    \(V_\Omega [S_{{\textbf{a}}},\mu ]\) is continuously differentiable on \(\partial \Omega\) for all \(\mu\) in \(C^{m,1}(\partial \Omega )\).

  3. (lll)

    \(M_{jl}[V_\Omega [S_{{\textbf{a}}},\cdot ]]\) is linear and continuous from \(C^{m,1}(\partial \Omega )\) to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) for all \(j,l\in \{1,\dots ,n\}\),

(cf., e.g., [8, Lem. 2.3]). Since \(C^{m,1}(\partial \Omega )\) is continuously embedded into \(C^{0,1}(\partial \Omega )\), statements (l), (ll) hold by case \(m=1\). We now prove statement (lll) by exploiting the formula (5.5) and the inductive assumption. Since the components of \(\nu\) are of class \(C^{m,1}\) and \(\Omega\) is of class \(C^{m+1,1}\) and accordingly of class \(C^{m+1,\alpha }\) for all \(\alpha \in ]0,1[\), the first two terms in the right-hand side of (5.5) define linear and continuous maps of the variable \(\mu\) from \(C^{m}(\partial \Omega )\) to \(C^{m,\omega _1(\cdot )}\) (cf. [8, Thm. 8.3 (i)]).

Since \(M_{jl}\) is continuous from \(C^{m,1}(\partial \Omega )\) to \(C^{m-1,1}(\partial \Omega )\) and the inductive assumption implies that \(V_\Omega [S_{{\textbf{a}}},\cdot ]\) is linear and continuous from \(C^{m-1,1}(\partial \Omega )\) to \(C^{m,\omega _1(\cdot )}\), we conclude that the right-hand side of (5.5) defines a linear and continuous map of the variable \(\mu\) from \(C^{m,1}(\partial \Omega )\) to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\). Hence, equality (5.5) implies the validity of statement (lll) and the proof is complete.\(\square\)

Analysis of the map \(Q_r\)

We are now ready to prove the following statement.

Theorem 6.1

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(\alpha \in ]0,1]\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{2,\alpha }\). Let \(j\in \{1,\dots ,n\}\). Then, the following statements hold.

  1. (i)

    If \(\alpha \in ]0,1[\), \(\beta \in ]0,\alpha ]\), then the bilinear map \(Q_j\left[ \cdot ,\cdot \right]\) from the space \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) to \(C^{1,\beta }(\partial \Omega )\) which takes a pair \((g,\mu )\) to \(Q_j\left[ g,\mu \right]\) is continuous (cf. (4.12)).

  2. (ii)

    If \(\alpha =1\), \(\beta =1\), then the bilinear map \(Q_j\left[ \cdot ,\cdot \right]\) from the space \(C^{1,1}(\partial \Omega )\times C^{0,1}(\partial \Omega )\) to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) which takes a pair \((g,\mu )\) to \(Q_j\left[ g,\mu \right]\) is continuous (cf. (4.12)).

Proof

We prove statements (i) and (ii) at the same time and make some appropriate comments when the two proofs present some difference. By the definition of norm in \(C^{1,\beta }(\partial \Omega )\) with \(\beta \in ]0,1[\) and in \(C^{1,\omega _1(\cdot )}(\partial \Omega )\), it suffices to show that

  1. (j)

    \(Q_j\) is bilinear and continuous from \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) to \(C^{0}(\partial \Omega )\).

  2. (jj)

    \(Q_j\left[ g,\mu \right]\) is continuously differentiable on \(\partial \Omega\) for all \((g,\mu )\) in \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\).

  3. (jjj)

    \({\textrm{grad}}_{\partial \Omega }Q_j\left[ \cdot ,\cdot \right]\) is bilinear and continuous from \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) to \(C^{0,\beta }(\partial \Omega ,{\mathbb {R}}^n)\) in case of statement (i) and to \(C^{0,\omega _1(\cdot )}(\partial \Omega , {\mathbb {R}}^n)\) in case of statement (ii),

(cf., e.g., [8, Lem. 2.3]). For a proof of (j), we refer to [8, Thm. 8.2 (i)]. Next, we set

$$K_j[g](x,y)\equiv (g(x)-g(y))\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)$$

for all \((x,y)\in (\partial \Omega )^2\setminus {\mathbb {D}}_{\partial \Omega }\) and \(g\in C^{1,\alpha }(\partial \Omega )\) and we note that

$$Q_j\left[ g,\mu \right] (x)=\int _{\partial \Omega }K_j[g](x,y)\mu (y)\,d\sigma _y \qquad \forall (g,\mu )\in C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )$$

and we turn to the proof of (jj). To do so, we resort to a classical differentiation Theorem in the form of [19, Thm. 6.2] and we turn to verify its assumptions. It is known that

$$\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\in {\mathcal {K}}_{n-1,n,1}((\partial \Omega )\times (\partial \Omega ))\subseteq {\mathcal {K}}_{n-1,n-1+\alpha ,\alpha }((\partial \Omega )\times (\partial \Omega ))$$

(cf. Lemma 3.4, [8, Lem. 4.3]). Then, the product Lemma [19, Lem. 3.4 (ii)] implies that

$$K_j[g]\in {\mathcal {K}}_{n-1-1,n-1,1-(1-1)} ((\partial \Omega )\times (\partial \Omega ))={\mathcal {K}}_{n-2,n-1,1} ((\partial \Omega )\times (\partial \Omega ))$$

for all \(g\in C^{0,1}(\partial \Omega )\) and that there exists \(c_1\in ]0,+\infty [\) such that

$$\begin{aligned} \Vert K_j[g]\Vert _{ {\mathcal {K}}_{n-2,n-1,1} ((\partial \Omega )\times (\partial \Omega ))}\le c_1\left\| \frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\right\| _{ {\mathcal {K}}_{n-1,n,1} ((\partial \Omega )\times (\partial \Omega ))}\Vert g\Vert _{C^{0,1}(\partial \Omega )} \end{aligned}$$
(6.2)

for all \(g\in C^{0,1}(\partial \Omega )\). Since \(\Omega\) is of class \(C^{2,\alpha }\) and (g, 1) belongs to \(C^{1,\alpha }(\partial \Omega )\times C^{1,\beta }(\partial \Omega )\), Theorem 8.3 of [8] implies that

$$\begin{aligned} \int _{\partial \Omega }K_j[g](\cdot ,y)\,d\sigma _y=Q_j[g,1]\in \left\{ \begin{array}{ll} C^{1,\alpha }(\partial \Omega )&{} \text {if}\ \alpha \in ]0,1[\,, \\ C^{1,\omega _1(\cdot )}(\partial \Omega )&{} \text {if}\ \alpha =1\,, \end{array} \right. \end{aligned}$$
(6.3)

for all \(g\in C^{1,\alpha }(\partial \Omega )\) and that

$$\begin{aligned}{} & {} Q_j[\cdot ,1]\ \text {is\ linear\ and\ continuous\ from }\ C^{1,\alpha }(\partial \Omega ) \nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad \text {to}\ \left\{ \begin{array}{ll} C^{1,\alpha }(\partial \Omega )&{} \text {if}\ \alpha \in ]0,1[, \\ C^{1,\omega _1(\cdot )}(\partial \Omega )&{} \text {if}\ \alpha =1. \end{array} \right. \end{aligned}$$
(6.4)

We also note that

$$K_j[g]\in C^1((\partial \Omega ){\setminus }\{y\})\qquad \forall y\in \partial \Omega \,,$$

for all \(g\in C^{1,\alpha }(\partial \Omega )\). Next, we compute the tangential gradient with respect to x of \(K_j[g]\). By the Leibnitz rule, we have

$$\begin{aligned}{} & {} {\textrm{grad}}_{\partial \Omega ,x} K_j[g](x,y) \nonumber \\{} & {} \qquad \qquad \quad ={\textrm{grad}}_{\partial \Omega ,x} g(x)\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y) +(g(x)-g(y)){\textrm{grad}}_{\partial \Omega ,x}\left( \frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\right) \end{aligned}$$
(6.5)

for all \((x,y)\in (\partial \Omega )^2\setminus {\mathbb {D}}_{\partial \Omega }\) and \(g\in C^{1,\alpha }(\partial \Omega )\). Since \(\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\) belongs to \({\mathcal {K}}_{n-1,n-1+\alpha ,\alpha }((\partial \Omega )\times (\partial \Omega ))\), \(n-1<n-1+\alpha\), and the components of \({\textrm{grad}}_{\partial \Omega ,x} g\) are \(\alpha\)-Hölder continuous, the product Proposition 3.6 (ii) implies that

$$\begin{aligned} ({\textrm{grad}}_{\partial \Omega ,x} g)_h(x)\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\in {\mathcal {K}}_{n-1,n-1+\alpha ,\alpha }((\partial \Omega )\times (\partial \Omega )) \end{aligned}$$
(6.6)

for all \(g\in C^{1,\alpha }(\partial \Omega )\) and \(h\in \{1,\dots ,n\}\). Since g is Lipschitz continuous, Lemma 3.8 implies that

$$\begin{aligned} (g(x)-g(y))\in {\mathcal {K}}_{-1,0,1}((\partial \Omega )\times (\partial \Omega )) \end{aligned}$$
(6.7)

for all \(g\in C^{1,\alpha }(\partial \Omega )\). Since \(\Omega\) is of class \(C^{2,\alpha }\), then it is also of class \(C^{1,1}\) and Lemma 4.13 implies that

$$\begin{aligned} \left( {\textrm{grad}}_{\partial \Omega ,x} \left( \frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y) \right) \right) _h\in {\mathcal {K}}_{n,n+1,1}((\partial \Omega )\times (\partial \Omega )) \end{aligned}$$
(6.8)

for all \(h\in \{1,\dots ,n\}\). Then, the product Theorem 3.5 (ii) implies that

$$\begin{aligned} (g(x)-g(y))\left( {\textrm{grad}}_{\partial \Omega ,x} \left( \frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y)\right) \right) _h\in {\mathcal {K}}_{n-1,n,1}((\partial \Omega )\times (\partial \Omega )) \end{aligned}$$
(6.9)

and that there exists \(c_2\in ]0,+\infty [\) such that

$$\begin{array}{l}\left\| (g(x)-g(y))\left( {\textrm{grad}}_{\partial \Omega ,x} \left( \frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y)\right) \right) _h\right\| _{{\mathcal {K}}_{n-1,n,1}((\partial \Omega )\times (\partial \Omega )) }\nonumber\\\le c_2\Vert g\Vert _{C^{0,1}(\partial \Omega )}\qquad \forall g\in C^{0,1}(\partial \Omega )\end{array}$$
(6.10)

for all \(h\in \{1,\dots ,n\}\). In particular, equality (6.5) and the memberships of (6.6), (6.9) imply that

$$({\textrm{grad}}_{\partial \Omega ,x} K_j[g])_h\in {\mathcal {K}}_{n-1,(\partial \Omega )\times (\partial \Omega )}\qquad \forall g\in C^{1,\alpha }(\partial \Omega )$$

for all \(h\in \{1,\dots ,n\}\). Then, \(\int _{\partial \Omega }K_j[g](\cdot ,y)\mu (y)\,d\sigma _y\) is continuously differentiable and

$$\begin{aligned}{} & {} {\textrm{grad}}_{\partial \Omega } \int _{\partial \Omega }K_j[g](x,y)\mu (y)\,d\sigma _y\nonumber \\{} & {} \qquad \qquad \quad =\int _{\partial \Omega }[{\textrm{grad}}_{\partial \Omega ,x}K_j[g](x,y)](\mu (y)-\mu (x))\,d\sigma _y \nonumber \\{} & {} \qquad \qquad \qquad +\mu (x){\textrm{grad}}_{\partial \Omega } \int _{\partial \Omega }K_j[g](x,y) \,d\sigma _y , \end{aligned}$$
(6.11)

for all \(x\in \partial \Omega\) and for all \((g,\mu )\in C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) (cf. [19, Thm. 6.2]) and the proof of (jj) is complete. We now turn to prove (jjj). By equalities (6.5) and (6.11), we have

$$\begin{aligned}{} & {} ({\textrm{grad}}_{\partial \Omega } \int _{\partial \Omega }K_j[g](x,y)\mu (y)\,d\sigma _y)_h\nonumber \\{} & {} \qquad \quad = ({\textrm{grad}}_{\partial \Omega ,x} g(x))_h\int _{\partial \Omega }(\mu (y)-\mu (x))\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y) \,d\sigma _y \nonumber \\{} & {} \quad \qquad \qquad +\int _{\partial \Omega }(g(x)-g(y))\left( {\textrm{grad}}_{\partial \Omega ,x}\left( \frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\right) \right) _h(\mu (y)-\mu (x)) \,d\sigma _y \nonumber \\{} & {} \quad \qquad \qquad +\mu (x)({\textrm{grad}}_{\partial \Omega } \int _{\partial \Omega }K_j[g](x,y) \,d\sigma _y)_h , \end{aligned}$$
(6.12)

for all \(x\in \partial \Omega\), for all \((g,\mu )\in C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) and \(h\in \{1,\dots ,n\}\). In order to prove statement (jjj) it suffices to show that each addendum in the right-hand side of formula (6.12), defines a bilinear and continuous map from \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) to \(C^{0,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). We first consider the first addendum. Since \(\Omega\) is of class \(C^{2,\alpha }\), [8, Thm. 8.2] implies that \(Q_j[\cdot ,1]\) is linear and continuous from \(C^{0,\beta }(\partial \Omega )\) to \(C^{0,\beta }(\partial \Omega )\) in case of statement (i) and from \(C^{0,1}(\partial \Omega )\) to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Since the components of \({\textrm{grad}}_{\partial \Omega ,x}\) are linear and continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{0,\alpha }(\partial \Omega )\) and the pointwise product is bilinear and continuous in (generalized) Hölder spaces (cf., e.g., [8, Lem. 2.5]), we deduce that the first addendum in the right-hand side of formula (6.12), defines a bilinear and continuous map from \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) to \(C^{0,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

Next, we consider the second addendum in the right-hand side of the formula (6.12), that is an integral operator with the kernel of (6.9). We plan to apply a result of [18, Prop. 6.3 (ii)]. Since \(Y\equiv \partial \Omega\) is a compact manifold of class \(C^1\) that is embedded in \({\mathbb {R}}^n\), Y can be proved to be strongly upper \((n-1)\)-Ahlfors regular with respect to Y in the sense of [18, (1.4)] (cf. [22, Rmk. 2]). Then, we set

$$s_1\equiv n-1\,,\quad s_2\equiv n\,,\quad s_3 \equiv 1\,.$$

and we note that

$$(n-1)>0 \,,\ \ \beta \in ]0,\alpha ]\subseteq ]0,1]\,,\ \ s_1\in [\beta ,(n-1)+\beta [\,,\ \ s_2\in [\beta ,+\infty [$$

and that

$$\begin{aligned}{} & {} s_2-\beta =n-\beta >n-1,\\{} & {} \qquad \qquad \qquad \qquad \qquad s_2=n<n-1+\beta +1=n-1+\beta +s_3 \quad \text {if}\ \beta <1, \\{} & {} s_2-\beta =n-\beta =n-1 \quad \text {if}\ \beta =1. \end{aligned}$$

Then, [18, Prop. 6.3 (ii) (b) and (bb)] implies that the map

$$\text {from}\ {\mathcal {K}}^\sharp _{n-1,n,1}((\partial \Omega )\times (\partial \Omega ))\times C^{0,\beta }(\partial \Omega )\ \text {to}\ \left\{ \begin{array}{ll} C^{0,\beta }(\partial \Omega )&{} \text {if}\ \beta \in ]0,1[\,, \\ C^{1,\omega _1(\cdot )}(\partial \Omega )&{} \text {if}\ \beta =1\,, \end{array} \right.$$

that takes a pair \((K,\mu )\) to \(\int _{\partial \Omega }K(\cdot ,y)(\mu (y)-\mu (x))\,d\sigma _y\) is bilinear and continuous. Thus, it suffices to show that the map

$$\text {from}\ C^{1,\alpha }(\partial \Omega )\ \text {to}\ {\mathcal {K}}^\sharp _{n-1,n,1}((\partial \Omega )\times (\partial \Omega ))$$

that takes g to the kernel in (6.9) is linear and continuous. By (6.10), we know that such a map is linear and continuous from \(C^{1,\alpha }(\partial \Omega )\) to the space \({\mathcal {K}}_{n-1,n,1}((\partial \Omega )\times (\partial \Omega ))\). Then, by Lemma A.2 of the Appendix, there exists \(c_{\Omega ,1}\in ]0,+\infty [\) such that

$$\begin{aligned}{} & {} \sup _{x\in \partial \Omega }\sup _{r\in ]0,+\infty [} \left| \int _{(\partial \Omega )\setminus {\mathbb {B}}_n(x,r)} (g(x)-g(y))\left( {\textrm{grad}}_{\partial \Omega ,x}\left( \frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\right) \right) _h \,d\sigma _y \right| \nonumber \\{} & {} \qquad \le \sup _{x\in \partial \Omega }\sup _{r\in ]0,+\infty [} \biggl | \int _{(\partial \Omega )\setminus {\mathbb {B}}_n(x,r)} (g(x)-g(y) + \left( {\textrm{grad}}_{\partial \Omega }g(x)\right) \cdot (y-x) ) \end{aligned}$$
$$\begin{aligned}{} & {} \qquad \qquad \times \left( {\textrm{grad}}_{\partial \Omega ,x}\left( \frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\right) \right) _h \,d\sigma _y \biggr | \nonumber \\{} & {} \qquad +\sup _{x\in \partial \Omega }\sup _{r\in ]0,+\infty [} \biggl | \int _{(\partial \Omega )\setminus {\mathbb {B}}_n(x,r)} \left( ({\textrm{grad}}_{\partial \Omega }g(x))\cdot (y-x) \right) \nonumber \\{} & {} \qquad \qquad \times ({\textrm{grad}}_{\partial \Omega ,x}\frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y))_h \,d\sigma _y \biggr | \nonumber \\{} & {} \qquad \le c_{\Omega ,1} \biggl (\sup _{\partial \Omega }|g| +\sup _{\partial \Omega }|{\textrm{grad}}_{\partial \Omega }g| \nonumber \\{} & {} \qquad \qquad +|{\textrm{grad}}_{\partial \Omega }g:\partial \Omega |_\alpha \biggr ) \left\| \left( {\textrm{grad}}_{\partial \Omega ,x}\left( \frac{\partial S_{ {\textbf{a}} }}{\partial x_{j}}(x-y)\right) \right) _h \right\| _{{\mathcal {K}}_{n,n+1,n}((\partial \Omega )\times (\partial \Omega ))} \nonumber \\{} & {} \quad \qquad \qquad \qquad \times \sup _{x\in \partial \Omega } \int _{\partial \Omega }|x-y|^{1+\alpha -n}\,d\sigma _y \nonumber \\{} & {} \qquad \qquad +\sum _{z=1}^n\sup _{x\in \partial \Omega }|({\textrm{grad}}_{\partial \Omega }g(x))_z| \nonumber \\{} & {} \qquad \qquad \times 2n \sup _{s\in \{1,\dots ,n\}}\sup _{x\in \partial \Omega } \sup _{r\in ]0,+\infty [} \left| \, \int _{(\partial \Omega )\setminus {\mathbb {B}}_{n}(x,r)} (x_z-y_z)\frac{\partial }{\partial x_s}\frac{\partial S_{{\textbf{a}}}}{\partial x_j}(x-y) \,d\sigma _y\right| \nonumber \end{aligned}$$
(6.13)

for all \(g\in C^{1,\alpha }(\partial \Omega )\) and \(h\in \{1,\dots ,n\}\) (see also (4.20), (6.8), the definition of tangential gradient and Lemma 4.18). Hence, the map

$$\text {from}\ C^{1,\alpha }(\partial \Omega )\ \text {to}\ {\mathcal {K}}^\sharp _{n-1,n,1}((\partial \Omega )\times (\partial \Omega ))$$

that takes g to the kernel in (6.9) is linear and continuous and accordingly the second addendum in the right-hand side of formula (6.12), defines a bilinear and continuous map from \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) to \(C^{0,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

Next, we consider the third addendum on the right-hand side of the formula (6.12). Since \({\textrm{grad}}_{\partial \Omega ,x}\) is linear and continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{0,\alpha }(\partial \Omega )\), the continuity of \(Q_j[\cdot ,1]\) as in (6.4) and the continuity of the pointwise product in generalized Hölder spaces (cf., e.g., [8, Lem. 2.5]) imply that the third addendum in the right-hand side of formula (6.12), defines a bilinear and continuous map from \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) to \(C^{0,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{0,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Hence, the proof of (jjj) and of the theorem is complete.\(\square\)

In the previous theorem, we have considered sets of class \(C^{2,\alpha }\). We are now ready to consider case \(C^{m,\alpha }\) by an inductive argument on \(m\ge 2\) as in the proof of [8, Thm. 8.3].

Theorem 6.14

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(\alpha \in ]0,1]\). Let \(m\in {\mathbb {N}}\), \(m\ge 2\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{m,\alpha }\). Let \(r\in \{1,\dots ,n\}\). Then, the following statements hold.

  1. (i)

    If \(\alpha \in ]0,1[\) and \(\beta \in ]0,\alpha ]\), then the bilinear map \(Q_r\left[ \cdot ,\cdot \right]\) from the space \(C^{m-1,\alpha }(\partial \Omega )\times C^{m-2,\beta }(\partial \Omega )\) to \(C^{m-1,\beta }(\partial \Omega )\) which takes a pair \((g,\mu )\) to \(Q_r\left[ g,\mu \right]\) is continuous (cf. (4.12)).

  2. (ii)

    If \(\alpha =1\) and \(\beta =1\), then the bilinear map \(Q_r\left[ \cdot ,\cdot \right]\) from the space \(C^{m-1,1}(\partial \Omega )\times C^{m-2,1}(\partial \Omega )\) to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) which takes a pair \((g,\mu )\) to \(Q_r\left[ g,\mu \right]\) is continuous (cf. (4.12)).

Proof

We prove statements (i) and (ii) at the same time and make some appropriate comments when the two proofs present some difference. We proceed by induction on m. Case \(m=2\) holds by Theorem 6.1. We now prove that if the statement holds for m, then it holds also for \(m+1\). Then, we now assume that \(\Omega\) is of class \(C^{m+1,\alpha }\) and we prove that \(Q_r\left[ \cdot ,\cdot \right]\) is continuous from \(C^{m,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\) to \(C^{m,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

By the definition of norm in \(C^{m,\beta }(\partial \Omega )\) with \(\beta \in ]0,1[\) and in \(C^{m,\omega _1(\cdot )}(\partial \Omega )\), it suffices to show that

  1. (j)

    \(Q_r\) is bilinear and continuous from \(C^{m,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\) to \(C^{0}(\partial \Omega )\).

  2. (jj)

    \(Q_r\left[ g,\mu \right]\) is continuously differentiable for all \((g,\mu )\) in \(C^{m,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\).

  3. (jjj)

    If \(j,l\in \{1,\dots ,n\}\), then \(M_{lj}\left[ Q_r\left[ \cdot ,\cdot \right] \right]\) is bilinear and continuous from \(C^{m,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\) to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii),

(cf., e.g., [8, Lem. 2.3]). Statements (j), (jj) hold by the continuous embedding of \(C^{m,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\) into \(C^{1,\alpha }(\partial \Omega )\times C^{0,\beta }(\partial \Omega )\) and by case \(m=2\). We now prove statement (jjj). We first note that if \((g,\mu )\) belongs to \(C^{m,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\), then assumption \(m\ge 2\) and [8, Lem. 8.1] imply that the following formula holds

$$M_{lj}\left[ Q_r\left[ g,\mu \right] \right] = P_{ljr}[g,\mu ]\,,$$

where

$$\begin{aligned}{} & {} P_{ljr}[g,\mu ](x)\\{} & {} \qquad \equiv \biggl \{\nu _l(x)Q_r\left[ ({\textrm{grad}}_{\partial \Omega }g)_j,\mu \right] (x) - \nu _j(x)Q_r\left[ ({\textrm{grad}}_{\partial \Omega }g)_l,\mu \right] (x) \biggr \} \\{} & {} \qquad + \biggl \{\nu _l(x)Q_r\left[ g, \sum _{s=1}^{n} M_{sj}[\sum _{h=1}^{n} \frac{ a_{sh}\nu _h}{ \nu ^{t}a^{(2)}\nu }\mu ]\right] (x) \\{} & {} \qquad \qquad - \nu _j(x)Q_r\left[ g, \sum _{s =1}^{n}M_{sl}[\sum _{h=1}^{n} \frac{a_{sh}\nu _h}{ \nu ^{t}a^{(2)}\nu } \mu ]\right] (x)\biggr \}\\{} & {} \qquad + \sum _{s,h=1}^{n} a_{sh} \nu _l(x) \biggl \{\biggr . Q_s\left[ \nu _j,\frac{M_{hr}[g]\mu }{ \nu ^{t}a^{(2)}\nu }\right] (x) \\{} & {} \qquad \qquad + Q_s\left[ g, M_{hr}[\frac{\nu _j\mu }{\nu ^{t}a^{(2)}\nu } ]\right] (x) \biggl .\biggr \} \\{} & {} \qquad - \sum _{s,h=1}^{n} a_{sh} \nu _j(x) \biggl \{\biggr . Q_s\left[ \nu _l,\frac{M_{hr}[g]\mu }{ \nu ^{t}a^{(2)}\nu }\right] (x)\\{} & {} \qquad \qquad + Q_s\left[ g, M_{hr}[\frac{\nu _l\mu }{ \nu ^{t}a^{(2)}\nu } ]\right] (x) \biggl .\biggr \}\\{} & {} \qquad -\sum _{t=1}^{n} a_{s}\biggl \{\biggr .\nu _l(x) Q_s\left[ g,\frac{\nu _j\nu _r}{\nu ^{t}a^{(2)}\nu }\mu \right] (x) \\{} & {} \qquad \qquad -\nu _j(x) Q_s\left[ g,\frac{\nu _l\nu _r}{ \nu ^{t}a^{(2)}\nu }\mu \right] (x) \biggl .\biggr \} \\{} & {} \qquad -a\left\{ g(x)\left[ \nu _l(x)v_\Omega [S_{ {\textbf{a}} }, \frac{\nu _j\nu _r}{ \nu ^{t}a^{(2)}\nu }\mu ](x) - \nu _j(x)v_\Omega [S_{ {\textbf{a}} }, \frac{\nu _l\nu _r}{ \nu ^{t}a^{(2)}\nu }\mu ](x) \right] \right. \\{} & {} \qquad - \left. \left[ \nu _l(x)v_\Omega [S_{ {\textbf{a}} }, g\frac{\nu _j\nu _r}{ \nu ^{t}a^{(2)}\nu }\mu ](x) - \nu _j(x)v_\Omega [S_{ {\textbf{a}} }, g\frac{\nu _l\nu _r}{ \nu ^{t}a^{(2)}\nu }\mu ](x) \right] \right\} \ \ \forall x\in \partial \Omega , \end{aligned}$$

for all \((g,\mu )\in C^{m,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\). We first prove that if \((g,\mu )\) belongs to \(C^{m,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\), then each term in the right-hand side of the equality that defines \(P_{ljr}[g,\mu ]\) belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Then, the proof of the continuity of \(P_{ljr}\) as in (jjj) follows the same lines and is accordingly omitted.

By the continuity of all the components of \({\textrm{grad}}_{\partial \Omega }\) from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\), we have \(({\textrm{grad}}_{\partial \Omega }g)_j \in C^{m-1,\alpha }(\partial \Omega )\).

By the continuity of the embedding of \(C^{m-1,\beta }(\partial \Omega )\) into \(C^{m-2,\beta }(\partial \Omega )\), we have \(\mu \in C^{m-2,\beta }(\partial \Omega )\).

By the inductive assumption on \(Q_r\), \(Q_r\left[ ({\textrm{grad}}_{\partial \Omega }g)_j,\mu \right]\) belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

By the membership of the components of \(\nu\) in \(C^{m,\alpha }(\partial \Omega )\subseteq C^{m-1,\alpha }(\partial \Omega )\), and by the continuity of the pointwise product

$$\text {from}\ C^{m-1,\alpha }(\partial \Omega )\times C^{m-1,\beta }(\partial \Omega )\ \text {to}\ C^{m-1,\beta }(\partial \Omega )$$

in case of statement (i) and

$$\text {from}\ C^{m-1,\alpha }(\partial \Omega )\times C^{m-1,\omega _1(\cdot )}(\partial \Omega )\ \text {to}\ C^{m-1,\omega _1(\cdot )}(\partial \Omega )$$

in case of statement (ii) (cf., e.g., [8, Lems. 2.4, 2.5]), the sum in the first pair of braces in the right-hand side of the equality that defines \(P_{ljr}[g,\mu ]\) belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

For the remaining terms the argument is similar and thus we merely outline it. Since the components of \(\nu\) belong to \(C^{m,\alpha }(\partial \Omega )\subseteq C^{m-1,\alpha }(\partial \Omega )\), the continuity of the pointwise product in Schauder spaces implies that

$$\left( g, \sum _{s=1}^{n} M_{sj}[\sum _{h=1}^{n} \frac{ a_{sh}\nu _h}{ \nu ^{t}a^{(2)}\nu }\mu ]\right) \in C^{m-1,\alpha }(\partial \Omega )\times C^{m-2,\beta }(\partial \Omega )\,.$$

Then, the inductive assumption on \(Q_r\) ensures that

$$Q_r\left[ g, \sum _{s=1}^{n} M_{sj}[\sum _{h=1}^{n} \frac{ a_{sh}\nu _h}{ \nu ^{t}a^{(2)}\nu }\mu ]\right]$$

belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Then, again the continuity of the pointwise product in Schauder spaces implies that the sum in the second pair of braces in the right-hand side of the equality that defines \(P_{ljr}[g,\mu ]\) belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

By the membership of the components of \(\nu\) in \(C^{m,\alpha }(\partial \Omega )\subseteq C^{m-1,\alpha }(\partial \Omega )\), by the continuity of \(M_{hr}\) from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\) and by the continuity of the embedding from \(C^{m-1,\beta }(\partial \Omega )\) to \(C^{m-2,\beta }(\partial \Omega )\), by the continuity of the pointwise product in Schauder spaces (cf., e.g., [8, Lems. 2.4, 2.5]), we have

$$\left( \nu _j,\frac{M_{hr}[g]\mu }{ \nu ^{t}a^{(2)}\nu }\right) \in C^{m-1,\alpha }(\partial \Omega )\times C^{m-2,\beta }(\partial \Omega )\,.$$

Then, the inductive assumption on \(Q_s\) ensures that

$$Q_s\left[ \nu _j,\frac{M_{hr}[g]\mu }{ \nu ^{t}a^{(2)}\nu }\right]$$

belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Similarly,

$$Q_s\left[ g, M_{hr}[\frac{\nu _j\mu }{\nu ^{t}a^{(2)}\nu } ]\right]$$

belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) and thus again by the continuity of the pointwise product in Schauder spaces, the term corresponding to the third pair of braces in the right-hand side of the equality that defines \(P_{ljr}[g,\mu ]\) belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

The proof for the term corresponding to the fourth pair of braces in the right-hand side of the equality that defines \(P_{ljr}[g,\mu ]\) is the same as that for the third pair.

By the membership of the components of \(\nu\) in \(C^{m,\alpha }(\partial \Omega )\subseteq C^{m-1,\alpha }(\partial \Omega )\), by the continuity of the embedding of \(C^{m,\alpha }(\partial \Omega )\) into \(C^{m-1,\alpha }(\partial \Omega )\), by the continuity of the embedding of \(C^{m-1,\beta }(\partial \Omega )\) into \(C^{m-2,\beta }(\partial \Omega )\), by the continuity of the pointwise product in Schauder spaces (cf., e.g., [8, Lems. 2.4, 2.5]) and by the inductive assumption on \(Q_s\), the term corresponding to the fifth pair of braces in the right-hand side of the equality that defines \(P_{ljr}[g,\mu ]\) belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

The membership of the components of \(\nu\) in \(C^{m,\alpha }(\partial \Omega )\subseteq C^{m-1,\alpha }(\partial \Omega )\), the continuity of the pointwise product in Schauder spaces (cf., e.g., [8, Lems. 2.4, 2.5]) and the continuity of the operator \(v_\Omega [S_{ {\textbf{a}} }, \cdot ]_{|\partial \Omega }\) from the space \(C^{m-1,\beta }(\partial \Omega )\) to \(C^{m,\beta }(\partial \Omega )\subseteq C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\subseteq C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) (cf. [8, Thm. 7.1], Theorem 5.2) imply that the term corresponding to the last pair of braces in the right-hand side of the equality that defines \(P_{ljr}[g,\mu ]\) belongs to \(C^{m-1,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Hence, the proof of (jjj) and of the theorem is complete.\(\square\)

Next, we prove the following extension of a corresponding statement of [8, Thm. 8.4].

Theorem 6.15

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(m\in {\mathbb {N}}\setminus \{0\}\). Let \(\alpha \in ]0,1]\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{m,\alpha }\). Then, the following statements hold.

  1. (i)

    If \(\alpha \in ]0,1[\) and \(\beta \in ]0,\alpha ]\), then the trilinear operator R from the space \(\left( C^{m-1,\alpha }(\partial \Omega )\right) ^{2}\times C^{m-2,\beta }(\partial \Omega )\) to \(C^{m-1,\beta }(\partial \Omega )\) that is delivered by the formula

    $$\begin{aligned}{} & {} R[g,h,\mu ] \equiv \sum _{r=1}a_{r} \left\{ Q_r[gh,\mu ]-g Q_r[h,\mu ] -Q_r[h,g\mu ] \right\} \nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \quad +a\left\{ gV_\Omega [ S_{ {\textbf{a}} },h\mu ] - h V_\Omega [ S_{ {\textbf{a}} },g\mu ] \right\} \qquad \text {on}\ \partial \Omega \, \end{aligned}$$
    (6.16)

    for all \((g,h,\mu )\in \left( C^{m-1,\alpha }(\partial \Omega )\right) ^{2}\times C^{m-2,\beta }(\partial \Omega )\) is continuous.

  2. (ii)

    If \(\alpha =1\) and \(\beta =1\), then the trilinear operator R from the space \(\left( C^{m-1,1}(\partial \Omega )\right) ^{2}\times C^{m-2,1}(\partial \Omega )\) to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) that is delivered by the formula (6.16) is continuous.

Proof

Since R is the composition of \(Q_r\) and of the single layer potential, Theorem 6.14 on the continuity of \(Q_r\) and the continuity of \(V_\Omega [S_{ {\textbf{a}} }, \cdot ]\) from \(C^{m-1,\beta }(\partial \Omega )\) to \(C^{m,\beta }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) (cf. [8, Thm. 7.1], Theorem 5.2) and the continuity of the pointwise product in (generalized) Schauder spaces (cf., e.g., [8, Lems. 2.4, 2.5]) imply the validity of the statement.\(\square\)

Analysis of the operator \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\)

We are now ready to prove the following statement.

Theorem 7.1

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(\alpha \in ]0,1]\). Let \(m\in {\mathbb {N}}\), \(m\ge 2\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{m,\alpha }\). Then, the following statements hold.

  1. (i)

    If \(\alpha \in ]0,1[\), then \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is linear and continuous from \(C^{m-1,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\).

  2. (ii)

    If \(\alpha =1\), then \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is linear and continuous from \(C^{m-1,1}(\partial \Omega )\) to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\).

Proof

We prove statements (i) and (ii) at the same time and make some appropriate comments when the two proofs present some difference. By [8, Thm. 9.1], \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\mu ]\) is continuously differentiable and the following formula holds for the tangential derivatives of \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\mu ]\)

$$\begin{aligned} M_{lj}[W_\Omega [{\textbf{a}},S_{{\textbf{a}}} ,\mu ]]=T_{lj}[\mu ] \end{aligned}$$
(7.2)

where

$$\begin{aligned}{} & {} T_{lj}[\mu ] \equiv W_\Omega [{\textbf{a}},S_{{\textbf{a}}},M_{lj}[\mu ] ] \nonumber \\{} & {} \qquad \qquad \qquad +\sum _{b,r=1}^{n}a_{br} \left\{ Q_b\left[ \nu _{l},M_{jr}[\mu ]\right] - Q_b\left[ \nu _{j}, M_{lr}[\mu ]\right] \right\} \nonumber \\{} & {} \qquad \qquad \qquad +\nu _{l} Q_j\left[ \nu \cdot a^{(1)},\mu \right] -\nu _{j} Q_l\left[ \nu \cdot a^{(1)},\mu \right] \nonumber \\{} & {} \qquad \qquad \qquad +\nu \cdot a^{(1)} \left\{ Q_l\left[ \nu _{j},\mu \right] - Q_j\left[ \nu _{l},\mu \right] \right\} \nonumber \\{} & {} \qquad \qquad \qquad -\nu \cdot a^{(1)} V_\Omega [S_{ {\textbf{a}} }, M_{lj}[\mu ]] +V_\Omega [S_{ {\textbf{a}} }, \nu \cdot a^{(1)}M_{lj}[\mu ]]\nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad +R[\nu _{l},\nu _{j},\mu ] \qquad {\textrm{on}}\ \partial \Omega , \end{aligned}$$
(7.3)

for all \(l,j\in \{1,\dots ,n\}\) and \(\mu \in C^1(\partial \Omega )\).

We now prove the statement by induction on \(m\ge 2\). We first consider case \(m=2\). By the definition of the norm in \(C^{1,\alpha }(\partial \Omega )\) and in \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) and by formula (7.2) it suffices to prove that the following two statements hold.

  1. (j)

    \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{0}(\partial \Omega )\).

  2. (jj)

    \(T_{lj}[\cdot ]\) is continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all l, \(j\in \{1,\dots ,n\}\),

(cf. [8, Lem. 2.3 (ii)]). Since \(\Omega\) is of class \(C^{2,\alpha }\), then \(\Omega\) is of class \(C^{1,\gamma }\) for all \(\gamma \in ]0,1[\) and thus \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous from \(L^\infty (\partial \Omega )\) to \(C^{0}(\partial \Omega )\) (cf., e.g., [8, Thm. 7.4]). Hence, (j) holds true. We now prove statement (jj) by exploiting the formula (7.3). Since \(\Omega\) is of class \(C^{2,\alpha }\), then the normal \(\nu\) belongs to \(C^{1,\alpha }(\partial \Omega ,{\mathbb {R}}^n)\). Then, Theorem 6.14 with \(m=2\) ensures that \(Q_l\left[ \nu ,\cdot \right]\) and \(Q_j\left[ \nu \cdot a^{(1)},\cdot \right]\) are continuous from \(C^{0,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii), and that \(Q_b\left[ \nu _{l},M_{jr}[\cdot ]\right]\) is continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all l, j, \(r\in \{1,\dots ,n\}\).

By [21, Thm. 1.1], \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{0,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Since \(M_{jr}\) is continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{0,\alpha }(\partial \Omega )\), then \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},M_{lj}[\mu ] ]\) is continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii), for all l, \(j\in \{1,\dots ,n\}\).

Since \(\Omega\) is of class \(C^{1,\alpha }\), [8, Thm. 7.1] and Theorem 5.2 imply that \(V_\Omega [S_{ {\textbf{a}} }, \cdot ]\) is continuous from \(C^{0,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Since \(M_{lj}\) is continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{0,\alpha }(\partial \Omega )\), then \(V_\Omega [S_{ {\textbf{a}} }, M_{lj}[\cdot ]]\) and \(V_\Omega [S_{ {\textbf{a}} }, \nu \cdot a^{(1)}M_{lj}[\mu ]]\) are continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all l, \(j\in \{1,\dots ,n\}\). Then, the membership of \(\nu\) in \(C^{1,\alpha }(\partial \Omega ,{\mathbb {R}}^n)\) and Theorem 6.15 imply that \(T_{lj}\) is continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all l, \(j\in \{1,\dots ,n\}\) and thus statement (jj) holds true.

Hence, we have proved statements (j) and (jj), and thus \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{1,\alpha }(\partial \Omega )\) to \(C^{2,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{2,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

We now assume that \(\Omega\) is of class \(C^{m+1,\alpha }\) and that the statement is true for \(m\ge 2\) and we turn to prove that \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m+1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m+1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). By the definition of norm in \(C^{m+1,\alpha }(\partial \Omega )\) and in \(C^{m+1,\omega _1(\cdot )}(\partial \Omega )\) and formula (7.2), it suffices to prove that the following statements hold true.

  1. (a)

    \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{0}(\partial \Omega )\).

  2. (aa)

    \(T_{lj}[\cdot ]\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all l, \(j\in \{1,\dots ,n\}\),

(cf. [8, Lem. 2.3 (ii)]). Since \(C^{m,\alpha }(\partial \Omega )\) is continuously embedded into \(C^{2,\alpha }(\partial \Omega )\), statement (a) follows by case \(m=2\). We now prove (aa). By the inductive assumption, \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{m-1,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii). Since \(M_{lj}[\cdot ]\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\), we conclude that \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},M_{lj}[\mu ] ]\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii), for all l, \(j\in \{1,\dots ,n\}\).

Since \(\Omega\) is of class \(C^{m+1,\alpha }\), then the normal \(\nu\) belongs to \(C^{m,\alpha }(\partial \Omega ,{\mathbb {R}}^n)\). Then, Theorem 6.14 ensures that \(Q_l\left[ \nu ,\cdot \right]\) and \(Q_r\left[ \nu \cdot a^{(1)},\cdot \right]\) are continuous from the space \(C^{m-1,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) and that \(Q_b\left[ \nu _{l},M_{jr}[\cdot ]\right]\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all l, j, \(r\in \{1,\dots ,n\}\).

Since \(\Omega\) is of class \(C^{m,\alpha }\), [8, Thm. 7.1] and Theorem 5.2 imply that \(V_\Omega [S_{ {\textbf{a}} }, \cdot ]\) is continuous from \(C^{m-1,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

Since \(M_{lj}\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\), then the operators \(V_\Omega [S_{ {\textbf{a}} }, M_{lj}[\cdot ]]\) and \(V_\Omega [S_{ {\textbf{a}} }, \nu \cdot a^{(1)}M_{lj}[\mu ]]\) are continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all l, \(j\in \{1,\dots ,n\}\). Then, the membership of \(\Omega\) in the class \(C^{m+1,\alpha }\), of \(\nu\) in \(C^{m,\alpha }(\partial \Omega ,{\mathbb {R}}^n)\) and Theorem 6.15 imply that \(T_{lj}\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all l, \(j\in \{1,\dots ,n\}\) and thus statement (aa) holds true.

Hence, we have proved the validity of (a), (aa) and \(W_\Omega [{\textbf{a}},S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{m,\alpha }(\partial \Omega )\) to \(C^{m+1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m+1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) and the proof is complete. \(\square\)

An integral operator associated with the conormal derivative of a single layer potential

Another relevant layer potential operator associated with the analysis of boundary value problems for the operator \(P[{\textbf{a}},D]\) is defined by

$$W_{*,\Omega }[{\textbf{a}}, S_{ {\textbf{a}} },\mu ](x)\equiv \int _{\partial \Omega }\mu (y)DS_{ {\textbf{a}} }(x-y)a^{(2)}\nu (x)\,d\sigma _{y}\qquad \forall x\in \partial \Omega$$

for all \(\mu \in C^{0}(\partial \Omega )\). We now show that Theorems 5.2, 6.14, 7.1, [8, Thm. 7.1] imply the validity of the following statement, that exploits an elementary formula for \(W_{*,\Omega }\) (cf., e.g., [8, Proof of Thm. 10.1]). We also mention that the following statement extends the corresponding result of Kirsch [15, Thm. 3.3 (b)] who has considered the case in which \(S_{ {\textbf{a}} }\) is the fundamental solution of the Helmholtz operator, \(n=3\), \(\alpha \in ]0,1[\).

Theorem 8.1

Let \({\textbf{a}}\) be as in (1.1), (1.2), (1.3). Let \(S_{ {\textbf{a}} }\) be a fundamental solution of \(P[{\textbf{a}},D]\). Let \(\alpha \in ]0,1]\). Let \(m\in {\mathbb {N}}\), \(m\ge 2\). Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^{n}\) of class \(C^{m,\alpha }\). Then, the following statements hold.

  1. (i)

    If \(\alpha \in ]0,1[\), then the operator \(W_{*,\Omega }[{\textbf{a}}, S_{ {\textbf{a}} },\cdot ]\) is linear and continuous from \(C^{m-2,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\).

  2. (ii)

    If \(\alpha =1\), then the operator \(W_{*,\Omega }[{\textbf{a}}, S_{ {\textbf{a}} },\cdot ]\) is linear and continuous from \(C^{m-2,1}(\partial \Omega )\) to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\).

Proof

We prove statements (i) and (ii) at the same time and make some appropriate comments when the two proofs present some difference. By a simple computation, we have

$$\begin{aligned}{} & {} W_{*,\Omega }[{\textbf{a}}, S_{ {\textbf{a}} },\mu ] =\sum _{b,r=1}^{n}a_{br} Q_b[\nu _{r},\mu ] \nonumber \\{} & {} \qquad \qquad \qquad \qquad \quad - W_\Omega [{\textbf{a}}, S_{{\textbf{a}}},\mu ] - V_\Omega [S_{{\textbf{a}}},(a^{(1)}\nu ) \mu ] \qquad \text {on}\ \partial \Omega \end{aligned}$$
(8.2)

for all \(\mu \in C^{0}(\partial \Omega )\) (cf. [8, (10.1)]).

By the membership of the components of \(\nu\) in \(C^{m-1,\alpha }(\partial \Omega )\), Theorem 6.14 implies that \(Q_b[\nu _{r},\cdot ]\) is continuous from \(C^{m-2,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) for all \(r\in \{1,\dots ,n\}\).

If \(m=2\), then \(\Omega\) is of class \(C^{2}\) and thus \(W_\Omega [{\textbf{a}}, S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{m-2,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii) (cf. [21, Thm. 1.1]).

If \(m>2\), then \(\Omega\) is of class \(C^{m-1,\alpha }\) and \(m-1\ge 2\). Then, Theorem 7.1 implies that \(W_\Omega [{\textbf{a}}, S_{{\textbf{a}}},\cdot ]\) is continuous from \(C^{m-2,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

By the continuity of the pointwise product in Schauder spaces (cf., e.g., [8, Lems. 2.4, 2.5]), the map from \(C^{m-2,\alpha }(\partial \Omega )\) to itself that takes \(\mu\) to \((a^{(1)}\nu ) \mu\) is continuous. Since \(\Omega\) is of class \(C^{m-1,\alpha }\), [8, Th. 7.1] and Theorem 5.2 imply that \(V_\Omega [ S_{{\textbf{a}}},\cdot ]\) is linear and continuous from \(C^{m-2,\alpha }(\partial \Omega )\) to \(C^{m-1,\alpha }(\partial \Omega )\) in case of statement (i) and to \(C^{m-1,\omega _1(\cdot )}(\partial \Omega )\) in case of statement (ii).

Then, formula (8.2) implies the validity of statement.\(\square\)