2.1 Model operators and SobolevSlobodetskii spaces in a cone
Pseudodifferential operators are locally defined by the wellknown formula
$$u(x)\longmapsto \int_{{\mathbf{R}}^{m}} \int_{{\mathbf{R}}^{m}}A(x,\xi )u(y)e^{i(xy)\cdot\xi}\,d\xi \,dy,\quad x\in{ \mathbf{R}}^{m}, $$
if M is a compact smooth manifold because one can use the ‘freezing coefficients principle’ or, in other words, a ‘local principle’. For a manifold with a smooth boundary we need a new local formula for defining the operator A: more precisely near inner points of M we use the usual formula, but near the boundary points we need another formula:
$$u(x)\longmapsto \int_{{\mathbf{R}}^{m}_{+}} \int_{{\mathbf{R}}^{m}}A(x,\xi )u(y)e^{i(xy)\cdot\xi}\,d\xi \,dy,\quad x\in{ \mathbf{R}}^{m}_{+}, $$
where \({\mathbf{R}}^{m}_{+}=\{x\in{\mathbf{R}}^{m}: x=(x',x_{m}), x_{m}>0\}\).
For invertibility of such an operator with symbol \(A(\cdot,\xi)\) not depending on a spatial variable x one can apply the theory of the classical Riemann boundary value problems for upper and lower complex halfplanes with a parameter \(\xi'=(\xi_{1},\ldots,\xi_{m1})\). This step was systematically studied in [2]. But if the boundary ∂M has at least one conical point, this approach is not effective.
A conical point at the boundary is such a point for which its neighborhood is diffeomorphic to the cone \(C^{a}_{+}=\{x\in{\mathbf{R}}^{m}: x_{m}>a \vert x' \vert , x'=(x_{1},\ldots,x_{m1}), a>0\}\), hence a local definition for pseudodifferential operator near the conical point is the following:
$$ u(x)\longmapsto \int_{C^{a}_{+}} \int_{{\mathbf{R}}^{m}}A(x,\xi )u(y)e^{i(xy)\cdot\xi}\,d\xi \,dy,\quad x\in C^{a}_{+}. $$
(1)
To describe a solvability picture for the model elliptic pseudodifferential equation with the operator (1)
$$ (Au) (x)=v(x),\quad x\in C^{a}_{+}, $$
(2)
with symbol \(A(\cdot,\xi)\) nondepending on a spatial variable x in multidimensional cone \(C^{a}_{+}=\{x\in{\mathbf{R}}^{2}: x_{2}>a \vert x_{1} \vert , a>0\}\) earlier we considered the special singular integral operator [4]
$$(K_{a}u) (x)=\frac{a}{2\pi^{2} }\lim_{\tau\to0+} \int_{{\mathbf {R}}^{2}}\frac{u(y)\,dy}{(x_{1}y_{1})^{2}a^{2}(x_{2}y_{2}+i\tau)^{2}}. $$
This operator served a conical singularity in the general theory of boundary value problems for elliptic pseudodifferential equations on manifolds with a nonsmooth boundary. The operator \(K_{a}\) is a convolution operator, and the parameter a is the size of an angle, \(x_{2}>a \vert x_{1} \vert \), \(a=\cot\alpha\).
To study the invertibility property for the operator (1) we have introduced the concept of the wave factorization for an elliptic symbol near a singular boundary point [3, 4] and using this property we have described Fredholm properties for equation (2). We use SobolevSlobodetskii spaces for studying these properties.
Definition 1
By definition the SobolevSlobodetskii space \(H_{s}({\mathbf{R}}^{m})\) consists of distributions u for which their Fourier transforms are locally integrable functions \(\tilde{u}(\xi)\) such that
$$ \Vert u \Vert _{s}^{2}= \int_{{\mathbf{R}}^{m}} \bigl\vert \tilde {u}(\xi) \bigr\vert ^{2}\bigl(1+ \vert \xi \vert \bigr)^{2s} \,d\xi< +\infty, \qquad \tilde{u}(\xi)= \int_{{\mathbf{R}}^{m}}u(x)e^{ix\cdot\xi}\,dx. $$
(3)
We will denote the Fourier image of the space \(H_{s}({\mathbf{R}}^{m})\) by \(\tilde{H}_{s}({\mathbf{R}}^{m})\). \(\tilde{H}_{s}({\mathbf{R}}^{m})\) and, consequently, \(H_{s}({\mathbf{R}}^{m})\) are Hilbert spaces with respect to the inner product
$$\langle u,v\rangle_{s}= \int_{{\mathbf{R}}^{m}}\tilde{u}(\xi)\overline{ \tilde{v}(\xi)}\bigl(1+ \vert \xi \vert \bigr)^{2s}\,d\xi, $$
and formula (3) defines the norm in the spaces \(H_{s}({\mathbf{R}}^{m})\) and \(\tilde{H}_{s}({\mathbf{R}}^{m})\).
If \(s=0\) then \(\tilde{H}_{0}({\mathbf{R}}^{m})=L_{2}({\mathbf{R}}^{m})\), and by virtue of Plancherel’s theorem \(H_{0}({\mathbf{R}}^{m})=F^{1}\tilde{H}_{0}({\mathbf{R}}^{m})=L_{2}({\mathbf{R}}^{m})\).
In the case \(s=n\) (\(n>0\), n integer) \(H_{n}({\mathbf{R}}^{m})\) consists of functions \(u(x)\) that are integrable with their square functions, for which their generalized derivatives \(\partial D^{k}u(x)\) under \(1\le \vert k \vert \le n\) are integrable with their square functions also. The norm (3) in this case is equivalent to the following norm:
$$\Vert u \Vert _{n}^{2}=\sum _{ \vert k \vert \le n} \int _{{\mathbf{R}}^{m}} \bigl\vert \partial D^{k}u(x) \bigr\vert ^{2}\,dx= \sum_{ \vert k \vert \le n} \frac{1}{(2\pi)^{m}} \int_{{\mathbf{R}}^{m}} \bigl\vert \xi^{k}\tilde{u}(\xi) \bigr\vert ^{2}\,d\xi. $$
In the case \(s=n\), \(n>0\), n integer, the distributions from \(H_{n}({\mathbf{R}}^{m})\) are derivatives of functions from \(L_{2}({\mathbf{R}}^{m})\) whose order is not higher than n.
By definition, the space \(H_{s}(C^{a}_{+})\) consists of distributions from \(H_{s}({{\mathbf{R}}}^{m})\), which support belongs to \(\overline{C^{a}_{+}}\). The norm in the space \(H_{s}(C^{a}_{+})\) is induced by the norm from \(H_{s}({\mathbf{R}}^{m})\). The righthand side v is chosen from the space \(H_{s\alpha}^{0}(C^{a}_{+})\), which is the space of distributions \(S'(C^{a}_{+})\), admitting the continuation on \(H_{s\alpha}({\mathbf {R}}^{m})\). The norm in the space \(H_{s\alpha}^{0}(C^{a}_{+})\) is defined
$$\Vert v \Vert ^{+}_{s\alpha}=\inf \Vert lv \Vert _{s\alpha}, $$
where the infimum is chosen from all continuations l.
2.2 Wave factorization and solvability
Let us return to equation (2). We will recall some of our preliminary results [3, 4]. The symbol \(\stackrel{*}{C^{a}_{+}}\) denotes a conjugate cone for \(C^{a}_{+}\):
$$\stackrel{*}{C^{a}_{+}}=\bigl\{ x\in{\mathbf{R}}^{2}: x=(x_{1},x_{2}), ax_{2}> \vert x_{1} \vert \bigr\} , $$
\(C^{a}_{}\equivC^{a}_{+}, T(C^{a}_{+})\) denotes a radial tube domain over the cone \(C^{a}_{+}\) [20], i.e. the domain in a complex space \({\mathbf{C}}^{2}\) of type \({\mathbf{R}}^{2}+iC^{a}_{+}\).
We consider symbols \(A(\xi)\) satisfying the condition
$$c_{1}\leq \bigl\vert A(\xi) \bigl(1+ \vert \xi \vert \bigr)^{\alpha} \bigr\vert \leq c_{2}, $$
which are elliptic, and the number \(\alpha\in{\mathbf{R}}\) is called an order of the operator A.
To describe the solvability picture for equation (2) we use the following.
Definition 2
Wave factorization with respect to the cone \(C^{a}_{+}\) for the symbol \(A(\xi)\) is called a representation in the form
$$A(\xi)=A_{\neq}(\xi)A_{=}(\xi), $$
where the factors \(A_{\neq}(\xi)\), \(A_{=}(\xi)\) must satisfy the following conditions:

(1)
\(A_{\neq}(\xi)\), \(A_{=}(\xi)\) are defined for all admissible values \(\xi\in{\mathbf{R}}^{2}\), without maybe the points \(\{\xi\in {\mathbf{R}}^{2}: \vert \xi_{1} \vert ^{2}=a^{2}\xi^{2}_{2}\}\);

(2)
\(A_{\neq}(\xi)\), \(A_{=}(\xi)\) admit an analytical continuation into radial tube domains \(T(\stackrel{*}{C^{a}_{+}})\), \(T(\stackrel{*}{C^{a}_{}})\), respectively, with estimates
$$\begin{gathered} \bigl\vert A_{\neq}^{\pm1}(\xi+i \tau) \bigr\vert \leq c_{1}\bigl(1+ \vert \xi \vert + \vert \tau \vert \bigr)^{\pm\kappa}, \\ \bigl\vert A_{=}^{\pm1}(\xii\tau) \bigr\vert \leq c_{2}\bigl(1+ \vert \xi \vert + \vert \tau \vert \bigr)^{\pm(\alpha\kappa)}, \forall \tau\in\stackrel{*}{C^{a}_{+}}. \end{gathered} $$
The number \(\kappa\in{\mathbf{R}}\) is called the index of the wave factorization.
For \(\vert \kappas \vert <1/2\) one has the existence and uniqueness theorem [3]. For this purpose we need a certain lemma.
Lemma 1
Let functions
\(B_{\ne}(\xi+i\tau)\), \(B_{=}(\xi+i\tau)\)
be analytical in
\(T(\stackrel{*}{C^{a}_{+}})\)
and
\(T(\stackrel{*}{C^{a}_{}})\)
and satisfy the estimates
$$\begin{gathered} \bigl\vert B_{\ne}(\xi+i\tau) \bigr\vert \le c_{1}\bigl(1+ \vert \xi \vert + \vert \tau \vert \bigr)^{\alpha},\quad \tau\in\stackrel{*}{C^{a}_{+}}, \\ \bigl\vert B_{=}(\xi+i\tau) \bigr\vert \le c_{2}\bigl(1+ \vert \xi \vert + \vert \tau \vert \bigr)^{\alpha},\quad \tau\in\stackrel{*}{C^{a}_{}}. \end{gathered} $$
Then the multiplication operator by the function
\(B_{\ne}(\xi)\)
boundedly acts from space
\(\tilde{H}_{s}(C_{+}^{a})\)
into
\(\tilde{H}_{s\alpha}(C_{+}^{a})\), and the multiplication operator by the function
\(B_{=}(\xi)\)
from space
\(\tilde{H}_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\)
into space
\(\tilde{H}_{s\alpha}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\).
Proof
The fact that multiplication operators by functions \(B_{\ne}(\xi)\), \(B_{=}(\xi)\) boundedly act from spaces \(\tilde{H}_{s}(C_{+}^{a})\), \(\tilde{H}_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) into space \(\tilde{H}_{s\alpha}({\mathbf{R}}^{2})\) is well known [2]. For clarity we denote \(u\in\tilde{H}_{s}(C_{+}^{a})\) by \(u_{+}\) and \(u\in\tilde{H}_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) by \(u_{}\). Let us show that \(B_{\ne}(\xi)\tilde{u}_{+}(\xi)\in\tilde{H}_{s\alpha}(C_{+}^{a})\) for any \(\tilde{u}_{+}\in\tilde{H}_{s}(C_{+}^{a})\).
The space \(\tilde{H}_{s}(C_{+}^{a})\) has an explicit description [3]: \(\tilde{u}_{+}\in\tilde{H}_{s}(C_{+}^{a})\) if and only if \(\tilde{u}_{+}(\xi+i\tau)\) is analytical in \(T(\stackrel{*}{C^{a}_{+}})\) and the quantity
$$\sup \int_{{\mathbf{R}}^{2}} \bigl\vert \tilde{u}_{+}(\xi+i\tau) \bigr\vert ^{2}\bigl(1+ \vert \xi \vert \bigr)^{2s}\,d\xi,\quad \tau \in\stackrel{*}{C^{a}_{+}}, $$
is finite and coincides with
$$\int_{{\mathbf{R}}^{2}} \bigl\vert \tilde{u}_{+}(\xi) \bigr\vert ^{2}\bigl(1+ \vert \xi \vert \bigr)^{2s}\,d\xi. $$
Then evidently, \(B_{\ne}(\xi+i\tau)\tilde{u}_{+}(\xi+i\tau)\) is analytical in \(T(\stackrel{*}{C^{a}_{+}})\) and
$$\begin{gathered} \sup_{\tau\in\stackrel{*}{C^{a}_{+}}} \int_{{\mathbf{R}}^{2}} \bigl\vert B_{\ne}(\xi+i\tau)\tilde{u}_{+}( \xi+i\tau) \bigr\vert ^{2} \bigl(1+ \vert \xi \vert \bigr)^{2(s\alpha)}\,d\xi \\ \quad = \int_{{\mathbf{R}}^{2}} \bigl\vert B_{\ne}(\xi)\tilde{u}_{+}(\xi) \bigr\vert ^{2} \bigl(1+ \vert \xi \vert \bigr)^{2(s\alpha)}\,d \xi\le c \int_{{\mathbf{R}}^{2}} \bigl\vert \tilde{u}(\xi) \bigr\vert ^{2} \bigl(1+ \vert \xi \vert \bigr)^{2s}\,d\xi, \end{gathered} $$
i.e., \(B_{\ne}(\xi)\tilde{u}_{+}(\xi)\in\tilde{H}_{s\alpha }(C_{+}^{a})\).^{Footnote 1}
Now let us consider \(B_{=}(\xi)\tilde{u}_{}(\xi)\). Let at first \(u_{}\in C_{0}^{\infty}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\). Of course \(F^{1}B_{=}\equiv b\) exists in the distribution sense and \(\operatorname{supp} b\subset\overline{C_{+}^{a}}\), as above. Then \(F^{1}(B_{=}\tilde{u}_{})=b*u_{}\). By the definition of a convolution
$$(b*u_{}) (x)= \bigl(b(y),\overline{u_{}(xy)} \bigr), $$
where \(\overline{u_{}(xy)}\) is considered as a function on y (x is fixed), and notation \(b(y)\) means that functional b acts on y, a variable. Let us show that \((b*u_{})(x)=0\) under \(x\in C_{+}^{a}\). Consider two cases: \(y\inC_{+}^{a}\) and \(y\notinC_{+}^{a}\). In the first case \(xy\in C_{+}^{a}\) and, thus \(u_{}(xy)=0\) because \(\operatorname{supp} u_{}(xy)\subset{\mathbf {R}}^{2}\setminus C_{+}^{a}\). In the second case \((b*u_{})\) vanishes because \(y\notin\operatorname{supp} b\).
Transfer to the general case \(u_{}\in H_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) is realized by virtue of the density of class \(C_{0}^{\infty}({\mathbf{R}}^{2}\setminus \overline{C_{+}^{a}})\) in space \(H_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\).
So, it was shown that \(B_{=}(\xi)\tilde{u}_{}(\xi)\in\tilde{H}_{s\alpha }({\mathbf{R}}^{2})\) and \(\operatorname{supp} F^{1}(B_{=}\tilde{u}_{})\subset ({\mathbf{R}}^{2}\setminus C_{+}^{a})\). Hence, \(B_{=}(\xi)\tilde{u}_{}(\xi)\in\tilde{H}_{s\alpha}({\mathbf{R}}^{2}\setminus \overline{C_{+}^{a}})\). □
As above we use the notation \(u_{+}\) for the function \(u\in H_{s}(C^{a}_{+})\).
Theorem 1
If the elliptic symbol
\(A(\xi)\)
admits wave factorization with respect to the cone
\(C^{a}_{+}\)
and
\(\vert \kappas \vert <1/2\), then equation (2) has a unique solution
\(u_{+}\in H_{s}(C^{a}_{+})\)
for an arbitrary righthand side
\(v\in H^{0}_{s\alpha}(C^{a}_{+})\),
$$\widetilde{u}_{+}(\xi)=A_{\neq}^{1}(\xi) (K_{a} \widetilde{lv}) (\xi), $$
where
lv
is an arbitrary continuation of
v
on the whole
\(H_{s\alpha }({\mathbf{R}}^{2})\).
A priori estimate holds:
$$\Vert u_{+} \Vert _{s}\le c \Vert v \Vert _{s\alpha}^{+}. $$
Proof
We give here this proof to explain the appearance of the operator \(K_{a}\) because it plays a crucial role in our studies.
Let us denote
$$ u_{}=lvAu_{+}. $$
(4)
Taking into account wave factorization after applying to (4) the Fourier transform we have
$$A_{\ne}(\xi)\tilde{u}_{+}(\xi)+ A_{=}^{1}(\xi)\tilde{u}_{}(\xi)= A_{=}^{1}(\xi)\tilde{lv}(\xi). $$
According to the properties of wave factorization elements \(A_{\ne}(\xi)\), \(A_{=}(\xi)\) we have \(A_{\ne}(\xi)\tilde{u}_{+}(\xi)\in\tilde{H}_{s\kappa}(C_{+}^{a})\), \(A_{=}^{1}(\xi)\tilde{u}_{}(\xi)\in\tilde{H}_{s\kappa} ({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) (because \(\tilde{u}_{}\in\tilde{H}_{s\alpha} ({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\)), \(A_{=}^{1}(\xi)\tilde{lv}(\xi)\in\tilde{H}_{s\kappa}({\mathbf{R}}^{2})\), where κ is index of wave factorization. Since \(\vert s\kappa \vert <1/2\), \(\tilde{H}_{s\kappa}({\mathbf {R}}^{2})\) admits a unique representation as a sum of two orthogonal subspaces \(\tilde{H}_{s\kappa}(C_{+}^{a})\) and \(\tilde{H}_{s\kappa} ({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) [3] so that
$$A_{\ne}\tilde{u}_{+}=K_{a}A_{=}^{1}\tilde{lv}, $$
and it implies
$$\tilde{u}_{+}=A_{\ne}^{1}K_{a}A_{=}^{1} \tilde{lv}. $$
A priori estimate is
$$\begin{aligned} \Vert u_{+} \Vert _{s}&= \Vert \tilde{u}_{+} \Vert _{s}\le c \bigl\Vert K_{a}A_{=}^{1}\tilde{lv} \bigr\Vert _{s\kappa} \\ &\le c \bigl\Vert A_{=}^{1}\tilde{lv} \bigr\Vert _{s\kappa}\le c \Vert \tilde{lv} \Vert _{s\kappa} \\ &=c \Vert \tilde{lv} \Vert _{s\alpha}\le c \Vert v \Vert _{s\alpha}^{+}, \end{aligned} $$
taking into account boundedness of operator \(K_{a}\) in \(\tilde{H}_{s}({\mathbf{R}}^{2})\) for \(\vert s \vert <1/2\) and boundedness of continuation operator l [3]. □
Remark 1
If \(\vert \kappas \vert >1/2\) there are additional conditions or solvability conditions for the righthand side to obtain a unique solvability for equation (2) in appropriate SobolevSlobodetskii spaces [3].