UDC 517.946

FUNDAMENTAL SOLUTIONS OF \(B\)-ELLIPTIC EQUATIONS

I. A. Kipriyanov, V. I. Kononenko

Introduction. Consider a linear differential operator of order \(2m\) with constant coefficients

\[ L = L \left( \frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}, B_y \right) = \]

\[ = \sum_{2j+\nu \leq 2m} \sum_{i_1,\ldots,i_\nu=1}^{n} a_j^{(i_1,\ldots,i_\nu)} \frac{\partial^\nu}{\partial x_{i_1}, \ldots, \partial x_{i_\nu}} B_y^j . \]

Here and below \(B_y^j\) denotes the iteration of the Bessel operator

\[ B_y = \frac{\partial^2}{\partial y^2} + \frac{k}{y} + \frac{\partial}{\partial y} \quad (k>0,\ y \geq 0). \]

Suppose that the operator \(L\) is \(B\)-elliptic [6], i.e., for every real vector
\(\alpha = (\alpha_1,\ldots,\alpha_n,\alpha_{n+1})\) the inequality

\[ \left| L_0 \left( i\alpha_1,\ldots,i\alpha_n,(i\alpha_{n+1})^2 \right) \right| \geq \delta |\alpha|^{2m}, \]

holds, where \(\delta\) is a positive number,

\[ |\alpha| = \sqrt{\sum_{i=1}^{n+1} \alpha_i^2}, \]

and \(L_0\) is the principal part of the operator \(L\).

For a broad class of right-hand sides \(f(x,y)\) we can solve the equation \(Lu=f\), if we find a fundamental solution. The present article is devoted chiefly to the construction of fundamental solutions of \(B\)-elliptic equations.

We briefly describe the contents of the paper. In § 1, spaces of basic and generalized functions are constructed. In § 2, the corresponding generalized function \(r^\lambda\) is studied in detail. In § 3, the mixed Fourier—Bessel transform of this function is found. The developed theory makes it possible in § 4, with the aid of the mixed Fourier—Bessel transform of generalized functions, to find a solution of the iterated equation

\[ \Delta_B^m u = \delta(x,y) \quad (y \geq 0), \]

where

\[ \Delta_B = \sum_{i=1}^{n} \frac{\partial^2}{\partial x_i^2} + B_y . \]

We note that in the special case \(m=1\) the fundamental solution was found by A. Weinstein (see [4]).

In § 5 an expansion of \(r^\lambda\) in terms of so-called weighted plane waves is constructed. With the aid of this expansion, in § 6 the fundamental solution is found in the general case.

§ 1. BASIC AND GENERALIZED FUNCTIONS

A complex-valued function \(\varphi(x,y)\) \(\bigl(x=(x_1,\ldots,x_n)\bigr)\) is called a basic function if it is infinitely differentiable, even in \(y\), and satisfies inequalities of the form

\[ \left|D_{x,y}^{q}\varphi(x,y)\right|\leq \frac{C_{q\nu}}{(1+r^2)^\nu}, \tag{1.1} \]

where \(r^2=\sum_{1}^{n}x_i^2+y^2\), for arbitrary \(q,\nu=0,1,2,\ldots\). Since \(\varphi(x,y)\) is even in \(y\), the Bessel operator

\[ B_y=\frac{\partial^2}{\partial y^2}+\frac{k}{y}\frac{\partial}{\partial y}, \tag{1.2} \]

may be applied to the basic functions any number of times, and inequalities of the form

\[ \left|D_x^q B_y^\mu \varphi(x,y)\right|\leq \frac{C_{q,\mu,\nu}}{(1+r^2)^\nu} \tag{1.3} \]

will again hold for arbitrary integers \(q\geq 0\), \(\mu\geq 0\), and \(\nu\geq 0\).

We shall call the set of all basic functions the basic space \(S_B\). It is clear that \(S_B\) is a linear space.

Convergence of a sequence \(\varphi_n\in S_B\) to zero is defined in the usual way [2]. A multiplier in the space \(S_B\) is any infinitely differentiable function \(f(x,y)\), even in \(y\), which, as do its derivatives, grows as \(r\to\infty\) no faster than some power of \(r\).

For basic functions, the direct and inverse mixed Fourier—Bessel transforms are defined by

\[ F[\varphi]=\psi(\sigma,\tau)=\int_{R_n}\int_0^\infty \varphi(x,y)e^{ix\sigma}j_\nu(y\tau)y^{2\nu+1}\,dy\,dx, \tag{1.4} \]

\[ F^{-1}[\psi]=\varphi(x,y)= \frac{1}{(2\pi)^n 2^{2\nu}\Gamma^2(\nu+1)} \times \int_{\mathring R_n}\int_0^\infty \psi(\sigma,\tau)e^{-ix\sigma}j_\nu(y\tau)\tau^{2\nu+1}\,d\tau\,d\sigma, \tag{1.5} \]

where \(\nu=\dfrac{k-1}{2}>-\dfrac12\).

These transforms have recently found wide application in the embedding theory for weighted classes [7].

It can be shown that the mixed Fourier—Bessel transform maps the space \(S_B\) of basic functions one-to-one onto itself,

\[ F[S_B]=F^{-1}[S_B]=S_B. \tag{1.6} \]

We now introduce the operation of convolution in the space of basic functions. By the convolution \(f * \varphi\) of the basic functions \(f(x,y)\) and \(\varphi(x,y)\) we shall mean the expression

\[ f * \varphi = \int_{R_n}\int_0^\infty T^{x,y}_{s,t} f(x,y)\,\varphi(s,t)\,t^k\,dt\,ds, \tag{1.7} \]

where the generalized-shift operator \(T^{x,y}_{s,t}\) is defined by the formula

\[ T^{x,y}_{s,t} f(x,y) = \frac{\Gamma(\nu+1)} {\Gamma\!\left(\frac12\right)\Gamma\!\left(\nu+\frac12\right)} \times \]

\[ {}\times \int_0^\pi f\!\left(x_1-s_1,\ldots,x_n-s_n, \sqrt{y^2+t^2-2yt\cos\alpha}\right) \sin^{2\nu}\alpha\,d\alpha . \tag{1.8} \]

Using the properties of the operator \(T_x^y\) (see [8]), we obtain the following properties of the operator \(T^{x,y}_{s,t}\), which will be used below:

\[ T^{x,y}_{s,t} j_\nu(\tau y)=j_\nu(\tau y)\,j_\nu(\tau t). \tag{1.9} \]

§ 2. The generalized function \(r^\lambda\)

We shall study the functional

\[ (r^\lambda,\varphi(x,y)) = \int_{R_n}\int_0^\infty r^\lambda \varphi(x,y)y^k\,dy\,dx, \tag{2.1} \]

where

\[ r^2=\sum_{i=1}^{2}x_i^2+y^2. \]

Since \(r^\lambda y^k\) is a homogeneous function of degree \(\lambda+k\), the integral (2.1) has meaning for \(\operatorname{Re}\lambda>-(n+k+1)\). For \(\operatorname{Re}\lambda\le -(n+k+1)\), we shall define the functional (2.1) by analytic continuation with respect to the parameter \(\lambda\).

Let us pass in the integral (2.1) to spherical coordinates according to the formulas

\[ y=r\cos\alpha_1, \]

\[ x_1=r\sin\alpha_1\cos\alpha_2, \]

\[ \cdots\cdots\cdots\cdots \]

\[ x_n=r\sin\alpha_1\cdots\sin\alpha_n. \]

Then \(dx\,dy=r^n\,dr\,d\Omega\), \(\cos\alpha_1>0\), since \(y>0\). Put

\[ \int_\Omega \cos^k\alpha_1\,\varphi(x,y)\,d\Omega = a_0 S_\varphi(r), \tag{2.2} \]

where

\[ a_0 = \int_\Omega \cos^k\alpha_1\,d\Omega = \frac{2\pi^{\frac n2}\Gamma\!\left(\frac{k+1}{2}\right)} {\Gamma\!\left(\frac{n+k+1}{2}\right)}. \]

Here \(\Omega\) is the half-sphere in the space \(R_{n+1}\), and \(S_\varphi(r)\) is the weighted mean of \(\varphi(x,y)\) over the half-sphere of radius \(r\). The functional (2.1) will now be written in the following form:

\[ (r^\lambda,\varphi)=a_0\int_0^\infty r^{\lambda+n+k}S_\varphi(r)\,dr . \tag{2.3} \]

The function \(S_\varphi(r)\) is infinitely differentiable for \(r>0\) and, as \(r\to\infty\), decreases faster than any power of \(1/r\). This follows from the analogous properties of the function \(\varphi(x,y)\).

Let us prove that \(S_\varphi(r)\) is infinitely differentiable also at \(r=0\). Indeed, expanding the function \(\varphi(x,y)\) by Taylor’s formula, we obtain

\[ a_0 S_\varphi(r)= \int_{\Omega}\cos^k\alpha_1 \left[ \varphi(0,0)+ \sum_{i=1}^{n+1}\frac{\partial\varphi(0,0)}{\partial x_i}x_i+ \right. \]

\[ \left. +\frac{1}{2} \sum_{i,j=1}^{n+1} \frac{\partial^2\varphi(0,0)}{\partial x_i\partial x_j}x_i x_j+\cdots \right]\,d\Omega . \tag{2.4} \]

For convenience we have denoted \(y=x_{n+1}\). It is easy to see that every term of the integrand sum (except the remainder term) containing \(x_i\) in an odd power vanishes after integration. Indeed, dividing the domain of integration into two domains \(x_i>0\) and \(x_i<0\), we obtain two integrals with opposite signs. For \(x_{n+1}\) the corresponding term is equal to zero even before integration, because \(\varphi(x,y)\) is an even function of \(y\). Thus we obtain

\[ S_\varphi(r)=\varphi(0,0)+b_1r^2+b_2r^4+\cdots+O(r^{2m}) . \tag{2.5} \]

From this expansion it follows that \(S_\varphi(r)\) has all derivatives at \(r=0\), and all its odd derivatives at \(r=0\) vanish. Consequently, the function \(S_\varphi(r)\) may be regarded as an even basic [2] function of the variable \(r\). Equality (2.3) can in this case be understood as the result of applying the functional \(a_0 x_+^\mu\) \((\mu=\lambda+n+k)\) to the basic function \(S_\varphi(x)\) (see [2]). This functional is extended to all \(\mu\), except \(\mu=-1,-2,\ldots\), where there are poles; moreover, the residue at the pole \(\mu=-m\) is equal to

\[ a_0 \frac{\left(((-1)^{m-1}\delta^{(m-1)}(x),S_\varphi(x))\right)}{(m-1)!} = a_0\frac{S_\varphi^{(m-1)}(0)}{(m-1)!}. \tag{2.6} \]

Since the odd derivatives of the function \(S_\varphi(r)\) vanish at \(r=0\), there remain poles at the points

\[ \mu=-1,-3,\ldots,-2p-1,\ldots \]

or

\[ \lambda=-\gamma,-(\gamma+2),\ldots,-(\gamma+2p),\ldots, \]

where \(\gamma=n+k+1\). Hence it follows that the residue of the function \((r^\lambda,\varphi)\) at \(\lambda=-\gamma-2p\) is equal to

\[ a_0\frac{(\delta^{(2p)}(x),S_\varphi(x))}{(2p)!} = a_0\frac{S^{(2p)}(0)}{(2p)!}. \tag{2.7} \]

The quantity \(S_\varphi^{(2p)}(0)\) can be expressed directly in terms of the function \(\varphi(x,y)\), bypassing its averaging. Put

\[ \Delta_B=\sum_{i=1}^{n}\frac{\partial^2}{\partial x_i^2}+B_y. \tag{2.8} \]

It is not hard to verify that

\[ \Delta_B(r^{\lambda+2})=(\lambda+2)(\lambda+\gamma)r^\lambda \tag{2.9} \]

for \(\operatorname{Re}\lambda>-\gamma\). For the remaining \(\lambda\) this formula remains valid by analytic continuation. Using this formula, we obtain

\[ r^\lambda= \frac{\Delta_B^p(r^{\lambda+2p})} {(\lambda+2)\cdots(\lambda+2p)(\lambda+\gamma)\cdots(\lambda+\gamma+2p-2)}. \tag{2.10} \]

The residue of the function \(r^\lambda\) at \(\lambda=-\gamma-2p\) can now be computed as the residue of the right-hand side at this value of \(\lambda\). But for any fundamental function \(\varphi(x,y)\) we have

\[ (\Delta_B^p r^{\lambda+2p},\varphi)=(r^{\lambda+2p},\Delta_B^p\varphi), \]

and the residue of the right-hand side at \(\lambda=-\gamma-2p\) is equal to \(a_0\Delta_B^p\varphi(0,0)\). Hence it follows that the residue of the function \((r^\lambda,\varphi)\) at \(\lambda=-\gamma-2p\) is equal to

\[ \frac{a_0\Delta_B^p\varphi(0,0)} {(-2p-\gamma+2)\cdots(-\gamma)(-2p)\cdots(-2)} = \frac{a_0(\Delta_B^p\delta(x,y),\varphi(x,y))} {2^p\cdot p!\,\gamma(\gamma+2)\cdots(\gamma+2p-2)}. \tag{2.11} \]

Thus, the residue of the generalized function \(r^\lambda\) at \(\lambda=-\gamma-2p\) is equal to
\([a_0\Delta_B^p\delta(x,y)]\times[2^p\cdot p!\,\gamma(\gamma+2)\cdots(\gamma+2p-2)]^{-1}\).
Comparing this expression with (2.7), we obtain

\[ S_\varphi^{(2p)}(0)= \frac{(2p)!\,\Delta_B^p\varphi(0,0)} {2^p\cdot p!\,\gamma(\gamma+2)\cdots(\gamma+2p-2)}. \tag{2.12} \]

We can now write the following Taylor expansion for the function \(S_\varphi(r)\):

\[ S_\varphi(r)=\varphi(0,0)+\frac{1}{2!}S_\varphi''(0)r^2+\cdots+ \frac{1}{(2k)!}S_\varphi^{(2k)}(0)r^{2k}+\cdots = \]

\[ =\sum_{k=0}^{m} \frac{\Delta_B^k\varphi(0,0)r^{2k}} {2^k k!\,\gamma(\gamma+2)\cdots(\gamma+2k-2)} +\cdots . \tag{2.13} \]

This is a formula of the type of Pizzetti’s formula [2].

Let us write the Taylor and Laurent series expansions for the function \(r^\lambda\). Using the expansion of the function \(x_+^\mu\) in a neighborhood of the regular point \(\mu_0\) into a Taylor series [2], we obtain

\[ (r^\lambda,\varphi)=a_0(x_+^\mu,S_\varphi(x))= a_0(x_+^{\mu_0},S_\varphi(x))+ \]

\[ +a_0'(\mu-\mu_0)(x_+^{\mu_0}\ln x_+,S_\varphi(x))+\cdots . \tag{2.14} \]

Since \(\lambda+k+n=\mu\), \(\lambda_0+k+n=\mu_0\), introducing the notation

\[ a_0\bigl(x_+^\mu \ln x_+,\, S_\varphi(x)\bigr)=(r^\lambda \ln r,\varphi) \]

and so on, we obtain

\[ r^\lambda=r^{\lambda_0}+(\lambda-\lambda_0)r^{\lambda_0}\ln r+ \frac{1}{2}(\lambda-\lambda_0)^2 r^{\lambda_0}\ln^2 r+\cdots \tag{2.15} \]

Let us now write the Laurent expansion of \(r^\lambda\) in neighborhoods of the point \(\lambda=-\gamma-2p\). Since \(\mu=-2p-1\), the expansion of \(x_+^\mu\) in neighborhoods of this point has the form

\[ x_+^\mu=\frac{\delta^{(2p)}(x)}{(2p)!(\mu+2p+1)} +x_+^{-2p-1}+(\mu+2p+1)x_+^{-2p-1}\ln x_+ +\cdots . \]

Thus, we have

\[ (r^\lambda,\varphi)=a_0\bigl(x_+^\mu,S_\varphi(x)\bigr)= \frac{a_0\bigl(\delta^{(2p)}(x),S_\varphi(x)\bigr)} {(2p)!(\lambda+\gamma+2p)}+ \]

\[ +\,a_0\bigl(x_+^{-2p-1},S_\varphi(x)\bigr) +(\lambda+\gamma+2p)a_0\bigl(x_+^{-2p-1}\ln x_+,S_\varphi(x)\bigr)+\cdots \tag{2.16} \]

Since

\[ \frac{a_0\bigl(\delta^{(2p)}(x),S_\varphi(x)\bigr)}{(2p)!} =b_p\Delta_B^p\varphi(0,0)=b_p(\Delta_B^p\delta,\varphi), \tag{2.17} \]

where

\[ b_p=\frac{a_0}{2^p p!\,\gamma(\gamma+2)\cdots(\gamma+2p-2)}, \]

then, setting

\[ a_0\bigl(x_+^{-2p-1},S_\varphi(x)\bigr)=(r^{-2p-\gamma},\varphi), \]

\[ a_0\bigl(x_+^{-2p-1}\ln x_+,S_\varphi(x)\bigr)=r^{-2p-\gamma}\ln r,\varphi) \]

and so on, we obtain the following expansion of \(r^\lambda\) in a neighborhood of \(\lambda=-2p-\gamma\):

\[ r^\lambda=\frac{b_p\Delta_B^p\delta(x,y)}{\lambda+\gamma+2p} +r^{-2p-\gamma}+ \]

\[ +(\lambda+\gamma+2p)r^{-2p-\gamma}\ln r +\frac{1}{2}(\lambda+\gamma+2p)^2 r^{-2p-\gamma}\ln^2 r+\cdots \tag{2.18} \]

For what follows it is convenient to normalize the generalized function \(r^\lambda\). We have

\[ (r^\lambda,e^{-r^2})=\int_{R_n^+}\int_0^\infty r^\lambda e^{-r^2}y^k\,dy\,dx= \]

\[ =a_0\int_0^\infty r^{\lambda+k+n}e^{-r^2}\,dr =\frac{a_0}{2}\Gamma\left(\frac{\lambda+\gamma}{2}\right). \]

The function

\[ \frac{2r^\lambda}{a_0\Gamma\left(\frac{\lambda+\gamma}{2}\right)} \]

is an entire analytic function of \(\lambda\). Con-

the value of this function at the special points of the numerator and denominator can be found as the ratio of the corresponding residues. Thus, we obtain

\[ \left. \frac{2r^\lambda}{a_0 \Gamma \left( \frac{\lambda+\gamma}{2} \right)} \right|_{\lambda=-\gamma-2p} = (-1)^p \frac{\Delta_B^p \delta(x,y)} {2^p \gamma(\gamma+2)\ldots(\gamma+2p-2)} . \tag{2.19} \]

In particular, for \(p=0\) we have

\[ \left. \frac{2r^\lambda}{a_0 \Gamma \left( \frac{\lambda+\gamma}{2} \right)} \right|_{\lambda=-\gamma} = \delta(x,y). \tag{2.20} \]

§ 3. THE FOURIER–BESSEL TRANSFORM OF THE GENERALIZED FUNCTION \(r^\lambda\)

Consider the integral

\[ J=\int_{R_n}\int_0^\infty r^\lambda e^{i\sigma x} j_\nu(\tau y)y^{2\nu+1}\,dy\,dx . \tag{3.1} \]

It converges for \(-\gamma<\operatorname{Re}\lambda<0\); moreover, this improper integral may be understood as the limit \(J=\lim_{R\to\infty}J_R\), where

\[ J_R=\iint_{Q_R} r^\lambda e^{i\sigma x} j_\nu(\tau y)y^{2\nu+1}\,dy\,dx . \tag{3.2} \]

The domain \(Q_R\) is defined by the inequalities:

\[ \sum_{i=1}^{n} x_i^2 \le R^2,\quad 0\le y \le \sqrt{R^2-\sum_{i=1}^{n}x_i^2}. \]

Passing in the integral (3.2) to spherical coordinates \(x,\varphi_1,\ldots,\varphi_{n-1}\) in the subspace \(x_1,\ldots,x_n\), we obtain

\[ \begin{aligned} J_R &= \int_0^R x^{n-1}\,dx \int_{\Omega} e^{ix|\sigma|\cos\psi}\,d\Omega \int_0^{\sqrt{R^2-x^2}} (x^2+y^2)^{\frac{\lambda}{2}} j_\nu(\tau y)y^{2\nu+1}\,dy \\ &= (2\pi)^{\frac n2-1}s^{1-\frac n2} \int_0^R dx \int_0^{\sqrt{R^2-x^2}} r^\lambda J_{\frac n2-1}(sx)j_\nu(\tau y)x^{\frac n2}y^{2\nu+1}\,dy, \end{aligned} \tag{3.3} \]

where \(s=|\sigma|=\sqrt{\sum_{i=1}^{n}\sigma_i^2}\), \(r^2=x^2+y^2\). Here we have used the following integral [1]:

\[ \int_{\Omega} e^{i\rho s\cos\psi}\,d\Omega = (2\pi)^{\frac n2-1}(\rho s)^{1-\frac n2} J_{\frac n2-1}(\rho s). \]

Since \(j_\nu(x)=2^\nu \Gamma(\nu+1)x^{-\nu}J_\nu(x)\), we obtain

\[ \begin{aligned} J_R &= (2\pi)^{\frac n2-1} 2^\nu \Gamma(\nu+1)s^{1-\frac n2}\tau^{-\nu} \int_0^R \int_0^{\sqrt{R^2-x^2}} r^\lambda J_{\frac n2-1} \\ &\qquad\qquad\qquad\qquad \times (sx)J_\nu(\tau y)x^{\frac n2}y^{\nu+1}\,dy\,dx . \end{aligned} \tag{3.4} \]

To compute this integral, introduce the polar coordinate system
\(x=r\cos\varphi,\ y=r\sin\varphi\); then we obtain

\[ J_R=C s^{1-\frac n2}\tau^{-\nu}\int_0^R r^{\lambda+\nu+2+\frac n2}\,dr \times \]
\[ {}\times \int_0^{\pi/2} J_\nu(\tau r\sin\varphi)J_{\frac n2-1}(sr\cos\varphi) \sin^{\nu+1}\varphi\,\cos^{\frac n2}\varphi\,d\varphi, \tag{3.5} \]

where
\[ C=(2\pi)^{\frac n2-1}2^\nu\Gamma(\nu+1). \]
The inner integral is Sonine’s second definite integral [1]; therefore we have

\[ J_R=C\rho^{-\lambda-\gamma}\int_0^R J_{\nu+\frac n2}(t)\, t^{\lambda+\nu+\frac n2+1}\,dt, \tag{3.6} \]

where \(\rho^2=s^2+\tau^2\). Thus, we obtain

\[ J=\lim_{R\to\infty}J_R =C\rho^{-\lambda-\gamma}\int_0^\infty J_{\nu+\frac n2}(t)\, t^{\lambda+\nu+\frac n2+1}\,dt. \tag{3.7} \]

The last integral is nothing other than Weber’s improper integral [1], absolutely convergent for
\(-\gamma<\operatorname{Re}\lambda<-\frac{\gamma}{2}-1\). Therefore, finally,

\[ F[r^\lambda]=C_\lambda\rho^{-\lambda-\gamma}, \tag{3.8} \]

where

\[ C_\lambda= 2^{\lambda+\gamma-2}\pi^{\frac n2-1} \Gamma\!\left(\frac{k+1}{2}\right) \frac{\Gamma\!\left(\frac{\lambda+\gamma}{2}\right)} {\Gamma\!\left(-\frac{\lambda}{2}\right)}. \]

Since both sides of equality (3.8) are analytic functions of \(\lambda\), it remains valid for all \(\lambda\).

In exactly the same way, integral (3.1) is computed for the case \(n=1\), except that instead of Sonine’s integral one uses Gegenbauer’s definite integral [1].

In the usual way, one can now obtain formulas for the Fourier–Bessel transform of generalized functions—the coefficients in the expansion of \(r^\lambda\) into Taylor and Laurent series. In particular, we have

\[ F[r^{-\gamma-2m}] = C_{-1}^{(\gamma+2m)}\rho^{2m}\ln\rho + C_0^{(\gamma+2m)}\rho^{2m}. \tag{3.9} \]

Here the numerical coefficients \(C_{-1}^{(\gamma+2m)}\) and \(C_0^{(\gamma+2m)}\) are the coefficients of the Laurent expansion of the function \(C_\lambda\) in a neighborhood of the point \(\lambda=-\gamma-2m\).

§ 4. ITERATED EQUATION \(\Delta_B^m u=\delta\)

We shall seek a solution of the equation

\[ \Delta_B^m u=\delta(x,y) \tag{4.1} \]

in the space \(S'_B\). Here

\[ \Delta_B=\sum_{i=1}^{n}\frac{\partial^2}{\partial x_i^2}+\frac{\partial^2}{\partial y^2}+\frac{k}{y}\frac{\partial}{\partial y}. \]

After applying the mixed Fourier–Bessel transform to both sides of equation (4.1), we obtain

\[ (-1)^m\rho^{2m}V=1, \tag{4.2} \]

where \(\rho^2=\sum_{i=1}^{n}\sigma_i^2+\tau^2\). If \(2m<\gamma\), then a solution of the last equation is the locally summable function

\[ V=\frac{(-1)^m}{\rho^{2m}}. \tag{4.3} \]

Now let \(\mu=-2m\) not be a pole of the analytic function \(r^\lambda\). Then, as is easy to verify, the functional

\[ V=(-1)^m\rho^{-2m}. \tag{4.4} \]

serves as a solution of equation (4.1).

Finally, let \(\lambda=-2m\) be a pole of the analytic function \(r^\lambda\). Consider the expansion of the function \(\rho^\lambda\) in a neighborhood of this point into a Laurent series:

\[ \rho^\lambda=\frac{a_{-1}}{\lambda+2m}+a_0+a_1(\lambda+2m)+\cdots \tag{4.5} \]

Multiplying this equality by \(\rho^{2m}\) and passing to the limit as \(\lambda\to -2m\), on the left-hand side we obtain unity, while on the right all terms, beginning with the third, become zero. Hence it follows that \(\rho^{2m}a_{-1}=0\), and therefore

\[ \rho^{2m}a_0=1. \tag{4.6} \]

Thus, in the present case a solution of equation (4.1) is the function

\[ V=(-1)^m\rho^{-2m}. \tag{4.7} \]

Let us now find the inverse Fourier–Bessel transform. In the first case, in accordance with formula (3.8), we have \(u=Cr^{2m-\gamma}\). We obtain the same result in the second case, also using formula (3.8). In the third case we use the formula

\[ F\left[(-1)^m\rho^{-2m}\right]=C_1r^{2m-\gamma}\ln r+C_2r^{2m-\gamma}. \tag{4.8} \]

In this case \(\gamma\) is even and \(2m\geq\gamma\). Therefore the second term \(C_2r^{2m-\gamma}\) is transformed by the operator \(\Delta_B^m\) into zero.

Thus, the required fundamental solution has the form

\[ u(x,y)= \begin{cases} C_1r^{2m-\gamma}\ln r, & 2m\geq\gamma \text{ and } \gamma \text{ is even},\\ C_2r^{2m-\gamma}, & \text{in the remaining cases}. \end{cases} \tag{4.9} \]

To obtain the fundamental solution with a singularity at an arbitrary point \((s,t)\), apply to the function \(u(x,y)\) the shift operator \(T_{s,t}^{x,y}\):

\[ u(x,y,s,t)=C_3\int_{0}^{\pi}\left[\sum_{i=1}^{n}(x_i-s_i)^2+y^2+t^2-2yt\cos\alpha\right]^{\frac{2m-\gamma}{2}}\times \]

\[ {}\times \ln \left|\sum_{i=1}^{n}(x_i-s_i)^2+y^2+t^2-2yt\cos\alpha\right|^{\frac12}\sin^{k-1}\alpha\,d\alpha, \tag{4.10} \]

\[ u(x,y,s,t)=C_4\int_{0}^{\pi} \left[\sum_{i=1}^{n}(x_i-s_i)^2+y^2+t^2-2yt\cos\alpha\right]^{\frac{2m-\gamma}{2}} \sin^{k-1}\alpha\,d\alpha . \]

§ 5. Expansion of the function \(r^\lambda\) into plane waves

Let us study the functional

\[ \left(|\omega x|_{B}^{\lambda},\varphi\right)= \int_{R_n}\int_{0}^{\infty}\left[\int_{0}^{\pi} \left|\sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}y\cos\alpha\right|^{\lambda} {}\times \sin^{k-1}\alpha\,d\alpha\right]\varphi(x,y)y^k\,dy\,dx, \tag{5.1} \]

generated by the function, locally summable for \(\operatorname{Re}\lambda>-1\),

\[ |\omega x|_{B}^{\lambda} = \int_{0}^{\pi} \left|\sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}y\cos\alpha\right|^{\lambda} \sin^{k-1}\alpha\,d\alpha . \tag{5.2} \]

Here \(\omega=(\omega_1,\ldots,\omega_{n+1})\) is an arbitrary vector on the unit sphere. The functional (5.1) can be analytically continued in the parameter \(\lambda\) into the left half-plane. Indeed, setting \(y\cos\alpha=z,\ y\sin\alpha=\tau\), we obtain a representation of the integral (5.1) in the form

\[ \left(|\omega x|_{B}^{\lambda},\varphi\right)= \int_{R_{n+1}} \left|\sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}z\right|^{\lambda} \varphi_0(x,z)\,dx\,dz, \tag{5.3} \]

where

\[ \varphi_0(x,z)=\int_{0}^{\infty}\tau^{k-1}\varphi\bigl(x,\sqrt{z^2+\tau^2}\bigr)\,d\tau . \]

The function \(\varphi_0(x,z)\) is infinitely differentiable and, as \(r\to\infty\), decreases faster than any power of \(1/r\), together with all its derivatives, i.e., belongs to the space \(S\) [2]. Therefore the right-hand side of (5.3) can be analytically continued to all \(\lambda\), except \(\lambda=-1,-3,-5,\ldots\), where it has simple poles. Thus the analytic continuation of the left-hand side is also constructed. It follows from this that the functional

\[ \frac{|\omega x|_{B}^{\lambda}}{\Gamma\left(\frac{\lambda+1}{2}\right)} \]

will be an entire analytic function of \(\lambda\).

Next, in an analogous way we can construct the analytic continuation of the functional

\[ \left(|\omega x|_{B}^{\lambda}\operatorname{sign}(\omega x),\varphi\right)= \int_{R_n}\int_{0}^{\infty}\left[\int_{0}^{\pi} \left|\sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}y\cos\alpha\right|^{\lambda} {}\times \]

\[ \times \operatorname{sign}\left(\sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}y\cos\alpha\right)\sin^{k-1}\alpha\,d\alpha \right]\varphi(x,y)y^k\,dy\,dx . \tag{5.4} \]

This functional, as a function of the parameter \(\lambda\), has poles at the points \(\lambda=-2,-4,\ldots\). The function

\[ \frac{|\omega x|_{B}^{\lambda}\operatorname{sign}(\omega x)} {\Gamma\left(\dfrac{\lambda+2}{2}\right)} \tag{5.5} \]

is an entire analytic function of the parameter \(\lambda\). For the generalized functions constructed, the following differentiation formulas are valid:

\[ \frac{\partial}{\partial x_i}|\omega x|_{B}^{\lambda} = \lambda\omega_i|\omega x|_{B}^{\lambda-1}\operatorname{sign}(\omega x), \]

\[ \frac{\partial}{\partial x_i}|\omega x|_{B}^{\lambda}\operatorname{sign}(\omega x) = \lambda\omega_i|\omega x|_{B}^{\lambda-1}, \tag{5.6} \]

\[ B_y|\omega x|_{B}^{\lambda} = \lambda(\lambda-1)\omega_{n+1}^{2}|\omega x|_{B}^{\lambda-2}. \]

For good \(\lambda\) these formulas are not difficult to obtain directly or by integration by parts; for arbitrary \(\lambda\) they remain valid by analytic continuation.

Let us note that the constructed functionals are continuous with respect to the vector \(\omega\). Hence it follows that the functional \(|\omega x|_{B}^{\lambda}\) can be integrated over the unit sphere \(|\omega|=1\) with the continuous weight \(|\omega_{n+1}|^k\), i.e., one can construct the functional

\[ \int_{\Omega}|\omega x|_{B}^{\lambda}|\omega_{n+1}|^k\,d\Omega . \tag{5.7} \]

Let us find an explicit expression for this functional. For this purpose, when \(\operatorname{Re}\lambda>-1\), we must compute the integral

\[ J= \int_{\Omega} \left[ \int_{0}^{\pi} \left| \sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}y\cos\varphi \right|^{\lambda} \sin^{k-1}\varphi\,d\varphi \right] |\omega_{n+1}|^k\,d\Omega . \tag{5.8} \]

If we set \(\omega_{n+1}\cos\varphi=\alpha_1\), \(\omega_{n+1}\sin\varphi=\alpha_2\), then the given integral can be written in the form

\[ J= \int_{\Omega_1} \left| \sum_{i=1}^{n}\omega_i x_i+\alpha_1 y \right|^{\lambda} |\alpha_2|^{k-1}\,d\Omega_1, \tag{5.9} \]

or

\[ J= \int_{-1}^{1} \frac{|\alpha_2|^{k-1}}{\sqrt{1-\alpha_2^2}}\,d\alpha_2 \int_{\Omega_2} \left| \sum_{i=1}^{n}\omega_i x_i+\alpha_1 y \right|^{\lambda} d\Omega_2, \tag{5.10} \]

where \(\Omega_1\) is the sphere \(\sum_{i=1}^{n}\omega_i^2+\alpha_1^2+\alpha_2^2=1\), \(d\Omega_1\) is the surface element of this sphere,

\[ \Omega_2 \]
is the sphere \(\sum_{i=1}^{n}\omega_i^2+\alpha_1^2=1-\alpha_2^2\), and \(d\Omega_2\) is the surface element of this sphere.

It is seen from this that the integral (5.7) is invariant with respect to rotation of the vector \((x,y)\). Now setting \(x_1=r,\quad x_2=\cdots=x_n=y=0\), as a result of the calculations we obtain

\[ \frac{1}{\pi^{\frac n2}\Gamma\left(\frac k2\right)\Gamma\left(\frac{\lambda+1}{2}\right)} \int_{\Omega}|\omega x|_B^\lambda|\omega_{n+1}|^k\,d\Omega = \frac{2r^\lambda}{\Gamma\left(\frac{\lambda+\gamma}{2}\right)}. \tag{5.11} \]

This equality, proved for \(\operatorname{Re}\lambda>-1\), remains valid for arbitrary \(\lambda\) by analytic continuation. This is, in our case, the basic expansion of the function \(r^\lambda\) in plane waves.

Let us consider the special case of this formula when \(\lambda=-\gamma\).
Let \(\gamma=n+k+1\) not be an odd number. Then we have

\[ \delta(x,y)=C_1(n,k)\int_{\Omega}|\omega x|_B^{-\gamma}|\omega_{n+1}|^k\,d\Omega, \tag{5.12} \]

where

\[ C_1(n,k)= \frac{\Gamma\left(\frac{n+k+1}{2}\right)} {2\pi^n\Gamma\left(\frac k2\right)\Gamma\left(\frac{k+1}{2}\right)\Gamma\left(\frac{-n-k}{2}\right)}. \]

Now let \(\gamma=n+k+1\) be an odd number. Put

\[ |\omega x|_B^{-\gamma} = \lim_{\lambda\to-\gamma} \frac{|\omega x|_B^\lambda}{\Gamma\left(\frac{\lambda+1}{2}\right)}. \]

Then we obtain

\[ \delta(x,y)=C_2(n,k)\int_{\Omega}|\omega x|_B^{-\gamma}|\omega_{n+1}|^k\,d\Omega, \tag{5.13} \]

where

\[ C_2(n,k)= \frac{\Gamma\left(\frac{n+k+1}{2}\right)} {2\pi^n\Gamma\left(\frac k2\right)\Gamma\left(\frac{k+1}{2}\right)}. \]

§ 6. FUNDAMENTAL SOLUTIONS OF \(B\)-ELLIPTIC EQUATIONS

Consider the equation

\[ L(D_x,B_y)u=a_0\delta(x,y). \tag{6.1} \]

Here \(L(D_x,B_y)\) is a linear differential operator of order \(2m\), with constant coefficients, of \(B\)-elliptic type [6].

To solve equation (6.1) we shall apply the following method (see, for example, [2]). We replace the right-hand side of equation (6.1) by the function

\[ \frac{2r^\lambda}{\Gamma\left(\frac{\lambda+\gamma}{2}\right)}, \]

and expand the latter in plane waves (see § 5), i.e., we consider the equation

\[ L(D_x,B_y)u= \frac{1}{\pi^{\frac n2}\Gamma\left(\frac k2\right)\Gamma\left(\frac{\lambda+1}{2}\right)} \int_{\Omega}|\omega x|_B^\lambda|\omega_{n+1}|^k\,d\Omega. \tag{6.2} \]

Obviously, if \(v_{\lambda,\omega}(x,y)\) is a solution of the equation

\[ L(D_x,B_y)v= \frac{1}{\pi^{\frac n2}\Gamma\left(\frac k2\right)\Gamma\left(\frac{\lambda+1}{2}\right)} |\omega x|_B^\lambda, \tag{6.3} \]

then the solution of equation (6.2) is written as follows:

\[ u_\lambda(x,y)=\int_\Omega v_{\lambda,\omega}(x,y)|\omega_{n+1}|^k\,d\Omega. \tag{6.4} \]

We shall seek the solution of equation (6.3) in the form

\[ v_{\lambda,\omega}(x,y)=\int_0^\pi w_{\lambda,\omega}\left(\sum_{i=1}^n \omega_i x_i+\omega_{n+1}y\cos\alpha\right) \sin^{k-1}\alpha\,d\alpha. \tag{6.5} \]

We have

\[ \frac{\partial v_{\lambda,\omega}(x,y)}{\partial x_i} = \int_0^\pi \omega_i w'_{\lambda,\omega}\left(\sum_{i=1}^n \omega_i x_i+\omega_{n+1}y\cos\alpha\right) \sin^{k-1}\alpha\,d\alpha, \]

\[ B_y v_{\lambda,\omega}(x,y) = \int_0^\pi \omega_{n+1}^2 w''_{\lambda,\omega}\left(\sum_{i=1}^n \omega_i x_i+\omega_{n+1}y\cos\alpha\right) \sin^{k-1}\alpha\,d\alpha. \tag{6.6} \]

Therefore we find that

\[ L(D_x,B_y)v_{\lambda,\omega}(x,y) = \int_0^\pi L\left(\omega_1\frac{d}{d\xi},\ldots,\omega_n\frac{d}{d\xi}, \omega_{n+1}^2\frac{d^2}{d\xi^2}\right) \]

\[ {}\times w_{\lambda,\omega}(\xi)\sin^{k-1}\alpha\,d\alpha. \tag{6.7} \]

We require that the function \(w_{\lambda,\omega}(\xi)\) be a solution of the following equation:

\[ L\left(\omega_1\frac{d}{d\xi},\ldots,\omega_n\frac{d}{d\xi}, \omega_{n+1}^2\frac{d^2}{d\xi^2}\right)w(\xi) = \]

\[ = \frac{|\xi|^\lambda} {\pi^{\frac n2}\Gamma\left(\frac k2\right)\Gamma\left(\frac{\lambda+1}{2}\right)}. \tag{6.8} \]

It is known [2] that there exists a generalized function \(w_{\lambda,\omega}(\xi)\) that is a solution of this equation for all \(\lambda\). By virtue of the \(B\)-ellipticity of the operator \(L(D_x,B_y)\), the modulus of the leading coefficient \(L_0(\omega_1,\ldots,\omega_n;\omega_{n+1}^2)\) of equation (6.8) has a positive minimum, and the remaining coefficients depend continuously on the vector \(\omega\). Consequently, the function \(w_{\lambda,\omega}(\xi)\) depends continuously (in the sense of generalized functions) on the vector \(\omega\).

Let us construct the functional

\[ \bigl(v_{\lambda,\omega}(x,y),\varphi(x,y)\bigr) = \bigl(w_{\lambda,\omega}(\xi),\varphi_0(x,z)\bigr), \tag{6.9} \]

where

\[ \xi=\sum_{i=1}^n \omega_i x_i+\omega_{n+1}z, \qquad \varphi_0(x,z)=\int_0^\infty \tau^{k-1}\varphi\left(x,\sqrt{z^2+\tau^2}\right)\,d\tau. \]

Here \(v_{\lambda,\omega}\in S'_B,\quad w_{\lambda,\omega}\in S'\). It can be verified that for good \(\lambda\) this definition of the function \(v_{\lambda,\omega}\) turns into equality (6.5). In addition, the formula
\[ \bigl(L(D_x,B_y)v_{\lambda,\omega}(x,y),\varphi(x,y)\bigr)= \]
\[ =\left(L\left(\omega_1{d\over d\xi},\ldots,\omega_n{d\over d\xi}, \omega_{n+1}^2{d^2\over d\xi^2}\right)w_{\lambda,\omega}(\xi),\varphi_0(x,z)\right). \tag{6.10} \]

Thus, we have constructed the function \(v_{\lambda,\omega}\) for all \(\lambda\). We now show that the function
\[ u(x,y)=\int_\Omega v_{-\gamma,\omega}(x,y)|\omega_{n+1}|^k\,d\Omega \tag{6.11} \]
solves the original problem. Indeed, we have
\[ \bigl(L(D_x,B_y)u,\varphi(x,y)\bigr) =\int_\Omega \bigl(L(D_x,B_y)v_{-\gamma,\omega},\varphi(x,y)\bigr)|\omega_{n+1}|^k\,d\Omega= \]
\[ =\int_\Omega \left(L\left(\omega_1{d\over d\xi},\ldots,\omega_n{d\over d\xi}, \omega_{n+1}^2{d^2\over d\xi^2}\right)w_{-\gamma,\omega}(\xi),\varphi_0(x,z)\right)|\omega_{n+1}|^k\,d\Omega= \]
\[ =\int_\Omega \bigl(C(n,k)|\xi|^{-\gamma},\varphi_0(x,z)\bigr)|\omega_{n+1}|^k\,d\Omega= \]
\[ =C(n,k)\int_\Omega \bigl(|\omega x|_{\overline B}^{-\gamma},\varphi(x,y)\bigr)|\omega_{n+1}|^k\,d\Omega= \]
\[ =\left(C(n,k)\int_\Omega |\omega x|_{\overline B}^{-\gamma}|\omega_{n+1}|^k\,d\Omega,\varphi(x,y)\right) =\bigl(a_0\delta(x,y),\varphi(x,y)\bigr), \tag{6.12} \]
as was required.

Let us consider in greater detail the case of a homogeneous equation. In this case equation (6.8) is written as follows:
\[ L(\omega_1,\ldots,\omega_n,\omega_{n+1}^2)w^{(2m)}(\xi) = {|\xi|^\lambda\over \pi^{n/2}\Gamma\left({k\over2}\right)\Gamma\left({\lambda+1\over2}\right)}. \tag{6.13} \]
As a particular solution of this equation one may take the function
\[ w(\xi)= {1\over \pi^{n/2}\Gamma\left({k\over2}\right)\Gamma\left({\lambda+1\over2}\right) L(\omega_1,\ldots,\omega_n,\omega_{n+1}^2)} \times \]
\[ \times\left( {|\xi|^{\lambda+2m}\over(\lambda+1)\cdots(\lambda+2m)} + \sum_{k=1}^{m} {\xi^{2m-2k}\over(2k-1)!(2m-2k)!(\lambda+2k)} \right). \tag{6.14} \]
Further, one can construct the functions \(v_{\lambda,\omega}(x,y)\) and \(u_\lambda(x,y)\) and pass to the limit as \(\lambda\to-\gamma\).

Let \(2m \geqslant \gamma\), and let \(\gamma\) not be an even number. Then the fundamental solution has the form

\[ u(x,y)=C(n,k)\int_{\Omega}\left[\int_{0}^{\pi}\left|\sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}y\cos\alpha\right|^{2m-\gamma}\sin^{k-1}\alpha\,d\alpha\right]\times \]

\[ {}\times \frac{|\omega_{n+1}|^k\,d\Omega}{L(\omega_1,\ldots,\omega_n,\omega_{n+1}^2)}. \tag{6.15} \]

Here the constant has the form

\[ C(n,k)= \frac{1}{ \pi^{\frac n2}\Gamma\left(\frac k2\right) \Gamma\left(\frac{-n-k}{2}\right)(1-\gamma)\cdots(2m-\gamma) } \tag{6.15′} \]

in the case of fractional \(k\), and

\[ C(n,k)= \frac{ (-1)^{\frac{n+k}{2}}\left(\frac{n+k}{2}\right)! }{ 2\pi^{\frac n2}\Gamma\left(\frac k2\right)(u+k)!(2m-\gamma)! } \]

in the case of odd \(\gamma\).

In the case of even \(\gamma\) we obtain

\[ u(x,y)=C(n,k)\int_{\Omega}\left[\int_{0}^{\pi} \left(\sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}y\cos\alpha\right)^{2m-\gamma}\times\right. \]

\[ \left.{}\times \ln\left|\sum_{i=1}^{n}\omega_i x_i+\omega_{n+1}y\cos\alpha\right| \sin^{k-1}\alpha\,d\alpha\right]\times \]

\[ {}\times \frac{|\omega_{n+1}|^k\,d\Omega}{L(\omega_1,\ldots,\omega_n,\omega_{n+1}^2)}. \tag{6.16} \]

Here

\[ C(n,k)= \frac{ (-1)^{\frac{n+k-1}{2}}(2m-\gamma)! }{ \pi^{\frac{n+1}{2}}\Gamma\left(\frac k2\right) 2^{n+k}\left(\frac{n+k-1}{2}\right)! }. \]

In both cases the fundamental solution is an ordinary function, continuous at the origin of coordinates.

Let now \(2m<\gamma\). Then the fundamental solution has the form

\[ u(x,y)=C(n,k)\int_{\Omega} |\omega x|_{B}^{2m-\gamma} \frac{|\omega_{n+1}|^k\,d\Omega}{L(\omega_1,\ldots,\omega_n,\omega_{n+1}^2)}. \tag{6.17} \]

Here \(C(n,k)\) is determined by formula (6.15′) in the case of fractional \(k\), and by the formula

\[ C(n,k)= \frac{ (-1)^{\frac{n+k+1}{2}}(n+k-2m)! }{ \pi^{\frac{n+1}{2}}\Gamma\left(\frac k2\right) 2^{n+k}\left(\frac{n+k-1}{2}\right)! }. \]

in the case of even \(\gamma\), and by the formula

\[ C(n,k)=\frac{(n+k-2m)!}{\pi^{\frac n2}\Gamma\left(\frac k2\right)(n+k)!} \]

in the case of odd \(\gamma\).

The fundamental solution expressed by formula (6.17) is also an ordinary (non-generalized) function, since the homogeneous function of degree \(2m-(n+k+1)\) is integrable in the space \(R_{n+1}\) with weight \(y^k\).

To obtain a fundamental solution with singularity at an arbitrary point \((s,t)\), one must apply the shift operator \(T^{s,t}_{x,y}\) to formulas (6.15)—(6.17).

It can be shown that in a neighborhood of the origin the fundamental solution admits the estimates:

\[ u(x,y)= \begin{cases} O\left(r^{2m-\gamma}\ln r\right), & \text{if } 2m\geq \gamma \text{ and } \gamma \text{ is even},\\ O\left(r^{2m-\gamma}\right), & \text{in all other cases}. \end{cases} \tag{6.18} \]

After application of the shift operator \(T^{s,t}_{x,y}\), the character of the singularity is smoothed, and the solution inside the domain \((y>0)\) behaves in the same way as the fundamental solution of an ordinary elliptic equation.

Note added in proof. In the recently published paper by R. J. Weinacht in Contributions to Differential Equations, 1964, vol. 3, a fundamental solution of the operator \(\Delta_B^m\) was also found by another method.

References

1. Watson G. N. A Treatise on the Theory of Bessel Functions. IL, 1949.
2. Gel'fand I. M., Shilov G. E. Generalized Functions and Operations on Them. Fizmatgiz, 1958.
3. Gel'fand I. M., Shilov G. E. Spaces of Test and Generalized Functions. Fizmatgiz, 1958.
4. Weinstein A. Trans. of the Amer. Math. Soc., 63, 2, 342—354, 1948.
5. Zhitomirskii Ya. M. Matem. sb., 36, no. 2, 1955, pp. 299—310.
6. Kipriyanov I. A. DAN SSSR, 158, No. 2, 275—278, 1964.
7. Kipriyanov I. A. Abstract of a doctoral dissertation. Moscow, 1964.
8. Levitan B. M. UMN, vol. VI, no. 2, 42, 102—143, 1951.

Received by the editors
August 30, 1966

Voronezh Technological Institute