A. N. BOGAEVSKII

HARMONIC FUNCTIONS ON \(GL(2)\)

(Presented by Academician I. G. Petrovskii, 24 VI 1963)

We consider the full complex linear group of second order \(G = GL(2)\) with elements

\[ g=\left\|\begin{matrix} g_1 & g_2\\ g_3 & g_4 \end{matrix}\right\|. \]

By the Laplace operator on it we shall mean the second-order differential operator, invariant under the operators of right and left translations,

\[ \mathcal L \det \bar g \left( \frac{\partial^2}{\partial g_1\,\partial g_4} - \frac{\partial^2}{\partial g_2\,\partial g_3} \right), \]

where

\[ \frac{\partial}{\partial g_k} = \frac{\partial}{\partial u_k} + i\frac{\partial}{\partial v_k}, \qquad g_k=u_k+iv_k. \]

More precisely, on \(G\) there are two Laplace operators, namely:

\[ \mathcal L_1=\operatorname{Re}\mathcal L,\qquad \mathcal L_2=\operatorname{Im}\mathcal L. \]

A twice differentiable function \(f(g)\) on \(G\) satisfying the two Laplace equations

\[ \mathcal L_1 f(g)=0,\qquad \mathcal L_2 f(g)=0, \]

is called harmonic.

Let \(G_0\) be the full complex unimodular linear group of second order with elements

\[ g_0=\left\|\begin{matrix} g_1^0 & g_2^0\\ g_3^0 & g_4^0 \end{matrix}\right\|, \qquad \det g_0=1, \]

and let \(A\) be the subgroup of \(G_0\) consisting of matrices of the form

\[ \hat a=\left\|\begin{matrix} 1 & a\\ 0 & 1 \end{matrix}\right\|. \]

The homogeneous space \(H=G_0/A\) can be interpreted as the space of linear elements of the complex plane and identified with the space of triangular unimodular matrices

\[ h= \left\|\begin{matrix} \xi^{-1} & 0\\ 0 & \xi \end{matrix}\right\| \left\|\begin{matrix} 1 & 0\\ z & 1 \end{matrix}\right\| = \left\|\begin{matrix} \xi^{-1} & 0\\ \xi z & \xi \end{matrix}\right\|^{*}, \]

in which the group \(G_0\) acts as follows:

\[ h= \left\|\begin{matrix} \xi^{-1} & 0\\ \xi z & \xi \end{matrix}\right\| \xrightarrow{\,g_0\,} h^{*}=h\circ g_0 = \left\|\begin{matrix} \xi^{-1}(zg_2^0+g_4^0)^{-1} & 0\\ \xi(zg_1^0+g_3^0) & \xi(zg_2^0+g_4^0) \end{matrix}\right\|. \]

We shall refer the matrices \(h\) to the parameters \(z,\xi\):

\[ h=(z,\xi), \qquad h\circ g_0= \left( \frac{zg_1^0+g_3^0}{zg_2^0+g_4^0}, \, \xi(zg_2^0+g_4^0) \right). \]

We extend the group of transformations \(G_0\) of the space \(H\) to the group \(G\), putting, for \(g\in G\),

\[ h\circ g= \left\|\begin{matrix} \xi^{-1}(zg_2+g_4)^{-1} & 0\\ \xi(zg_1+g_3) & \xi(zg_2+g_4) \end{matrix}\right\| \quad \text{or} \quad h\circ g= \left( \frac{zg_1+g_3}{zg_2+g_4}, \, \xi(zg_2+g_4) \right). \]

In what follows we consider functions \(\varphi(h_1,h_2)=\varphi(z_1,z_2;\xi_1,\xi_2)\) of two linear elements \(h_1=(z_1,\xi_1)\), \(h_2=(z_2,\xi_2)\), satisfying

\[ \text{* For more details on this, see }(^{1,\,2}). \]

condition

\[ \varphi(\hat{\delta}h_1,\hat{\delta}h_2)=|\delta|^{-4}\varphi(h_1,h_2), \qquad \text{where }\quad \hat{\delta}= \begin{pmatrix} \delta^{-1}&0\\ 0&\delta \end{pmatrix}, \]

or, more explicitly,

\[ \varphi(z_1,z_2;\delta\xi_1,\delta\xi_2) = |\delta|^{-4}\varphi(z_1,z_2;\xi_1,\xi_2). \tag{1} \]

Let \(\widetilde{\varphi}(h_1,h_2)\) be such that \(\widetilde{\varphi}(z_1,z_2;1,\xi)\) is infinitely differentiable in all its (six) real arguments and

\[ \widetilde{\varphi}(z_1,z_2;1,\xi)\equiv 0 \]

whenever at least one of the inequalities

\[ |z_1|>N,\qquad |z_2|>N,\qquad |\xi|>N,\qquad |\xi|<\varepsilon, \]

is satisfied, where \(N,\varepsilon\) are some positive numbers.

A function \(\varphi(h_1,h_2)\) representable as a sum with a finite number of terms

\[ \varphi(h_1,h_2) = \varphi_1(h_1\circ g_1',h_2\circ g_2') + \varphi_2(h_1\circ g_1'',h_2\circ g_2'') +\cdots, \]

where \(\varphi_1(h_1,h_2),\varphi_2(h_1,h_2),\ldots\) are functions of precisely the type just described, will be called finite.

Starting from a finite \(\varphi(h_1,h_2)\), construct a function \(f(g)\) on \(G\) by the formula

\[ f(g)=|\xi|^4\int \varphi\left(z,\frac{zg_1+g_3}{zg_2+g_4};\xi,\xi(zg_2+g_4)\right)\,dz, \tag{2} \]

where \(dz=dx\,dy,\ z=x+iy\), and the integral is taken over the whole plane of the complex variable \(z\) (in view of (1) it does not depend on \(\xi\)).

Introducing the notation \(dh=|\xi|^4dz\), formula (2) can be rewritten in the form

\[ f(g)=\int\varphi(h,h\circ g)\,dh. \tag{3} \]

The integral (3) is a relative invariant, namely,

\[ \int \varphi(h\circ \widetilde{g},h\circ \widetilde{g}g)\,dh = |\det \widetilde{g}|^{-2} \int \varphi(h,h\circ g)\,dh. \]

The function \(f(g)\), obtained by means of (2), (3), is harmonic.

Define the norm of \(\varphi(h_1,h_2)\) by the formula

\[ \|\varphi\|^2 = \int |\delta|^2 \left|\varphi(h_1,\hat{\delta}h_2)\right|^2 \,dh_1\,dh_2\,d\delta \qquad (d\delta=d\delta_1\,d\delta_2,\ \delta=\delta_1+i\delta_2). \tag{4} \]

The integral (4) does not depend on \(\xi_1,\xi_2\) and is a relative invariant, namely:

\[ \int |\delta|^2 \left|\varphi(h_1\circ g_1,\hat{\delta}h_2\circ g_2)\right|^2 \,dh_1\,dh_2\,d\delta = \]

\[ = |\det g_1|^{-2}|\det g_2|^{-2} \int |\delta|^2 \left|\varphi(h_1,\hat{\delta}h_2)\right|^2 \,dh_1\,dh_2\,d\delta. \]

Denote by \(\Phi\) the Hilbert space of functions \(\varphi(h_1,h_2)\) with finite norm (4).

The set \(\widetilde{\Phi}\) of finite \(\varphi(h_1,h_2)\) is contained in \(\Phi\) and is everywhere dense in it with respect to the norm (4).

Let \(\Gamma_0\) be the linear space of infinitely differentiable harmonic functions \(f(g)\) on \(G\) (it is not assumed that they are obtained from \(\varphi\)) satisfying the condition:

\[ \left| \frac{\partial^{k_1+k_2}}{\partial\gamma^{k_1}\partial\overline{\gamma}^{k_2}} \frac{\partial^{p_1+p_2+q_1+q_2}} {\partial\alpha^{p_1}\partial\overline{\alpha}^{p_2} \partial\beta^{q_1}\partial\overline{\beta}^{q_2}} f(\hat{a}\sigma\hat{b}) \right| < \frac{C(a,b)} {\left(1+|\alpha|^2+|\beta|^2+|\gamma|^2+|\alpha\beta|^2\right)^{\frac{k_1+k_2+1}{2}+\varepsilon}} \]

for any fixed \(\hat{a},\hat{b}\) and \(k_1+k_2+p_1+p_2+q_1+q_2\leqslant 2\). Here

\[ \sigma= \begin{pmatrix} \alpha&0\\ \gamma&\beta \end{pmatrix}, \qquad \hat{a}= \begin{pmatrix} 1&a\\ 0&1 \end{pmatrix}, \qquad \hat{b}= \begin{pmatrix} 1&b\\ 0&1 \end{pmatrix}, \]

\[ \frac{\partial}{\partial\alpha} = \frac{\partial}{\partial \operatorname{Re}\alpha} +i\frac{\partial}{\partial \operatorname{Im}\alpha}, \qquad \frac{\partial}{\partial\beta} = \frac{\partial}{\partial \operatorname{Re}\beta} +i\frac{\partial}{\partial \operatorname{Im}\beta}, \qquad \frac{\partial}{\partial\gamma} = \frac{\partial}{\partial \operatorname{Re}\gamma} +i\frac{\partial}{\partial \operatorname{Im}\gamma}. \]

I. For \(f(g)\in \Gamma_0\) the integral

\[ \|f\|^2=\frac{|\det g_1|^2|\det g_2|^2}{4\pi^2} \int\left\{\left|\frac{\partial}{\partial u}f(g_1\sigma g_2)\right|^2+ \left|\frac{\partial}{\partial v}f(g_1\sigma g_2)\right|^2\right\}\,d\sigma \tag{5} \]

\[ (\gamma=u+iv,\qquad d\sigma=d\alpha\,d\beta\,d\gamma) \]

is finite and does not depend on \(g_1,g_2\).

II. The linear space \(\Gamma_0\) with norm (5) is mapped isomorphically and isometrically onto a certain everywhere dense in \(\Phi\) linear space \(\Phi_0\), which includes the linear space \(\widetilde{\Phi}\) of finite \(\varphi(h_1,h_2)\), by means of the mutually inverse formulas:

\[ f(g)=\int \varphi(h,h\circ g)\,dh, \tag{6} \]

\[ \varphi(h_1,h_2)=\frac{1}{4\pi^2}\int \left[L_{z_0}f\left(h_1^{-1}\hat{\zeta}\hat{z}_0h_2\right)\right]_{z_0=0}\,d\zeta, \tag{7} \]

where

\[ \hat{\zeta}=\left\|\begin{matrix}\zeta&0\\0&1\end{matrix}\right\|, \qquad \hat{z}_0=\left\|\begin{matrix}1&0\\z_0&1\end{matrix}\right\|, \qquad L_{z_0}=\frac{\partial^2}{\partial x_0^2}+\frac{\partial^2}{\partial y_0^2}, \]

\[ z_0=x_0+iy_0,\qquad d\zeta=dt\,dw,\qquad \zeta=t+iw. \]

Moreover, as was stated, \(\|f\|^2=\|\varphi\|^2\). The integral

\[ \varphi(h_1,h_2)=\frac{1}{4\pi^2}|\det g_1|^2 \int\left[L_{z_0}f\left(g_1(h_1\circ g_1)^{-1}\hat{\zeta}\hat{z}_0(h_2\circ g_2^{-1})g_2\right)\right]_{z_0=0}\,d\zeta \tag{8} \]

does not depend on \(g_1,g_2\), so that (8) generalizes formula (7).

III. In view of what was said in the preceding paragraph, and as formulas (7), (6) show, the function \(f(g)\in\Gamma_0\) is completely determined by its values on the triangular group \(\Sigma\) of matrices

\[ \sigma=\left\|\begin{matrix}\alpha&0\\ \gamma&\beta\end{matrix}\right\| \]

(the argument of \(f\) in (7) is triangular).

It is also completely determined by its values on any surface \(g_1\Sigma g_2\) (see (8), (6)), obtained by a two-sided shift of the group \(\Sigma\).

For every infinitely smooth function \(f_0(\sigma)\) on \(\Sigma\), satisfying the condition

\[ \left| \frac{\partial^{k_1+k_2}}{\partial\gamma^{k_1}\partial\overline{\gamma}^{k_2}} \frac{\partial^{p_1+p_2+q_1+q_2}}{\partial\alpha^{p_1}\partial\overline{\alpha}^{p_2}\partial\beta^{q_1}\partial\overline{\beta}^{q_2}} \right| < \frac{C}{\left(1+|\alpha|^2+|\beta|^2+|\gamma|^2+|\alpha\beta|^2\right)^{\frac{k_1+k_2+1}{2}+\varepsilon}}, \]

\[ k_1+k_2+p_1+p_2+q_1+q_2\leq 2 \]

there exists a harmonic \(f(g)\) such that

\[ f(\sigma)\equiv f_0(\sigma). \]

This function is given by the formula

\[ f(g)=-\frac{1}{4\pi^2}\int\left\{\int \left[L_{z_0}f_0\left(h^{-1}\hat{\zeta}\hat{z}_0h\circ g\right)\right]_{z_0=0}\,d\zeta\right\}\,dh. \]

The question of whether \(f(g)\) belongs to the space \(\Gamma_0\) remains open for the present.

Gorky State University
named after N. I. Lobachevsky

Received
7 V 1963

CITED LITERATURE

\(^{1}\) I. M. Gel'fand, M. I. Graev, Tr. Mosk. matem. obshch., 8, 321 (1959).
\(^{2}\) I. M. Gel'fand, M. A. Naimark, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 36 (1950).