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Asgeirsson’s mean value theorem and related identities. (English) Zbl 1019.35022
L. Schwartz observed the equivalence of the mean value property of a harmonic function, i.e. \[ f_R\ast u=0\text{ for } u \text{ with } \Delta_xu=0,\;f_R=2R\delta\bigl(|x|^2-R^2\bigr)-R^{n-1}\omega_n\delta, \] with the (unique) existence of a distribution \(\mu_R\) with compact support such that \(\Delta_x\mu_R=f_R.\) Explicitly, \(\mu_R=\text{const}(|x|^{2-n}-R^{2-n})\chi^0_+ (R-|x|),\) where \(\chi_{\pm}^a\) denotes the analytic distribution-valued function \(a\mapsto t_{\pm}^a/{\Gamma(a+1)}\), \( a\in\mathbb C\) [Théorie des distributions, Nouv. éd., Hermann, Paris, p. 217 and (VI, 10; 29), p. 218 (1996; Zbl 0149.09501)].
Generalizing he proves that, for \(H\in\mathcal E',\) \(H\ast u=0\) for all \(u\) with \(\Delta_x u=0,\) is equivalent to the existence of \(\mu\in\mathcal E'\) with \(\Delta_x\mu=H.\) Generalizing further he remarks that, for a partial differential operator \(P(\partial)\) with constant coefficients, \(H\in\mathcal E'\) fulfills \(H\ast u=0\) for all \(u\in\mathcal D'\) with \(P(\partial)u=0,\) if and only if there exists \(\mu\in\mathcal E'\) with \(P(\partial)\mu=H\) (p. 218).
In Chapter 2 of the paper under review, the author proves the classical mean value theorem of L. Asgeirsson [Math. Ann. 113, 321-346 (1937; Zbl 0015.01804)], i.e., \[ f_R\ast u=0\text{ for } u \text{ with } (\Delta_x-\Delta_y)u=0,\quad x\in\mathbb R^\nu,\;y\in\mathbb R^\nu, \] \[ f_R=2R\delta\bigl(|x|^2-R^2\bigr)\otimes\delta_y-\delta_x\otimes 2R\delta\bigl(|y|^2-R^2\bigr), \] by constructing explicitly the corresponding distribution \(\mu_R\in\mathcal E'\) fulfilling \((\Delta_x-\Delta_y)\mu_R=f_R.\)
Denoting by \((-\frac 1{16})A_R(x,y)\) the quartic polynomial which, for appropriate \(R,|x|,|y|,\) expresses the square of the area of a triangle with these sides, and by \(K_R\) the set \(\{(x,y)\in\mathbb R^{2\nu}\mid |x|+|y|\leq R\},\) the result is (Theorem 2.1): \(\mu_R=\frac 12\chi_+^{\frac{1-\nu}2}( \frac\pi{4R^2}A_R),\) restricted to \(\mathcal E'(K_R).\)
Herein, \(\mu_R\) is defined as pullback of \(\chi_+^a\) by \(\frac\pi{4R^2}A_R,\) suitably regularized at the singular set \(\{(x,y)\in\mathbb R^{2\nu}: |x|\cdot|y|=0\), \(|x|+|y|=R\}.\)
Let us observe that, alternatively, \(\mu_R\) can be defined as the value at \(\lambda=\frac{1-\nu}2\) of the distribution-valued function \(\lambda\mapsto T_\lambda=\frac 12\chi^0_+(R-|x|-|y|)\chi_+^\lambda (\frac\pi{4R^2}A_R),\) which is analytically continued to the whole complex plane, except for the series of simple poles \(-\frac{1+\nu}2, -\frac{3+\nu}2,\cdots,\) by means of the recurrence relation \[ (\Delta_x-\Delta_y)T_\lambda=\frac{2\pi}{R^2}\bigl(|x|^2-|y|^2\bigr) (\nu+2\lambda-1)T_{\lambda-1}. \] A first main purpose of the paper is the generalization of Asgeirsson’s Theorem with respect to the quadratic forms \(|x|^2,|y|^2.\) The result is stated in Theorem 4.2.:
Let \(Q_1(x),\;x\in\mathbb R^\nu\), \(Q_2(y)\), \(y\in\mathbb R^\nu,\) be nondegenerate real quadratic forms and \(Q_1^\ast,Q_2^\ast\) the dual quadratic forms defined by the inverses of the symmetric matrices representing \(Q_1,Q_2.\) Analogously to \(A_R,\) define the quartic polynomial \(T_R\) by the formula \[ T_R(x,y)=\bigl[Q_1(x)-Q_2(y)\bigr]^2-2R^2\bigl[Q_1(x)+Q_2(y)\bigr]+R^4. \] Then the distributions \(\chi_\pm^{\frac{1-\nu}2}(\frac\pi{4R^2}T_R),\) defined outside the characteristic surface \(T_R^{-1}(0),\) can be extended to distributions \(U_\pm\) fulfilling \[ \bigl[Q_1^\ast(\partial_x)-Q_2^\ast(\partial_y)\bigr]U_\pm=c_\pm^12R\delta \bigl(Q_1(x)-R^2\bigr)\otimes\delta_y +c_\pm^2\delta_x\otimes 2R\delta\bigl(Q_2(y)-R^2)), \] \[ c_\pm^1=\frac 4{\sqrt{|\text{det} Q_2|}}\begin{cases} \sin\bigl(\pi(1+\nu_2^-)/2\bigr),\\ -\sin(\pi\nu_2^+/2),\end{cases} \] \[ c_\pm^2=\frac 4{\sqrt{|\text{det} Q_1|}} \begin{cases} -\sin\bigl(\pi(1+\nu_1^-)/2\bigr),\\ \sin(\pi\nu_1^+/2),\end{cases} \] wherein \((\nu_j^+,\nu_j^-)\) denote the signatures of \(Q_j, j=1,2.\)
The key point is the extension which relies on the existence of a kind of a generalized fundamental solution – stated as Theorem 3.3. Roughly speaking, if \(Q(x,\partial_x)\) denotes a second-order differential operator with \(\mathcal C^\infty\)-coefficients and \(F\) is a real-valued \(\mathcal C^\infty\)-function on a manifold \(X\) such that \(S=\{x\in X: F(x)=0\), \(F'(x)=0\}\) is a submanifold of codimension \(\nu>2\) with \(\text{rk} F''(x) =\nu,\) and \(Q(x,\partial_x)\chi_\pm^{1-\nu/2}(F)\) vanishes outside \(S,\) then \(\chi_\pm^{1-\nu/2}(F)\) can be extended to X and fulfill there \(Q(x,\partial_x) \chi_\pm^{1-\nu/2}(F)=\text{const} \delta_{S,F}.\) Therein \(\delta_{S,F}\) is the Dirac distribution on \(S,\) carefully defined in Proposition 3.2.
A second purpose of the paper is the study of the invariance of the distribution
\(\frac 12\chi^{\frac{1-\nu}2}(\frac \pi{4R^2}T_R)\) under Kelvin transformations \(u\mapsto\widetilde u,\) defined by \[ \widetilde u(x,y)= -\biggl(\frac{R^2}{Q_1(x)-Q_2(y)}\biggr)^{\nu-1}u \biggl(\frac{(R^2x,R^2y)}{Q_1(x)-Q_2(y)}\biggr). \] Finally, the possibility of constructing solutions of the type \(\chi_\pm^{-\nu/2}(\frac\pi{2a}P)\) for polynomials \(P\) other than \(T_R\) is examined and a theorem is proven for a particular polynomial \(P\) of third degree describing the surface which results from \(T_R^{-1}(0)\) by an inversion \((x,y)\mapsto\frac{(R^2x,R^2y)}{Q_1(x)-Q_2(y)}.\)
Some minor misprints can easily be corrected (p. 379, last and 17th line from the bottom, p. 386, 17th line from the bottom: replace \(R+s\) by \(R-s;\) p. 379 and p. 386, 16th line from the bottom: replace \(s>-R\) by \(s<R).\)

MSC:
35C05 Solutions to PDEs in closed form
35A08 Fundamental solutions to PDEs
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