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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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##### References:
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