zbMATH — the first resource for mathematics

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 \(s<R).\)

35C05 Solutions to PDEs in closed form
35A08 Fundamental solutions to PDEs
Full Text: DOI
[1] Friedlander, G., Simple progressive solutions of the wave equation, Proc. Cambridge philos. soc., 43, 360-373, (1947) · Zbl 0029.04101
[2] Hörmander, L., The analysis of linear partial differential operators I, (1983), Springer-Verlag Berlin/New York
[3] Hörmander, L., Local P-convexity, J. anal. math., 80, 101-141, (2000) · Zbl 0968.35026
[4] Lewy, H., Extension of Huyghen’s principle to the ultrahyperbolic equation, Ann. mat. pura appl. (4), 39, 63-64, (1955) · Zbl 0068.07902
[5] M. Riesz, A Special Characteristic Surface—A New Relativistic Model for a Particle?, (L. Gårding and L. Hörmander, Eds.), Marcel Riesz Collected Papers, pp. 848-858, Springer-Verlag, Berlin/New York, 1988.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.