Mean value properties of the Laplacian via spectral theory

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis z squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi ({z^2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an even entire function of temperate exponential type, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a selfadjoint realization of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative normal upper Delta plus c left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">- \Delta + c\,(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Laplace-Beltrami operator on a Riemannian manifold, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi \,(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the operator given by spectral theory. A Paley-Wiener theorem on the support of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi \,(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is proved, and is used to show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L u equals lamda u"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Lu = \lambda u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a suitable domain implies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis upper L right-parenthesis u equals phi left-parenthesis lamda right-parenthesis u"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi \,(L)\,u = \phi \,(\lambda )\,u</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as well as a generalization of Àsgeirsson’s theorem. A concrete realization of the operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi \,(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given in the case of a compact Lie group or a noncompact symmetric space with complex isometry group.

Locations

• Transactions of the American Mathematical Society - View - PDF