Linear topological invariants for kernels of differential operators by shifted fundamental solutions

Abstract We characterize the condition $$(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions $$(P\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$(P\overline{\overline{\Omega }})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mover> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for distributional kernels are characterized in a similar way. By lifting theorems for Fréchet spaces and (PLS)-spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamental solutions. As an application, we give a new proof of the fact that the space $$\{ f \in {\mathscr {E}}(X) \, | \, P(D)f = 0\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> <mml:mspace/> <mml:mo>|</mml:mo> <mml:mspace/> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> satisfies $$(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for any differential operator P ( D ) and any open convex set $$X \subseteq {\mathbb {R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> .