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- W2108970619 abstract "Fix <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 less-than-or-equal-to p greater-than q less-than-or-equal-to n minus 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi>q</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0 leq p > q leq n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G left-parenthesis p comma n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G(p,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G left-parenthesis q comma n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G(q,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the affine Grassmann manifolds of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>- and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-planes in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript n> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We investigate the Radon transform <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper R Superscript left-parenthesis q comma p right-parenthesis Baseline colon upper C Superscript normal infinity Baseline left-parenthesis upper G left-parenthesis q comma n right-parenthesis right-parenthesis right-arrow upper C Superscript normal infinity Baseline left-parenthesis upper G left-parenthesis p comma n right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>R</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {R}^{(q,p)} : C^{infty } (G(q,n)) to C^{infty } (G(p,n))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with the inclusion incidence relation. For the generic case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dimension upper G left-parenthesis q comma n right-parenthesis greater-than dimension upper G left-parenthesis p comma n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>></mml:mo> <mml:mi>dim</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>dim G(q,n) > dim G(p,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p plus q greater-than n> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>p+q > n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we will show that the range of this transform is given by smooth functions on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G left-parenthesis p comma n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G(p,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p plus q equals n> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>p+q =n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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- W2108970619 date "2004-04-16" @default.
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- W2108970619 title "Dual Radon transforms on affine Grassmann manifolds" @default.
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