Title: 𝐿^{𝑝}-𝐿^{𝑝’} estimates for overdetermined Radon transforms
Abstract: We prove several variations on the results of F. Ricci and G. Travaglini (2001), concerning <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p Baseline minus upper L Superscript p prime"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L^{p}-L^{p’}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounds for convolution with all rotations of arc length measure on a fixed convex curve in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R} ^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Estimates are obtained for averages over higher-dimensional convex (nonsmooth) hypersurfaces, smooth <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional surfaces, and nontranslation-invariant families of surfaces. We compare Ricci and Travaglini’s approach, based on average decay of the Fourier transform, with an approach based on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> boundedness of Fourier integral operators, and show that essentially the same geometric condition arises in proofs using the two techniques.