Improved $$L^p$$ bounds for Bochner–Riesz operators associated with rough convex domains in the plane

Abstract We consider generalized Bochner–Riesz operators associated with planar convex domains. We construct domains which yield improved $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> bounds for the Bochner–Riesz over previous results of Seeger–Ziesler and Cladek. The constructions are based on the existence of large $$B_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> sets due to Bose–Chowla, and large $$\Lambda (p)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> sets due to Bourgain.