Power weight inequalities for spherical maximal functions

Abstract

Abstract This paper is about spherical maximal functions with general dilation sets acting on functions in weighted $$L^p(|x|^\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> spaces. Aside from endpoint cases, a complete description of the allowable ranges of p , $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> is given in terms of the Legendre–Assouad function of the dilation set. This settles, up to endpoints, an open problem of Duoandikoetxea and Seijo.