Sobolev regularity of polar fractional maximal functions

Publication date: 12 October 2019

Abstract: We study the Sobolev regularity on the sphere

m a t h b b S^{d}

of the uncentered fractional Hardy-Littlewood maximal operator

at the endpoint

p = 1

, when acting on polar data. We first prove that if

,

and

f

is a polar

W^{1, 1} (m a t h b b S^{d})

function, we have

|

abla widetilde{mathcal{M}}_{�eta}f|_qlesssim_{d,�eta}| abla f|_1. We then prove that the map fmapsto �ig |

abla widetilde{mathcal{M}}_{�eta}f �ig |

is continuous from

W^{1, 1} (m a t h b b S^{d})

to

L^{q} (m a t h b b S^{d})

when restricted to polar data. Our methods allow us to give a new proof of the continuity of the map

from

W_{e x t r a d}^{1, 1} (m a t h b b R^{d})

to

L^{q} (m a t h b b R^{d})

. Moreover, we prove that a conjectural local boundedness for the centered fractional Hardy-Littlewood maximal operator

implies the continuity of the map

from

W^{1, 1}

to

L^{q}

, in the context of polar functions on

m a t h b b S^{d}

and radial functions on

m a t h b b R^{d}

.

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