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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>โ„“</mml:mi> <mml:mi>p</mml:mi> </mml:msup></mml:math>-improving for discrete spherical averages

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>โ„“</mml:mi> <mml:mi>p</mml:mi> </mml:msup></mml:math>-improving for discrete spherical averages

We initiate the theory of โ„“ p -improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove โ„“ p -improving estimates for the discrete spherical averages and some of their generalizations. As an application of our โ„“ p -improving inequalities for the dyadic discrete spherical โ€ฆ