Connections between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mi mathvariant="script">S</mml:mi> </mml:math> -operators and restriction estimates for spheres over finite fields

In this paper, we introduce a new operator, S , which is closely related to the restriction problem for spheres in F q d , the d -dimensional vector space over the finite field F q with q elements. The S operator is considered as a specific operator that maps functions on F q d to functions on F q d + 1 . We explore a relationship between the boundedness of the S operator and the restriction estimate for spheres in F q d . Consequently, using this relationship, we prove that the L 2 restriction conjectures for spheres hold in all dimensions when the test functions are restricted to homogeneous functions of degree zero.