Estimates for Fourier sums and eigenvalues of integral operators via multipliers on the sphere

Abstract

We provide estimates for weighted Fourier sums of integrable functions defined on the sphere when the weights originate from a multiplier operator acting on the space where the function belongs. That implies refined estimates for weighted Fourier sums of integrable kernels on the sphere that satisfy an abstract Hölder condition based on a parameterized family of multiplier operators defining an approximate identity. This general estimation approach includes an important class of multiplier operators, namely, that defined by convolutions with zonal measures. The estimates are used to obtain decay rates for the eigenvalues of positive integral operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis upper S Superscript m Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2(S^m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and generated by a kernel satisfying the Hölder condition based on multiplier operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis upper S Superscript m Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2(S^m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .