Results on weighted norm inequalities for multipliers

Abstract

Weighted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm inequalities are derived for multiplier operators on Euclidean space. The multipliers are assumed to satisfy conditions of the Hörmander-Mikhlin type, and the weight functions are generally required to satisfy conditions more restrictive than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which depend on the degree of differentiability of the multiplier. For weights which are powers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue x EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left | x \right |</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, sharp results are obtained which indicate such restrictions are necessary. The method of proof is based on the function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript number-sign"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>f</mml:mi> <mml:mi mathvariant="normal">#</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{f^\# }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of C. Fefferman and E. Stein rather than on Littlewood-Paley theory. The method also yields results for singular integral operators.