Littlewood-Paley and multiplier theorems on weighted $L\sp{p}$\ spaces

Abstract

The Littlewood-Paley operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma left-parenthesis f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma (f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for functions <italic>f</italic> defined on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">R</mml:mtext> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\textbf {R}}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is shown to be a bounded operator on certain 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> spaces. The weights satisfy an <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> condition over the class of all <italic>n</italic>-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hörmander’s Multiplier Theorem.

Locations

• Transactions of the American Mathematical Society - View - PDF