Extrapolation for multilinear compact operators and applications
Mingming Cao, Andrea Olivo, Kôzô Yabuta
Abstract
This paper is devoted to studying the Rubio de Francia extrapolation for multilinear compact operators. It allows one to extrapolate the compactness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from just one space to the full range of weighted spaces, whenever an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -linear operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript ModifyingAbove p With right-arrow comma ModifyingAbove r With right-arrow"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>p</mml:mi> <mml:mo stretchy="false"> → </mml:mo> </mml:mover> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo stretchy="false"> → </mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">A_{\vec {p}, \vec {r}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the limited range of the <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:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> scale. To show extrapolation theorems above, by means of a new weighted Fréchet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. To prove the latter, we also need to build a weighted interpolation theorem in mixed-norm Lebesgue spaces. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Calderón-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calderón commutators, and commutators of Riesz transforms related to Schrödinger operators.