Matrix-weighted Besov spaces

2002/08/07 by Svetlana Roudenko · 8 citations
Mathematics · #Advanced Harmonic Analysis Research #Mathematical Analysis and Transform Methods #Holomorphic and Operator Theory

paper · pdf · doi:10.1090/s0002-9947-02-03096-9

Abstract

Nazarov, Treil and Volberg defined matrix <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:msub> <mml:mi>A</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> weights and extended the theory of weighted norm inequalities on <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> to the case of vector-valued functions. We develop some aspects of Littlewood-Paley function space theory in the <italic>matrix weight setting</italic> . In particular, we introduce matrix- weighted homogeneous Besov spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper B With dot Subscript p Superscript alpha q Baseline left-parenthesis upper W right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex"> B^{α q}_p(W)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and matrix-weighted sequence Besov spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove b With dot Subscript p Superscript alpha q Baseline left-parenthesis upper W right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>b</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex"> b^{α q}_p(W)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , as well as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove b With dot Subscript p Superscript alpha q Baseline left-parenthesis left-brace upper A Subscript upper Q Baseline right-brace right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>b</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> <mml:mi>q</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>Q</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex"> b^{α q}_p(\A_Q\)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript upper Q"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>Q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are reducing operators for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Under any of three different conditions on the weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we prove the norm equivalences <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar ModifyingAbove f With right-arrow double-vertical-bar Subscript ModifyingAbove upper B With dot Sub Subscript p Sub Superscript alpha q Subscript left-parenthesis upper W right-parenthesis Baseline almost-equals double-vertical-bar left-brace ModifyingAbove s With right-arrow Subscript upper Q Baseline right-brace Subscript upper Q Baseline double-vertical-bar Subscript ModifyingAbove b With dot Sub Subscript p Sub Superscript alpha q Subscript left-parenthesis upper W right-parenthesis Baseline almost-equals double-vertical-bar left-brace ModifyingAbove s With right-arrow Subscript upper Q Baseline right-brace Subscript upper Q Baseline double-vertical-bar Subscript ModifyingAbove b With dot Sub Subscript p Sub Superscript alpha q Subscript left-parenthesis left-brace upper A Sub Subscript upper Q Subscript right-brace right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"

Citations

• Vector A₂ weights and a Hardy-Littlewood maximal function
• Weighted norm inequalities for the conjugate function and Hilbert transform

Cited by

• Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols
• Matrix-weighted Besov-type and Triebel–Lizorkin-type spaces I: A_p-dimensions of matrix weights and φ-transform characterizations
• On off-diagonal operators in matrix-weighted spaces
• Matrix weighted estimates on spaces of homogeneous type
• Matrix-Weighted Besov Spaces Associated with Non-isotropic Dilations
• Matrix-Weighted Campanato Spaces: Duality and Calderón--Zygmund Operators
• Recent results on matrix weighted norm inequalities
• A matrix weighted T1 theorem for matrix kernelled Calderon Zygmund operators - I

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