2013/09/24 by David Dos Santos Ferreira, Wolfgang Staubach · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Holomorphic and Operator Theory #advanced mathematical theories
paper · doi:10.1090/memo/1074
We investigate the global continuity 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">Lp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of left-bracket 1 comma normal infinity right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p∈ [1,∞ ]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain necessary non-degeneracy conditions. In this context we also prove the optimal global <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> boundedness result for Fourier integral operators with non-degenerate phase functions and the most general smooth Hörmander class amplitudes i.e. those in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript rho comma delta Superscript m"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ϱ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> δ </mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">Sm \varrho , δ </mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho comma delta element-of left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi> ϱ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> δ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\varrho , δ ∈ [0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also prove the very first results concerning the continuity of smooth and rough Fourier integral operators on 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:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">Lp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript w Superscript p"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>w</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">Lwp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1> p > ∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w element-of upper A Subscript p Baseline comma"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">w∈ Ap,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (i.e. the Muckenhoupt weights) for operators with rough and smooth amplitudes and phase functions satisfying a suitable rank condition. These results are shown to be optimal for operators with amplitudes in classical Hörmander classes and can also be given a geometrically invariant formulation. The weighted results are in turn applied to prove, for the first time, weighted and unweighted estimates for the commutators of Fourier integral operators with functions of bounded mean oscillation BMO, estimates on weighted Triebel-Lizorkin spaces, and finally global unweighted and local weighted estimates for the solutions of the Cauchy problem for <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> </i