The 𝑊^{𝑠,𝑝}-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

Abstract

In this paper, we investigate the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript s comma p"> <mml:semantics> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">W^{s,p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript a Baseline equals negative normal upper Delta plus StartFraction a Over StartAbsoluteValue x EndAbsoluteValue squared EndFraction comma a greater-than-or-equal-to minus StartFraction left-parenthesis d minus 2 right-parenthesis squared Over 4 EndFraction comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>+</mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mi>a</mml:mi> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>a</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mo> − </mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \mathcal L_a=-\Delta +\tfrac {a}{|x|^2}, \quad a\geq -\tfrac {(d-2)^2}{4}, \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript s comma p"> <mml:semantics> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">W^{s,p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bounded for certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">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="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which depend on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As corollaries, we solve some open problems associated with the operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript a"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>a</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal L_a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which include the dispersive estimates and the local smoothing estimates in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear

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