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• W3137984385 abstract "We study operators of the form <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T f left-parenthesis x right-parenthesis equals psi left-parenthesis x right-parenthesis integral f left-parenthesis gamma Subscript t Baseline left-parenthesis x right-parenthesis right-parenthesis upper K left-parenthesis t right-parenthesis d t comma> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} Tf(x)= psi (x) int f(gamma _t(x))K(t),dt, end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma Subscript t Baseline left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma _t(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a real analytic function of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis t comma x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(t,x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mapping from a neighborhood of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis 0 comma 0 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(0,0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript upper N Baseline times double-struck upper R Superscript n> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {R}^N times mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript n> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma 0 left-parenthesis x right-parenthesis identical-to x> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≡<!-- ≡ --></mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma _0(x)equiv x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=psi left-parenthesis x right-parenthesis element-of upper C Subscript c Superscript normal infinity Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>c</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>psi (x) in C_c^infty (mathbb {R}^n)</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=upper K left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>K(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a “multi-parameter singular kernel” with compact support in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript upper N> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; for example when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>K(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma Subscript t Baseline left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma _t(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in the single-parameter case when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>K(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Calderón-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for 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>-boundedness of such operators. This paper shows that when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma Subscript t Baseline left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma _t(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is real analytic, the sufficient conditions of Street and Stein are also necessary for 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>-boundedness 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>, for all such kernels <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W3137984385 created "2021-03-29" @default.
• W3137984385 creator A5058046068 @default.
• W3137984385 date "2022-09-02" @default.
• W3137984385 modified "2023-10-13" @default.
• W3137984385 title "Real analytic multi-parameter singular Radon transforms: Necessity of the Stein-Street condition" @default.
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• W3137984385 doi "https://doi.org/10.1090/tran/8715" @default.
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