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• W2907815512 abstract "In 1980, Roe proved that if a doubly-infinite sequence <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace f Subscript k Baseline right-brace> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{f_k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of functions on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Subscript k plus 1 Baseline equals left-parenthesis d f Subscript k Baseline slash d x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f_{k+1}=(df_{k}/dx)</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=StartAbsoluteValue f Subscript k Baseline left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-equal-to upper M> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>|f_{k}(x)|leq M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k equals 0 comma plus-or-minus 1 comma plus-or-minus 2 comma midline-horizontal-ellipsis> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>k=0,pm 1,pm 2,cdots</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=x element-of double-struck upper R> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>xin mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f 0 left-parenthesis x right-parenthesis equals a sine left-parenthesis x plus phi right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</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>a</mml:mi> <mml:mi>sin</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>φ<!-- φ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f_0(x)=asin (x+varphi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <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> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi> <mml:semantics> <mml:mi>φ<!-- φ --></mml:mi> <mml:annotation encoding=application/x-tex>varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are real constants. This result was extended to <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> by Strichartz in 1993, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d slash d x> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>d/dx</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was substituted by the Laplacian on <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>. While it is plausible that this theorem extends to other Riemannian manifolds or Lie groups, Strichartz showed that the result holds true for Heisenberg groups, but fails for hyperbolic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-space. This negative result can indeed be extended to any Riemannian symmetric space of noncompact type. We observe that this failure is rooted in the <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>-dependence 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>-spectrum of the Laplacian on the hyperbolic spaces. Taking this into account we shall prove that for all rank one Riemannian symmetric spaces of noncompact type, and more generally for the harmonic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N upper A> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>NA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> groups, the theorem actually holds true when uniform boundedness is replaced by uniform “almost <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”. In addition we shall see that for the symmetric spaces this theorem can be used to characterize the Poisson transforms of <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> functions on the boundary, which somewhat resembles the original theorem of Roe on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2907815512 date "2013-09-26" @default.
• W2907815512 modified "2023-10-13" @default.
• W2907815512 title "Characterization of almost 𝐿^{𝑝}-eigenfunctions of the Laplace-Beltrami operator" @default.
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