2010/05/31 by Jonathan Bober, Emanuel Carneiro, Kevin Hughes +2 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Bounded function #Dimension (graph theory) #Function (biology) #Holomorphic and Operator Theory #Maximal function #Maximal operator #Nonlinear Partial Differential Equations #Operator (biology) #math.FA #msc:42B25 #msc:46E35
paper · pdf · doi:10.1090/s0002-9939-2011-11008-6
published as Proc. Amer. Math. Soc. 140 (2012), 1669-1680 · V4 - Proof of Lemma 3 updated
openalex publication_date 2011/09/01 · arxiv created 2014/12/28 · arxiv updated 2014/12/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
In this paper we prove a discrete version of Tanaka’s theorem for the Hardy-Littlewood maximal operator in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , both in the non-centered and centered cases. For the non-centered maximal operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>M</mml:mi> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove that, given a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon double-struck upper Z right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </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">f: \mathbb Z → \mathbb R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of bounded variation, <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V a r left-parenthesis upper M overTilde f right-parenthesis less-than-or-equal-to upper V a r left-parenthesis f right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:mi>Var</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>M</mml:mi> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≤ </mml:mo> <mml:mi>Var</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname Var(\widetilde M f) ≤ \operatorname Var(f),</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="upper V a r left-parenthesis f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Var</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname Var(f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> represents the total variation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For the centered maximal operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove that, given a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon double-struck upper Z right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </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">f: \mathbb Z → \mathbb R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of script l Superscript 1 Baseline left-parenthesis double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f ∈ ℓ 1(\mathbb Z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V a r left-parenthesis upper M f right-parenthesis less-than-or-equal-to upper C double-vertical-bar f double-vertical-bar Subscript script l Sub Superscript 1 Subscript left-parenthesis double-struck upper Z right-parenthesis Baseline period">