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• W4313133049 abstract "Abstract We mainly study the dimension-free <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> {L}^{p} -inequality of the truncated maximal operator <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mi>a</m:mi> </m:mrow> </m:msubsup> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:munder> <m:mrow> <m:mi>sup</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:munder> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mo>∣</m:mo> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>∣</m:mo> </m:mrow> </m:mfrac> <m:mfenced open=∣ close=∣> <m:mrow> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:munder> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>t</m:mi> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant=normal>d</m:mi> <m:mi>y</m:mi> <m:mspace width=-0.25em /> </m:mrow> </m:mfenced> <m:mo>,</m:mo> </m:math> {M}_{n}^{a}fleft(x)=mathop{sup }limits_{tgt 0}frac{1}{| {B}_{a}^{1}| }left|mathop{int }limits_{{B}_{a}^{1}}fleft(x-ty){rm{d}}yhspace{-0.25em}right|, where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>:</m:mo> <m:mi>a</m:mi> <m:mo>≤</m:mo> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>∣</m:mo> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {B}_{a}^{1}=left{x:ale | x| le 1right} . When <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>a</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:math> 0le alt 1 , we prove that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>‖</m:mo> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mi>a</m:mi> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=double-struck>R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mi>C</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>‖</m:mo> <m:mi>f</m:mi> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=double-struck>R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:math> Vert {M}_{n}^{a}{Vert }_{{L}^{p}left({{mathbb{R}}}^{n})}le Cleft(p)Vert f{Vert }_{{L}^{p}left({{mathbb{R}}}^{n})} for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mi>n</m:mi> <m:mspace width=0.1em /> <m:mtext>/</m:mtext> <m:mspace width=0.1em /> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> pgt nhspace{0.1em}text{/}hspace{0.1em}left(n-1) . When <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>a</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> a=1 , we prove that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>‖</m:mo> <m:msubsup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=double-struck>R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mi>C</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>‖</m:mo> <m:mi>f</m:mi> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=double-struck>R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:math> Vert {M}_{n}^{1}{Vert }_{{L}^{p}left({{mathbb{R}}}^{n})}le Cleft(p)Vert f{Vert }_{{L}^{p}left({{mathbb{R}}}^{n})} for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> pge 2 ." @default.
• W4313133049 created "2023-01-06" @default.
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• W4313133049 date "2022-01-01" @default.
• W4313133049 modified "2023-09-25" @default.
• W4313133049 title "The dimension-free estimate for the truncated maximal operator" @default.
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