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- W2899484521 abstract "Abstract In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msubsup> <m:mi>ℳ</m:mi> <m:mi>β</m:mi> <m:mo>+</m:mo> </m:msubsup> </m:math> $ mathcal{M}_{beta}^{+} $ and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msubsup> <m:mi>ℳ</m:mi> <m:mi>β</m:mi> <m:mo>-</m:mo> </m:msubsup> </m:math> $ mathcal{M}_{beta}^{-} $ map <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ℝ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> $ W^{1,p}(mathbb{R}) $ into <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ℝ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> $ W^{1,q}(mathbb{R}) $ with 1 < p <∞, 0≤β<1/ p and q = p /(1- pβ ), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msubsup> <m:mi>M</m:mi> <m:mi>β</m:mi> <m:mo>+</m:mo> </m:msubsup> </m:math> $ M_{beta}^{+} $ and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msubsup> <m:mi>M</m:mi> <m:mi>β</m:mi> <m:mo>-</m:mo> </m:msubsup> </m:math> $ M_{beta}^{-} $ from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi mathvariant=normal>ℓ</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> $ ell^{1}(mathbb{Z}) $ to <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>BV</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> $ {rm BV}(mathbb{Z}) $ . Here <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>BV</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> $ {rm BV}(mathbb{Z}) $ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously." @default.
- W2899484521 created "2018-11-09" @default.
- W2899484521 creator A5068917403 @default.
- W2899484521 date "2018-10-01" @default.
- W2899484521 modified "2023-09-27" @default.
- W2899484521 title "On the regularity of one-sided fractional maximal functions" @default.
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