article · Proceedings of the American Mathematical Society · 2014
Maximal potentials, maximal singular integrals, and the spherical maximal function
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Abstract
We introduce a notion of maximal potentials and we prove that they form bounded operators from <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> to the homogeneous Sobolev space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper W With dot Superscript 1 comma p"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\dot {W}^{1,p}</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="n slash left-parenthesis n minus 1 right-parenthesis greater-than p greater-than n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n/(n-1)>p>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.
Cite this paper
Hajłasz, P., & Liu, Z. (2014). Maximal potentials, maximal singular integrals, and the spherical maximal function. Proceedings of the American Mathematical Society, 142(11), 3965–3974. https://doi.org/10.1090/s0002-9939-2014-12129-0
Hajłasz, Piotr, and Zhuomin Liu. “Maximal potentials, maximal singular integrals, and the spherical maximal function.” Proceedings of the American Mathematical Society, vol. 142, no. 11, 2014, pp. 3965–74, https://doi.org/10.1090/s0002-9939-2014-12129-0.
Hajłasz, Piotr, and Zhuomin Liu. 2014. “Maximal potentials, maximal singular integrals, and the spherical maximal function.” Proceedings of the American Mathematical Society 142 (11): 3965–74. https://doi.org/10.1090/s0002-9939-2014-12129-0.
Hajłasz, P. and Liu, Z. (2014) ‘Maximal potentials, maximal singular integrals, and the spherical maximal function’, Proceedings of the American Mathematical Society, 142(11), pp. 3965–3974. Available at: https://doi.org/10.1090/s0002-9939-2014-12129-0.
@article{hajasz2014maximal,
title = {{Maximal potentials, maximal singular integrals, and the spherical maximal function}},
author = {Hajłasz, Piotr and Liu, Zhuomin},
journal = {Proceedings of the American Mathematical Society},
volume = {142},
number = {11},
pages = {3965--3974},
year = {2014},
doi = {10.1090/s0002-9939-2014-12129-0}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.