TY - JOUR
T1 - New directions in harmonic analysis on L1
AU - Spector, Daniel
N1 - Funding Information:
This paper is based upon a series of lectures the author gave at Washington University in St. Louis, the University of Michigan, the University of Napoli Federico II, Purdue University, Notre Dame University, Indiana University, St. Louis University, Rutgers University, and Georgetown University while on sabbatical at Washington University in St. Louis. It is a pleasure to thank Steven Krantz and the mathematics department at Washington University in St. Louis for hosting him during the undertaking of this work, to José Pastrana for his questions that gave further impetus for the present structure, to Cody Stockdale for discussions concerning Calderón and Zygmund’s decomposition lemma, to Jean Van Schaftingen for comments on a draft of the manuscript, and to Bogdan Raiţǎ for his clarifications concerning a number of points in Section 7 . Needless to say that I remain responsible for the remaining shortcomings. The author is supported in part by the Taiwan Ministry of Science and Technology under research grants 105-2115-M-009-004-MY2 , 107-2918-I-009-003 and 107-2115-M-009-002-MY2 .
Funding Information:
This paper is based upon a series of lectures the author gave at Washington University in St. Louis, the University of Michigan, the University of Napoli Federico II, Purdue University, Notre Dame University, Indiana University, St. Louis University, Rutgers University, and Georgetown University while on sabbatical at Washington University in St. Louis. It is a pleasure to thank Steven Krantz and the mathematics department at Washington University in St. Louis for hosting him during the undertaking of this work, to José Pastrana for his questions that gave further impetus for the present structure, to Cody Stockdale for discussions concerning Calderón and Zygmund's decomposition lemma, to Jean Van Schaftingen for comments on a draft of the manuscript, and to Bogdan Raiţǎ for his clarifications concerning a number of points in Section 7. Needless to say that I remain responsible for the remaining shortcomings. The author is supported in part by the Taiwan Ministry of Science and Technology under research grants 105-2115-M-009-004-MY2, 107-2918-I-009-003 and 107-2115-M-009-002-MY2.
Publisher Copyright:
© 2019 Elsevier Ltd
PY - 2020/3
Y1 - 2020/3
N2 - The study of what we now call Sobolev inequalities has been studied for almost a century in various forms, while it has been eighty years since Sobolev's seminal mathematical contributions. Yet there are still things we do not understand about the action of integral operators on functions. This is no more apparent than in the L1 setting, where only recently have optimal inequalities been obtained on the Lebesgue and Lorentz scale for scalar functions, while the full resolution of similar estimates for vector-valued functions is incomplete. The purpose of this paper is to discuss how some often overlooked estimates for the classical Poisson equation give an entry into these questions, to present the state of the art of what is known, and to survey some open problems on the frontier of research in the area.
AB - The study of what we now call Sobolev inequalities has been studied for almost a century in various forms, while it has been eighty years since Sobolev's seminal mathematical contributions. Yet there are still things we do not understand about the action of integral operators on functions. This is no more apparent than in the L1 setting, where only recently have optimal inequalities been obtained on the Lebesgue and Lorentz scale for scalar functions, while the full resolution of similar estimates for vector-valued functions is incomplete. The purpose of this paper is to discuss how some often overlooked estimates for the classical Poisson equation give an entry into these questions, to present the state of the art of what is known, and to survey some open problems on the frontier of research in the area.
KW - L estimates
KW - Riesz potentials
KW - Sobolev inequalities
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U2 - 10.1016/j.na.2019.111685
DO - 10.1016/j.na.2019.111685
M3 - Article
AN - SCOPUS:85075198625
SN - 0362-546X
VL - 192
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
M1 - 111685
ER -