An optimal Sobolev embedding for L1

Daniel Spector

doi:10.1016/j.jfa.2020.108559

An optimal Sobolev embedding for L¹

Daniel Spector^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

30 Citations (Scopus)

Abstract

In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant C=C(α,d)>0 such that ‖I_αF‖_{L^d/(d−α),1(R^d;R^d)}≤C‖F‖_L¹(R^d;R^d) for all fields F∈L¹(R^d;R^d) such that curlF=0 in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime p=1 of the well-established results for p>1.

Original language	English
Article number	108559
Journal	Journal of Functional Analysis
Volume	279
Issue number	3
DOIs	https://doi.org/10.1016/j.jfa.2020.108559
Publication status	Published - 2020 Aug 15

Keywords

L-type estimates
Riesz potentials
Sobolev embeddings

ASJC Scopus subject areas

Analysis

Access to Document

10.1016/j.jfa.2020.108559

Cite this

@article{19ec1dbd2b7c45f580d6f986f237a18f,

title = "An optimal Sobolev embedding for L1 ",

abstract = "In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant C=C(α,d)>0 such that ‖IαF‖Ld/(d−α),1(Rd;Rd)≤C‖F‖L1(Rd;Rd) for all fields F∈L1(Rd;Rd) such that curlF=0 in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime p=1 of the well-established results for p>1.",

keywords = "L-type estimates, Riesz potentials, Sobolev embeddings",

author = "Daniel Spector",

note = "Publisher Copyright: {\textcopyright} 2020 The Author",

year = "2020",

month = aug,

day = "15",

doi = "10.1016/j.jfa.2020.108559",

language = "English",

volume = "279",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "3",

}

TY - JOUR

T1 - An optimal Sobolev embedding for L1

AU - Spector, Daniel

PY - 2020/8/15

Y1 - 2020/8/15

N2 - In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant C=C(α,d)>0 such that ‖IαF‖Ld/(d−α),1(Rd;Rd)≤C‖F‖L1(Rd;Rd) for all fields F∈L1(Rd;Rd) such that curlF=0 in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime p=1 of the well-established results for p>1.

AB - In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant C=C(α,d)>0 such that ‖IαF‖Ld/(d−α),1(Rd;Rd)≤C‖F‖L1(Rd;Rd) for all fields F∈L1(Rd;Rd) such that curlF=0 in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime p=1 of the well-established results for p>1.

KW - L-type estimates

KW - Riesz potentials

KW - Sobolev embeddings

UR - https://www.scopus.com/pages/publications/85083314321

UR - https://www.scopus.com/pages/publications/85083314321#tab=citedBy

U2 - 10.1016/j.jfa.2020.108559

DO - 10.1016/j.jfa.2020.108559

M3 - Article

AN - SCOPUS:85083314321

SN - 0022-1236

VL - 279

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 3

M1 - 108559

ER -