Some remarks on L1 embeddings in the subelliptic setting

Steven G. Krantz; Marco M. Peloso; Daniel Spector

doi:10.1016/j.na.2020.112149

Some remarks on L¹ embeddings in the subelliptic setting

Steven G. Krantz, Marco M. Peloso, Daniel Spector^*

^*Corresponding author for this work

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

3 Citations (Scopus)

Abstract

In this paper we establish an optimal Lorentz estimate for the Riesz potential in the L¹ regime in the setting of a stratified group G: Let Q≥2 be the homogeneous dimension of G and I_α denote the Riesz potential of order α on G. Then, for every α∈(0,Q), there exists a constant C=C(α,Q)>0 such that ‖I_αf‖_{L^{Q∕(Q−α),1}(G)}≤C‖XI₁f‖_L¹(G)for all f∈C_c^∞(G) such that XI₁f∈L¹(G), where X denotes the horizontal gradient.

Original language	English
Article number	112149
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	202
DOIs	https://doi.org/10.1016/j.na.2020.112149
Publication status	Published - 2021 Jan
Externally published	Yes

Keywords

L regime
Lorentz spaces
Sobolev embeddings
Stratified group
Subelliptic estimates

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.na.2020.112149

Cite this

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title = "Some remarks on L1 embeddings in the subelliptic setting",

abstract = "In this paper we establish an optimal Lorentz estimate for the Riesz potential in the L1 regime in the setting of a stratified group G: Let Q≥2 be the homogeneous dimension of G and Iα denote the Riesz potential of order α on G. Then, for every α∈(0,Q), there exists a constant C=C(α,Q)>0 such that ‖Iαf‖LQ∕(Q−α),1(G)≤C‖XI1f‖L1(G)for all f∈Cc∞(G) such that XI1f∈L1(G), where X denotes the horizontal gradient.",

keywords = "L regime, Lorentz spaces, Sobolev embeddings, Stratified group, Subelliptic estimates",

author = "Krantz, {Steven G.} and Peloso, {Marco M.} and Daniel Spector",

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

year = "2021",

month = jan,

doi = "10.1016/j.na.2020.112149",

language = "English",

volume = "202",

journal = "Nonlinear Analysis, Theory, Methods and Applications",

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T1 - Some remarks on L1 embeddings in the subelliptic setting

AU - Krantz, Steven G.

AU - Peloso, Marco M.

AU - Spector, Daniel

PY - 2021/1

Y1 - 2021/1

N2 - In this paper we establish an optimal Lorentz estimate for the Riesz potential in the L1 regime in the setting of a stratified group G: Let Q≥2 be the homogeneous dimension of G and Iα denote the Riesz potential of order α on G. Then, for every α∈(0,Q), there exists a constant C=C(α,Q)>0 such that ‖Iαf‖LQ∕(Q−α),1(G)≤C‖XI1f‖L1(G)for all f∈Cc∞(G) such that XI1f∈L1(G), where X denotes the horizontal gradient.

AB - In this paper we establish an optimal Lorentz estimate for the Riesz potential in the L1 regime in the setting of a stratified group G: Let Q≥2 be the homogeneous dimension of G and Iα denote the Riesz potential of order α on G. Then, for every α∈(0,Q), there exists a constant C=C(α,Q)>0 such that ‖Iαf‖LQ∕(Q−α),1(G)≤C‖XI1f‖L1(G)for all f∈Cc∞(G) such that XI1f∈L1(G), where X denotes the horizontal gradient.

KW - L regime

KW - Lorentz spaces

KW - Sobolev embeddings

KW - Stratified group

KW - Subelliptic estimates

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