Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces
Let m ≥ 2 , λ > 1 and define the multilinear Littlewood–Paley–Stein operators by g λ , μ ∗ ( f → ) ( x ) = ( ∬ R + n + 1 ( t t + | x - y | ) m λ | ∫ ( R n ) κ s t ( y , z → ) ∏ i = 1 κ f i ( z i ) d μ ( z 1 ) ⋯ d μ ( z κ ) | 2 d μ ( y ) d t t m + 1 ) 1 / 2 . In this paper, our main aim is to inve...
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Veröffentlicht in: | The Journal of Geometric Analysis 2021-09, Vol.31 (9), p.9295-9337 |
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container_title | The Journal of Geometric Analysis |
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creator | Cao, Mingming Xue, Qingying |
description | Let
m
≥
2
,
λ
>
1
and define the multilinear Littlewood–Paley–Stein operators by
g
λ
,
μ
∗
(
f
→
)
(
x
)
=
(
∬
R
+
n
+
1
(
t
t
+
|
x
-
y
|
)
m
λ
|
∫
(
R
n
)
κ
s
t
(
y
,
z
→
)
∏
i
=
1
κ
f
i
(
z
i
)
d
μ
(
z
1
)
⋯
d
μ
(
z
κ
)
|
2
d
μ
(
y
)
d
t
t
m
+
1
)
1
/
2
.
In this paper, our main aim is to investigate the boundedness of
g
λ
,
μ
∗
on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that
g
λ
,
μ
∗
is bounded from
L
p
1
(
μ
)
×
⋯
×
L
p
κ
(
μ
)
to
L
p
(
μ
)
under certain weak type assumptions. The multilinear non-convolution type kernels
s
t
only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón–Zygmund type kernels and the measures
μ
are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. This was done by considering first a sufficient condition for the strong type boundedness of
g
λ
,
μ
∗
based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions. |
doi_str_mv | 10.1007/s12220-020-00491-2 |
format | Article |
fullrecord | <record><control><sourceid>gale_proqu</sourceid><recordid>TN_cdi_proquest_journals_2562844835</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><galeid>A707340385</galeid><sourcerecordid>A707340385</sourcerecordid><originalsourceid>FETCH-LOGICAL-c309t-78c4d9a9be9e8315a8b0716ae72c3d36c94f391779a5f5e7a97a3359c7d98dfb3</originalsourceid><addsrcrecordid>eNp9kM1KAzEQxxdRsFZfwNOC56352Gw2x1L8pFqhCt5Cmp2tW7ZJTbJIb76Db-iTmGUFbzIMMxn-v0nyT5JzjCYYIX7pMSEEZahPlAuckYNkhBkT8UheD2OPGMoKQYrj5MT7TRQVNOej5P6ha0PTNgaUS-dNCC18WFt9f349qRb2sS4DNCZd7MCpYJ1PrUkfrcne7NauwYDtfLrcKQ3-NDmqVevh7LeOk5frq-fZbTZf3NzNpvNMUyRCxkudV0KJFQgoKWaqXCGOCwWcaFrRQou8pgJzLhSrGXAluKKUCc0rUVb1io6Ti2Hvztn3DnyQG9s5E6-UhBWkzPOSsqiaDKp1_IdsTG2DUzpGBdtGWwN1E-dTjjjNES17gAyAdtZ7B7XcuWar3F5iJHuT5WCyRH32JksSITpAPorNGtzfW_6hfgDCpYED</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2562844835</pqid></control><display><type>article</type><title>Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces</title><source>SpringerNature Journals</source><creator>Cao, Mingming ; Xue, Qingying</creator><creatorcontrib>Cao, Mingming ; Xue, Qingying</creatorcontrib><description>Let
m
≥
2
,
λ
>
1
and define the multilinear Littlewood–Paley–Stein operators by
g
λ
,
μ
∗
(
f
→
)
(
x
)
=
(
∬
R
+
n
+
1
(
t
t
+
|
x
-
y
|
)
m
λ
|
∫
(
R
n
)
κ
s
t
(
y
,
z
→
)
∏
i
=
1
κ
f
i
(
z
i
)
d
μ
(
z
1
)
⋯
d
μ
(
z
κ
)
|
2
d
μ
(
y
)
d
t
t
m
+
1
)
1
/
2
.
In this paper, our main aim is to investigate the boundedness of
g
λ
,
μ
∗
on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that
g
λ
,
μ
∗
is bounded from
L
p
1
(
μ
)
×
⋯
×
L
p
κ
(
μ
)
to
L
p
(
μ
)
under certain weak type assumptions. The multilinear non-convolution type kernels
s
t
only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón–Zygmund type kernels and the measures
μ
are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. This was done by considering first a sufficient condition for the strong type boundedness of
g
λ
,
μ
∗
based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.</description><identifier>ISSN: 1050-6926</identifier><identifier>EISSN: 1559-002X</identifier><identifier>DOI: 10.1007/s12220-020-00491-2</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Abstract Harmonic Analysis ; Convex and Discrete Geometry ; Differential Geometry ; Dynamical Systems and Ergodic Theory ; Fourier Analysis ; Geometry ; Global Analysis and Analysis on Manifolds ; Kernels ; Mathematics ; Mathematics and Statistics ; Operators</subject><ispartof>The Journal of Geometric Analysis, 2021-09, Vol.31 (9), p.9295-9337</ispartof><rights>Mathematica Josephina, Inc. 2020</rights><rights>COPYRIGHT 2021 Springer</rights><rights>Mathematica Josephina, Inc. 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c309t-78c4d9a9be9e8315a8b0716ae72c3d36c94f391779a5f5e7a97a3359c7d98dfb3</cites><orcidid>0000-0001-8327-5066</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12220-020-00491-2$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12220-020-00491-2$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Cao, Mingming</creatorcontrib><creatorcontrib>Xue, Qingying</creatorcontrib><title>Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces</title><title>The Journal of Geometric Analysis</title><addtitle>J Geom Anal</addtitle><description>Let
m
≥
2
,
λ
>
1
and define the multilinear Littlewood–Paley–Stein operators by
g
λ
,
μ
∗
(
f
→
)
(
x
)
=
(
∬
R
+
n
+
1
(
t
t
+
|
x
-
y
|
)
m
λ
|
∫
(
R
n
)
κ
s
t
(
y
,
z
→
)
∏
i
=
1
κ
f
i
(
z
i
)
d
μ
(
z
1
)
⋯
d
μ
(
z
κ
)
|
2
d
μ
(
y
)
d
t
t
m
+
1
)
1
/
2
.
In this paper, our main aim is to investigate the boundedness of
g
λ
,
μ
∗
on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that
g
λ
,
μ
∗
is bounded from
L
p
1
(
μ
)
×
⋯
×
L
p
κ
(
μ
)
to
L
p
(
μ
)
under certain weak type assumptions. The multilinear non-convolution type kernels
s
t
only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón–Zygmund type kernels and the measures
μ
are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. This was done by considering first a sufficient condition for the strong type boundedness of
g
λ
,
μ
∗
based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.</description><subject>Abstract Harmonic Analysis</subject><subject>Convex and Discrete Geometry</subject><subject>Differential Geometry</subject><subject>Dynamical Systems and Ergodic Theory</subject><subject>Fourier Analysis</subject><subject>Geometry</subject><subject>Global Analysis and Analysis on Manifolds</subject><subject>Kernels</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operators</subject><issn>1050-6926</issn><issn>1559-002X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kM1KAzEQxxdRsFZfwNOC56352Gw2x1L8pFqhCt5Cmp2tW7ZJTbJIb76Db-iTmGUFbzIMMxn-v0nyT5JzjCYYIX7pMSEEZahPlAuckYNkhBkT8UheD2OPGMoKQYrj5MT7TRQVNOej5P6ha0PTNgaUS-dNCC18WFt9f349qRb2sS4DNCZd7MCpYJ1PrUkfrcne7NauwYDtfLrcKQ3-NDmqVevh7LeOk5frq-fZbTZf3NzNpvNMUyRCxkudV0KJFQgoKWaqXCGOCwWcaFrRQou8pgJzLhSrGXAluKKUCc0rUVb1io6Ti2Hvztn3DnyQG9s5E6-UhBWkzPOSsqiaDKp1_IdsTG2DUzpGBdtGWwN1E-dTjjjNES17gAyAdtZ7B7XcuWar3F5iJHuT5WCyRH32JksSITpAPorNGtzfW_6hfgDCpYED</recordid><startdate>20210901</startdate><enddate>20210901</enddate><creator>Cao, Mingming</creator><creator>Xue, Qingying</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>IAO</scope><orcidid>https://orcid.org/0000-0001-8327-5066</orcidid></search><sort><creationdate>20210901</creationdate><title>Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces</title><author>Cao, Mingming ; Xue, Qingying</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c309t-78c4d9a9be9e8315a8b0716ae72c3d36c94f391779a5f5e7a97a3359c7d98dfb3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Abstract Harmonic Analysis</topic><topic>Convex and Discrete Geometry</topic><topic>Differential Geometry</topic><topic>Dynamical Systems and Ergodic Theory</topic><topic>Fourier Analysis</topic><topic>Geometry</topic><topic>Global Analysis and Analysis on Manifolds</topic><topic>Kernels</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operators</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cao, Mingming</creatorcontrib><creatorcontrib>Xue, Qingying</creatorcontrib><collection>CrossRef</collection><collection>Gale Academic OneFile</collection><jtitle>The Journal of Geometric Analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cao, Mingming</au><au>Xue, Qingying</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces</atitle><jtitle>The Journal of Geometric Analysis</jtitle><stitle>J Geom Anal</stitle><date>2021-09-01</date><risdate>2021</risdate><volume>31</volume><issue>9</issue><spage>9295</spage><epage>9337</epage><pages>9295-9337</pages><issn>1050-6926</issn><eissn>1559-002X</eissn><abstract>Let
m
≥
2
,
λ
>
1
and define the multilinear Littlewood–Paley–Stein operators by
g
λ
,
μ
∗
(
f
→
)
(
x
)
=
(
∬
R
+
n
+
1
(
t
t
+
|
x
-
y
|
)
m
λ
|
∫
(
R
n
)
κ
s
t
(
y
,
z
→
)
∏
i
=
1
κ
f
i
(
z
i
)
d
μ
(
z
1
)
⋯
d
μ
(
z
κ
)
|
2
d
μ
(
y
)
d
t
t
m
+
1
)
1
/
2
.
In this paper, our main aim is to investigate the boundedness of
g
λ
,
μ
∗
on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that
g
λ
,
μ
∗
is bounded from
L
p
1
(
μ
)
×
⋯
×
L
p
κ
(
μ
)
to
L
p
(
μ
)
under certain weak type assumptions. The multilinear non-convolution type kernels
s
t
only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón–Zygmund type kernels and the measures
μ
are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. This was done by considering first a sufficient condition for the strong type boundedness of
g
λ
,
μ
∗
based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s12220-020-00491-2</doi><tpages>43</tpages><orcidid>https://orcid.org/0000-0001-8327-5066</orcidid></addata></record> |
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issn | 1050-6926 1559-002X |
language | eng |
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source | SpringerNature Journals |
subjects | Abstract Harmonic Analysis Convex and Discrete Geometry Differential Geometry Dynamical Systems and Ergodic Theory Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Kernels Mathematics Mathematics and Statistics Operators |
title | Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces |
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