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BEGIN:VEVENT
SUMMARY:Eric Sawyer (McMaster University)
DTSTART;VALUE=DATE-TIME:20221219T010000Z
DTEND;VALUE=DATE-TIME:20221219T020000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/1
DESCRIPTION:Title: Stability of two weight norm inequalities and $T1$ and bump t
heorems for Sobolev and $L^{p}$ spaces with doubling measures and Calderon
-Zygmund operators.\nby Eric Sawyer (McMaster University) as part of N
CTS Conference on Fractional Integrals and Related Phenomena in Analysis\n
\nLecture held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nI
n joint work with M. Alexis\, J.-L. Luna Garcia\, and I.\nUriarte-Tuero\,
we consider stability of two weight norm inequalities under\nbiLipschitz p
ushforward of the measures. We investigate the dichotomy that\nstability i
s generally proved using bump characterizations\, while\ninstability is ge
nerally proved using testing characterizations\, and we\ndiscuss open ques
tions that require an interplay of bump conditions and\ntesting conditions
.\n\nIn joint work with Brett Wick\, we characterize two weight norm inequ
alities\nfor Calder\\`{o}n-Zygmund operators from one weighted space to an
other in\nterms of testing conditions\, when the measures are doubling. We
extend an\nearlier result of Michel Alexis\, the speaker and Ignacio Uria
rte-Tuero for $%\nL^{2}$ spaces\, to $L^{2}$-Sobolev spaces of small order
\, and to $L^{p}$\nspaces with $1< p < \\infty $. In the case $p\\neq 2$\,
we use variants of the\nquadratic Muckenhoupt conditions and weak bounded
ness properties introduced\nby Hyt\\"{o}nen and Vuorinen. In particular\,
this proves their conjecture for\nthe Hilbert transform in the case of dou
bling measures.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kabe Moen (University of Alabama)
DTSTART;VALUE=DATE-TIME:20221219T023000Z
DTEND;VALUE=DATE-TIME:20221219T033000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/2
DESCRIPTION:Title: New pointwise bounds for rough operators with applications to
Sobolev inequalities\nby Kabe Moen (University of Alabama) as part of
NCTS Conference on Fractional Integrals and Related Phenomena in Analysis
\n\nLecture held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\
nThe classical Gagliardo-Nirenberg-Sobolev (GNS) inequality states that th
e space $\\dot W^{1\,p}$ embeds in $L^{p^*}$ for $1\\leq pThe rectangular fractional integral operators\nby Hitoshi
Tanaka (National University Corporation Tsukuba University of Technology)
as part of NCTS Conference on Fractional Integrals and Related Phenomena
in Analysis\n\nLecture held in Room 515 in the Cosmology Building\, NTU.\n
\nAbstract\nWith rectangular doubling weight\, a generalized Hardy-Littlew
ood-Sobolev inequality for rectangular fractional integral operators is ve
rified.\nThe result is a nice application of $M$-linear embedding theorem
for dyadic rectangles obtained in H. Tanaka and K. Yabuta\, The $n$-linear
embedding theorem for dyadic rectangles\,\nAnn. Acad. Sci. Fenn. Math\, 4
4 (2019)\, 29-39 and H. Tanaka and K. Yabuta\, Two-weight norm inequalitie
s for product fractional integral operators\, Bull. Sci Math.\, 166 (2021)
102940\,1-18.\n\nFor a positive integer $N$\, let $0< \\alpha < N$.\nFor
the rectangular doubling weight $\\mu$ on $\\mathbb{R}^{N}$\, define the r
ectangular the fractional integral operator $R_{\\alpha}^{\\mu}$ by\n\\[\n
R_{\\alpha}^{\\mu}f(x)\n:=\n\\int_{\\mathbb{R}^{N}}\n\\mu(R(x\,y))^{\\frac
{\\alpha}{N}-1}\nf(y)\\\,{\\rm d}\\mu(y)\,\n\\quad x\\in\\mathbb{R}^{N}\,\
n\\]\n\nwhere $R(x\,y)$ stands for the minimal rectangle\, with respect to
inculusion\, which contains two deferent points $x$ and $y$ and has their
sides parallel to the cordinate axes.\nWe have the following theorem the
proof of which is our goal.\n\nTheorem:\nFor $1< p < q < \\infty$ and $\\f
rac{1}{q}=\\frac{1}{p}-\\frac{\\alpha}{N}$\, a generalized Hardy-Littlewoo
d-Sobolev inequality \n\\[\n\\|R_{\\alpha}^{\\mu}f\\|_{L^q(\\mu)}\n\\lesss
im\n\\|f\\|_{L^p(\\mu)}\n\\]\nholds for all $f\\in L^p(\\mu)$.\n\nThe case
$\\mu \\equiv 1$ and restricting rectangles to cubes\, the theorem is jus
t the Hardy-Littlewood-Sobolev inequality which is one of the most fundame
ntal norm inequality of real variable harmonic analysis.\nThe case $\\mu$
is a doubling weight and restricts rectangles to cubes\, the theorem was s
tudied in Stein's book (E.M. Stein\, Harmonic Analysis: Real-Variable Meth
ods\, Orthogonality and Oscillatory Integrals\, Princeton University Press
\, 1993).\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Perez (University of the Basque Country and BCAM)
DTSTART;VALUE=DATE-TIME:20221219T073000Z
DTEND;VALUE=DATE-TIME:20221219T083000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/4
DESCRIPTION:Title: Fractional Poincar\\'e-Sobolev inequalities: improvements of
a theorem by Bourgain\, Brezis and Mironescu\nby Carlos Perez (Univers
ity of the Basque Country and BCAM) as part of NCTS Conference on Fraction
al Integrals and Related Phenomena in Analysis\n\nLecture held in Room 515
in the Cosmology Building\, NTU.\n\nAbstract\nIn this lecture we will dis
cuss some recent results concerning fractional Poincar\\'e-Sobolev inequal
ities which improve some celebrated results by Bourgain-Brezis-Minorescu.
In particular we will provide extensions of the classical Gagliardo esti
mates related to the classical isoperimetric inequalities. Also\, these re
sults provide improvements of well-known results by Fabes-Kenig-Serapioni
central in the study of the regularity of the solutions of degenerate ell
iptic PDE. Our approach is based on methods from Harmonic Analysis. \n\nTh
is is part of two joint works with\, Ritva Hurri-Syrj\\"{a}nen\, Javier Ma
rt{\\'i}nez-Perales\, and Antti~V. V\\"{a}h\\"{a}kangas\, and a more rece
nt one with Julian and Kim Myyryl\\"ainen and Julian Weigt.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hiroki Saito (Nihon University College of Science and Technology)
DTSTART;VALUE=DATE-TIME:20221220T010000Z
DTEND;VALUE=DATE-TIME:20221220T020000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/5
DESCRIPTION:Title: Some embedding inequalities for weighted Sobolev and Besov sp
aces\nby Hiroki Saito (Nihon University College of Science and Technol
ogy) as part of NCTS Conference on Fractional Integrals and Related Phenom
ena in Analysis\n\nLecture held in Room 515 in the Cosmology Building\, NT
U.\n\nAbstract\nIn this talk\,\nI will discuss two embedding inequalities
\nfor the weighted Sobolev space and \nthe weighted homogeneous endpoint B
esov space\nby using the weighted Hausdorff capacity.\nFormerly\,\nAdams p
roved the following inequality:\nfor any $k\\in{\\mathbb N}\, 1\\leq k < n
$\,\n\\[\n\\int_{{\\mathbb R}^n}\n|f|\\\,{\\rm d} H^{n-k}\n\\le\nC\n\\|\\n
abla^{k}f\\|_{L^1}\,\n\\quad\nf\\in C_{0}^{\\infty}({\\mathbb R}^n)\,\n\\]
\nwhere $H^{d}\, 0 < d < n$\, is the Hausdorff capacity of dimension $d$.\
nXiao extended Adams' inequality to fractional derivatives\nby using the h
omogeneous endpoint Besov spaces $\\dot{B}_{11}^{s}$.\nTo establish the we
ighted theory\,\nI will determine the dual spaces of weighted Choquet spac
es\n$L^{1}(H^{d}_{w})$.\nMore precisely\,\nit is established that\nthe dua
l of $L^{1}(H^{d}_{w})$\ncan be identified with\nthe set of all Radon meas
ures $\\mu$ satisfying\n\\[\n\\sup_{ r > 0 }\n\\frac{ |\\mu| (B(x\,r)) }{
r^{d-n}\\int_{B(x\,r)} w(y) \\\, {\\rm d} y}\n < \\infty.\n\\]\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fernando Lopez-Garcia (Cal Poly Pomona)
DTSTART;VALUE=DATE-TIME:20221220T023000Z
DTEND;VALUE=DATE-TIME:20221220T033000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/6
DESCRIPTION:Title: A local-to-global method applied to Korn and other weighted i
nequalities\nby Fernando Lopez-Garcia (Cal Poly Pomona) as part of NCT
S Conference on Fractional Integrals and Related Phenomena in Analysis\n\n
Lecture held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nIn
this talk\, we will discuss a certain local-to-global technique\, which st
rongly relies on some weighted discrete Hardy-type inequalities on trees\,
and its applications to inequalities in weighted Sobolev spaces. The weig
hted Korn inequality and other inequalities on bounded euclidean domains a
re some of the applications. The weights considered here are powers of the
distance to the boundary of the domain. If time permits\, we will discuss
a sufficient condition on the exponents of the weights in terms of the As
souad dimension of the boundary.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitriy Stolyarov (St. Petersburg State University)
DTSTART;VALUE=DATE-TIME:20221220T060000Z
DTEND;VALUE=DATE-TIME:20221220T070000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/7
DESCRIPTION:Title: Hardy--Littlewood--Sobolev inequality for $p=1$\nby Dmitr
iy Stolyarov (St. Petersburg State University) as part of NCTS Conference
on Fractional Integrals and Related Phenomena in Analysis\n\nLecture held
in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nI will speak abo
ut the limiting case of the Hardy--Littlewood--Sobolev inequality for $p=1
$. While the naive extension of HLS to $p=1$ fails and the example that br
eaks the endpoint inequality is given by approximations of a delta measure
\, there are a few options how to obtain a correct inequality in the limit
case. One of them\, suggested by the work of Bourgain--Brezis\, Van Schaf
tingen\, and others\, is to exclude the delta measures by imposing a linea
r translation and dilation invariant constraint on the functions in questi
on. Another\, suggested by Maz'ja\, is based on adding certain non-lineari
ty to the inequality. I will survey new results in this direction.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bogdan Raita (Scuola Normale Superiore)
DTSTART;VALUE=DATE-TIME:20221220T073000Z
DTEND;VALUE=DATE-TIME:20221220T083000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/8
DESCRIPTION:Title: On Compensation Phenomena for Concentration Effects\nby B
ogdan Raita (Scuola Normale Superiore) as part of NCTS Conference on Fract
ional Integrals and Related Phenomena in Analysis\n\nLecture held in Room
515 in the Cosmology Building\, NTU.\n\nAbstract\nWe study compensation ph
enomena for fields satisfying both a pointwise and a linear differ-\nentia
l constraint. This effect takes the form of nonlinear elliptic estimates\,
where constraining the values of the field to lie in a cone compensates f
or the lack of ellipticity of the differential operator. We give a series
of new examples of this phenomenon for a geometric class of cones and oper
ators such as the divergence or the curl. One of our main findings is that
the maximal gain of integrability is tied to both the differential operat
or and the cone\, contradicting in particular a recent conjecture from arX
iv:2106.03077. This extends the recent theory of compensated integrability
due to D. Serre. In particular\, we find a new family of integrands that
are Div-quasiconcave under convex constraints. Joint work with A. Guerra a
nd M. Schrecker.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Jaye (Georgia Tech)
DTSTART;VALUE=DATE-TIME:20221221T010000Z
DTEND;VALUE=DATE-TIME:20221221T020000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/9
DESCRIPTION:Title: Removable sets for the fractional Laplacian\nby Benjamin
Jaye (Georgia Tech) as part of NCTS Conference on Fractional Integrals and
Related Phenomena in Analysis\n\nLecture held in Room 515 in the Cosmolog
y Building\, NTU.\n\nAbstract\nWe consider the following question: Suppos
e\, for some $\\alpha\\in (0\,1)$\, that $u$ solves the equation $(-\\Del
ta)^{\\alpha}u=0$ outside of a compact set $E$\, and that $u$ is Lipschit
z continuous. How small does $E$ have to be to redefine $u$ to solve $(-
\\Delta)^{\\alpha}u=0$ on a neighborhood of $E$? We shall answer this q
uestion by reducing it to a result about fractional Riesz transforms\, wh
ich was solved in joint work with F. Nazarov\, M.-C. Reguera\, and X. Tols
a.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ji Li (Macquarie University)
DTSTART;VALUE=DATE-TIME:20221221T023000Z
DTEND;VALUE=DATE-TIME:20221221T033000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/10
DESCRIPTION:Title: Flag Hardy space theory—a complete answer to a question by
E.M. Stein.\nby Ji Li (Macquarie University) as part of NCTS Conferen
ce on Fractional Integrals and Related Phenomena in Analysis\n\nLecture he
ld in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nIn 1999\, Was
hington University in Saint Louis hosted a conference on Harmonic Analysis
to celebrate the 70th birthday of G. Weiss. In his talk in flag singular
integral operators\, E. M. Stein asked ``What is the Hardy space theory in
the flag setting?'' In our recent paper\, we characterise completely a fl
ag Hardy space on the Heisenberg group. It is a proper subspace of the cla
ssical one-parameter Hardy space of Folland and Stein that was studied by
Christ and Geller. Our space is useful in several applications\, including
the endpoint boundedness for certain singular integrals associated with t
he Sub-Laplacian on Heisenberg groups\, and representations of flag BMO fu
nctions.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritva Hurri-Syrjanen (University of Helsinki)
DTSTART;VALUE=DATE-TIME:20221221T060000Z
DTEND;VALUE=DATE-TIME:20221221T070000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/11
DESCRIPTION:Title: On Choquet integrals and Poincar\\'e-Sobolev inequalities\nby Ritva Hurri-Syrjanen (University of Helsinki) as part of NCTS Confer
ence on Fractional Integrals and Related Phenomena in Analysis\n\nLecture
held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nWe discuss
the Poincar\\'e and Poincar\\'e-Sobolev inequalities in terms of Choquet
integrals with respect to the Hausdorff content.\nAlso\, we consider Tr
udinger's inequality in this context.\nThis talk is based on joint work wi
th Petteri Harjulehto.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Franz Gmeineder (University of Konstanz)
DTSTART;VALUE=DATE-TIME:20221221T073000Z
DTEND;VALUE=DATE-TIME:20221221T083000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/12
DESCRIPTION:Title: Traces via potentials for $\\mathrm{L}^{1}$-based function s
paces\nby Franz Gmeineder (University of Konstanz) as part of NCTS Con
ference on Fractional Integrals and Related Phenomena in Analysis\n\nLectu
re held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nIn this
talk we discuss two novel approaches to optimal Besov trace estimates for
$\\mathrm{L}^{1}$-based function spaces. Here the differentiability \nis n
ot determined by full $k$-th order gradients\, but only certain differenti
al expressions belonging to $\\mathrm{L}^{1}$. The approaches discussed in
this talk are equally new for the classical Uspenskii theorem on sharp Be
sov traces in the higher order gradient case. By the results discussed in
this talk\, we also obtain an in some sense optimal generalization of Aro
nszajn’s classical coercive inequalities to the $\\mathrm{L}^{1}$-framew
ork. \n\nJoint work with L. Diening (Bielefeld) and J. Van Schaftingen (U
Louvain) & B.\nRaita (U Pisa).\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Lacey (Georgia Tech)
DTSTART;VALUE=DATE-TIME:20221222T010000Z
DTEND;VALUE=DATE-TIME:20221222T020000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/13
DESCRIPTION:Title: Commutators\, Schatten Classes and Besov Spaces\nby Mich
ael Lacey (Georgia Tech) as part of NCTS Conference on Fractional Integral
s and Related Phenomena in Analysis\n\nLecture held in Room 515 in the Cos
mology Building\, NTU.\n\nAbstract\nWe revisit classical results of Peller
\, and Jansson Wolf\, concerning Schatten class estimates for commutators.
The latter are quantitative compactness estimates\, characterized in ter
ms of the symbol of the commutator being in Besov spaces. At the endpoint
\, there is an connection to a quantized derivative introduced by Connes.
We report on recent work that has extended the classical results in a num
ber of directions\, including generalized Riesz transforms\, weighted and
two-weighted situations. Joint work with several\, \nincluding Peng Chen\
, Zhijie Fan\, Ji Li\, Manasa N. Vempati\, and Brett Wick.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phuc Cong Nguyen (Louisiana State University)
DTSTART;VALUE=DATE-TIME:20221222T023000Z
DTEND;VALUE=DATE-TIME:20221222T033000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/14
DESCRIPTION:Title: Capacitary inequalities and the spherical and Hardy-Littlewo
od maximal functions on Choquet spaces\nby Phuc Cong Nguyen (Louisiana
State University) as part of NCTS Conference on Fractional Integrals and
Related Phenomena in Analysis\n\nLecture held in Room 515 in the Cosmology
Building\, NTU.\n\nAbstract\nWe discuss the boundedness of the spherical
and Hardy-Littlewood maximal functions on $L^q$ type spaces defined via
Choquet integrals associate to Sobolev capacities. We also present a capa
citary inequality of Maz'ya type which resolves a problem proposed by D. A
dams.\nThis talk is based on joint work with Keng Hao Ooi.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Augusto Ponce (UCLouvain)
DTSTART;VALUE=DATE-TIME:20221222T060000Z
DTEND;VALUE=DATE-TIME:20221222T070000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/15
DESCRIPTION:Title: The uncharted territory of \\(W^{\\alpha\, 1}\\) Sobolev spa
ces\nby Augusto Ponce (UCLouvain) as part of NCTS Conference on Fracti
onal Integrals and Related Phenomena in Analysis\n\nLecture held in Room 5
15 in the Cosmology Building\, NTU.\n\nAbstract\nI will address properties
of the fractional Sobolev space \\(W^{\\alpha\, 1}(\\mathbb{R}^{d})\\) fo
r any exponent \\(\\alpha > 0\\) which cannot be answered by classical rep
resentation formulas from Harmonic Analysis.\nThey can be handled instead
in terms of a strong capacitary inequality which is based itself on a geom
etric boxing inequality that connects the Hausdorff content of dimension \
\(d-\\alpha\\) and the fractional perimeter of order \\(0 < \\alpha < 1\\)
.\nThese results have been obtained in collaboration with D. Spector (Nati
onal Taiwan Normal University).\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brett Wick (Washington University in Saint Louis)
DTSTART;VALUE=DATE-TIME:20221222T073000Z
DTEND;VALUE=DATE-TIME:20221222T083000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/16
DESCRIPTION:Title: Wavelet Representation of Singular Integral Operators\nb
y Brett Wick (Washington University in Saint Louis) as part of NCTS Confer
ence on Fractional Integrals and Related Phenomena in Analysis\n\nLecture
held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nIn this tal
k\, we'll discuss a novel approach to the representation of singular integ
ral operators of Calderón-Zygmund type in terms of continuous model opera
tors. The representation is realized as a finite sum of averages of wavele
t projections of either cancellative or noncancellative type\, which are
themselves Calderón-Zygmund operators. Both properties are out of reach f
or the established dyadic-probabilistic technique. Unlike their dyadic cou
nterparts\, our representation reflects the additional kernel smoothness o
f the operator being analyzed. Our representation formulas lead natura
lly to a new family of T1 theorems on weighted Sobolev spaces whose smooth
ness index is naturally related to kernel smoothness. In the one parameter
case\, we obtain the Sobolev space analogue of the $A_2$ theorem\; that i
s\, sharp dependence of the Sobolev norm of T on the weight characteristic
is obtained in the full range of exponents. As an additional application\
, it is possible to provide a proof of the commutator theorems of Calderó
n-Zygmund operators with BMO functions.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu Jia Zhai (Clemson University)
DTSTART;VALUE=DATE-TIME:20221223T010000Z
DTEND;VALUE=DATE-TIME:20221223T020000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/17
DESCRIPTION:Title: Generic Leibniz rules and Leibniz-type estimates\nby Yu
Jia Zhai (Clemson University) as part of NCTS Conference on Fractional Int
egrals and Related Phenomena in Analysis\n\nLecture held in Room 515 in th
e Cosmology Building\, NTU.\n\nAbstract\nWe will introduce a class of mult
ilinear inequalities\, in particular Leibniz rules\, that can be perceived
as a generalization of the classical Leibniz rule or product rule. Some c
ases have been studied by Kato-Ponce\, Bourgain-Li\, Oh-Wu and Muscalu et
al. We will resolve this problem and prove the generic Leibniz rules and L
eibniz-type estimates.\nWe will also discuss their connections with the mu
ltilinear singular integral operators. If time permits\, we will provide t
he main idea of the proof\, which combines tools from multilinear harmonic
analysis and the commutator estimates originally introduced by Bourgain-L
i. This is joint with C. Benea.\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Cruz-Uribe (University of Alabama)
DTSTART;VALUE=DATE-TIME:20221223T023000Z
DTEND;VALUE=DATE-TIME:20221223T033000Z
DTSTAMP;VALUE=DATE-TIME:20230331T084811Z
UID:Fractional_Integrals/18
DESCRIPTION:Title: Matrix weighted norm inequalities for fractional integrals\nby David Cruz-Uribe (University of Alabama) as part of NCTS Conference
on Fractional Integrals and Related Phenomena in Analysis\n\nLecture held
in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nIn the scalar c
ase\, Carlos P\\'erez proved that a\nsufficient condition for the fraction
al integral $I_\\alpha$\,\n$0< \\alpha < n $\, to satisfy\n\n\\[ I_\\alpha
: L^p(v) \\rightarrow L^q(u)\, \\]\n\n$1< p \\leq q < \\infty$\, is that
the weights $u$ and $v$ satisfy \n\n\n\\[ \\sup_Q |Q|^{\\frac{\\alpha}{n}
+\\frac{1}{q}-\\frac{1}{p}}\n \\|u^{\\frac{1}{q}}\\|_{A\,Q} \\|v^{-\\frac
{1}{p}}\\|_{B\,Q} < \\infty\, \\]\n\nwhere the supremum is taken over all
cubes $Q$ in $\\mathbb{R}^n$\,\n$\\|\\cdot\\|_{A\,Q}$ and $\\|\\cdot\\|_{B
\,Q}$ are normalized local Orlicz\nnorms\, and the Young functions $A$ and
$B$ satisfy certain growth\nconditions. This was one of the first exampl
es of the so-called $A_p$\nbump conditions for two-weight norm inequalitie
s.\n\n\n\nIn this talk we discuss a generalization of this result to the s
etting\nof matrix weights. We showed that if $d\\times d$\, self-adjoint\
,\npositive semi-definite matrices $U$ and $V$ satisfy\n\n\\[ \\sup_Q |Q|^
{\\frac{\\alpha}{n}+\\frac{1}{q}-\\frac{1}{p}}\n \\big\\| \\|\n U^{\\fra
c{1}{q}}(x)V^{-\\frac{1}{p}} (y)\\|_{B_x\,Q}\\big\\|_{A_y\,Q} <\n \\infty
\, \\]\n\nthen\n\n\\[ I_\\alpha : L^p(V) \\rightarrow L^q(U). \\]\n\nWe al
so proved the analogous result for a generalization of the\nfractional max
imal operator to the matrix weighted setting. These\nresults were the fir
st two-weight inequalities proved for matrix weights.\n\n\nWe will provide
some background on matrix weights and then discuss the key ideas in\nthe
proofs of our results. If time permits we will also discuss very recent w
ork on a\nmatrix weighted\, weak-type endpoint estimate for $I_\\alpha$. \
n\n\nResults in this talk are joint work with Kabe Moen (UA)\, Josh\nIsral
owitz (SUNY Albany)\, and Brandon\nSweeting (UA).\n
LOCATION:https://researchseminars.org/talk/Fractional_Integrals/18/
END:VEVENT
END:VCALENDAR