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BEGIN:VEVENT
SUMMARY:Eric Sawyer (McMaster University)
DTSTART:20221219T010000Z
DTEND:20221219T020000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/1/">Stability of two weight norm inequalities and $T1$ and
  bump theorems for Sobolev and $L^{p}$ spaces with doubling measures and C
 alderon-Zygmund operators.</a>\nby Eric Sawyer (McMaster University) as pa
 rt of NCTS Conference on Fractional Integrals and Related Phenomena in Ana
 lysis\n\nLecture held in Room 515 in the Cosmology Building\, NTU.\n\nAbst
 ract\nIn joint work with M. Alexis\, J.-L. Luna Garcia\, and I.\nUriarte-T
 uero\, we consider stability of two weight norm inequalities under\nbiLips
 chitz pushforward of the measures. We investigate the dichotomy that\nstab
 ility is generally proved using bump characterizations\, while\ninstabilit
 y is generally proved using testing characterizations\, and we\ndiscuss op
 en questions that require an interplay of bump conditions and\ntesting con
 ditions.\n\nIn joint work with Brett Wick\, we characterize two weight nor
 m inequalities\nfor Calder\\`{o}n-Zygmund operators from one weighted spac
 e to another in\nterms of testing conditions\, when the measures are doubl
 ing. We extend an\nearlier result of Michel Alexis\, the speaker and Ignac
 io Uriarte-Tuero for $%\nL^{2}$ spaces\, to $L^{2}$-Sobolev spaces of smal
 l order\, and to $L^{p}$\nspaces with $1< p < \\infty $. In the case $p\\n
 eq 2$\, we use variants of the\nquadratic Muckenhoupt conditions and weak 
 boundedness properties introduced\nby Hyt\\"{o}nen and Vuorinen. In partic
 ular\, this proves their conjecture for\nthe Hilbert transform in the case
  of doubling measures.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kabe Moen (University of Alabama)
DTSTART:20221219T023000Z
DTEND:20221219T033000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/2/">New pointwise bounds for rough operators with applicat
 ions to Sobolev inequalities</a>\nby Kabe Moen (University of Alabama) as 
 part of NCTS Conference on Fractional Integrals and Related Phenomena in A
 nalysis\n\nLecture held in Room 515 in the Cosmology Building\, NTU.\n\nAb
 stract\nThe classical Gagliardo-Nirenberg-Sobolev (GNS) inequality states 
 that the space $\\dot W^{1\,p}$ embeds in $L^{p^*}$ for $1\\leq p<n$ and $
 p^*=\\frac{np}{n-p}$.  The usual method follows from the pointwise bound $
 |f|\\leq c_nI_1(|\\nabla f|)$ where $I_1$ is the fractional integral opera
 tor of order 1.  In this talk we will study when this can be extended to a
 n operator\, namely\, the pointwise bound $|Tf|\\leq c_TI_1(|\\nabla f|)$.
   We will show that this pointwise bound holds for several prominent opera
 tors in harmonic analysis\, including\, exotic maximal functions and rough
  singular integral operators.  As a consequence\, we derive several new $\
 \dot W^{1\,p}\\rightarrow L^q$ mapping for these operators.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hitoshi Tanaka (National University Corporation Tsukuba University
  of Technology)
DTSTART:20221219T060000Z
DTEND:20221219T070000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/3/">The rectangular fractional integral operators</a>\nby 
 Hitoshi Tanaka (National University Corporation Tsukuba University of Tech
 nology) as part of NCTS Conference on Fractional Integrals and Related Phe
 nomena in Analysis\n\nLecture held in Room 515 in the Cosmology Building\,
  NTU.\n\nAbstract\nWith rectangular doubling weight\, a generalized Hardy-
 Littlewood-Sobolev inequality for rectangular fractional integral operator
 s is verified.\nThe result is a nice application of $M$-linear embedding t
 heorem for dyadic rectangles obtained in H. Tanaka and K. Yabuta\, The $n$
 -linear embedding theorem for dyadic rectangles\,\nAnn. Acad. Sci. Fenn. M
 ath\, 44 (2019)\, 29-39 and H. Tanaka and K. Yabuta\, Two-weight norm ineq
 ualities 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 $\\mathbb{R}^{N}$\, defin
 e the rectangular the fractional integral operator $R_{\\alpha}^{\\mu}$ by
 \n\\[\nR_{\\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 res
 pect to inculusion\, which contains two deferent points $x$ and $y$ and ha
 s their sides parallel to the cordinate axes.\nWe have the following theor
 em the proof of which is our goal.\n\nTheorem:\nFor $1< p < q < \\infty$ a
 nd $\\frac{1}{q}=\\frac{1}{p}-\\frac{\\alpha}{N}$\, a generalized Hardy-Li
 ttlewood-Sobolev inequality \n\\[\n\\|R_{\\alpha}^{\\mu}f\\|_{L^q(\\mu)}\n
 \\lesssim\n\\|f\\|_{L^p(\\mu)}\n\\]\nholds for all $f\\in L^p(\\mu)$.\n\nT
 he case $\\mu \\equiv 1$ and restricting rectangles to cubes\, the theorem
  is just the Hardy-Littlewood-Sobolev inequality which is one of the most 
 fundamental norm inequality of real variable harmonic analysis.\nThe case 
 $\\mu$ is a doubling weight and restricts rectangles to cubes\, the theore
 m was studied in Stein's book (E.M. Stein\, Harmonic Analysis: Real-Variab
 le Methods\, Orthogonality and Oscillatory Integrals\, Princeton Universit
 y Press\, 1993).\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Perez (University of the Basque Country and BCAM)
DTSTART:20221219T073000Z
DTEND:20221219T083000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/4/">Fractional Poincar\\'e-Sobolev inequalities: improveme
 nts of a theorem by Bourgain\, Brezis and Mironescu</a>\nby Carlos Perez (
 University of the Basque Country and BCAM) as part of NCTS Conference on F
 ractional Integrals and Related Phenomena in Analysis\n\nLecture held in R
 oom 515 in the Cosmology Building\, NTU.\n\nAbstract\nIn this lecture we w
 ill discuss some recent results concerning fractional Poincar\\'e-Sobolev 
 inequalities which improve some celebrated results by Bourgain-Brezis-Mino
 rescu.   In particular we will provide extensions of the classical Gagliar
 do estimates related to the classical isoperimetric inequalities. Also\, t
 hese results provide improvements of well-known results by Fabes-Kenig-Ser
 apioni central  in the study of the regularity of the solutions of degener
 ate elliptic PDE. Our approach is based on methods from Harmonic Analysis.
  \n\nThis is part of two joint works with\, Ritva Hurri-Syrj\\"{a}nen\, Ja
 vier Mart{\\'i}nez-Perales\, and  Antti~V. V\\"{a}h\\"{a}kangas\, and a mo
 re recent one with Julian and Kim Myyryl\\"ainen and Julian Weigt.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hiroki Saito (Nihon University College of Science and Technology)
DTSTART:20221220T010000Z
DTEND:20221220T020000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/5
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/5/">Some embedding inequalities for weighted Sobolev and B
 esov spaces</a>\nby Hiroki Saito (Nihon University College of Science and 
 Technology) as part of NCTS Conference on Fractional Integrals and Related
  Phenomena in Analysis\n\nLecture held in Room 515 in the Cosmology Buildi
 ng\, NTU.\n\nAbstract\nIn this talk\,\nI will discuss two embedding inequa
 lities \nfor the weighted Sobolev space and \nthe weighted homogeneous end
 point Besov space\nby using the weighted Hausdorff capacity.\nFormerly\,\n
 Adams proved the following inequality:\nfor any $k\\in{\\mathbb N}\, 1\\le
 q k < n$\,\n\\[\n\\int_{{\\mathbb R}^n}\n|f|\\\,{\\rm d} H^{n-k}\n\\le\nC\
 n\\|\\nabla^{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 dimensio
 n $d$.\nXiao extended Adams' inequality to fractional derivatives\nby usin
 g the homogeneous endpoint Besov spaces $\\dot{B}_{11}^{s}$.\nTo establish
  the weighted theory\,\nI will determine the dual spaces of weighted Choqu
 et spaces\n$L^{1}(H^{d}_{w})$.\nMore precisely\,\nit is established that\n
 the dual of $L^{1}(H^{d}_{w})$\ncan be identified with\nthe set of all Rad
 on measures $\\mu$ 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://stable.researchseminars.org/talk/Fractional_Integrals/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fernando Lopez-Garcia (Cal Poly Pomona)
DTSTART:20221220T023000Z
DTEND:20221220T033000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/6
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/6/">A local-to-global method applied to Korn and other wei
 ghted inequalities</a>\nby Fernando Lopez-Garcia (Cal Poly Pomona) as part
  of NCTS Conference on Fractional Integrals and Related Phenomena in Analy
 sis\n\nLecture held in Room 515 in the Cosmology Building\, NTU.\n\nAbstra
 ct\nIn this talk\, we will discuss a certain local-to-global technique\, w
 hich strongly relies on some weighted discrete Hardy-type inequalities on 
 trees\, and its applications to inequalities in weighted Sobolev spaces. T
 he weighted Korn inequality and other inequalities on bounded euclidean do
 mains are 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 Assouad dimension of the boundary.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitriy Stolyarov (St. Petersburg State University)
DTSTART:20221220T060000Z
DTEND:20221220T070000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/7
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/7/">Hardy--Littlewood--Sobolev inequality for $p=1$</a>\nb
 y Dmitriy Stolyarov (St. Petersburg State University) as part of NCTS Conf
 erence on Fractional Integrals and Related Phenomena in Analysis\n\nLectur
 e held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nI will sp
 eak about the limiting case of the Hardy--Littlewood--Sobolev inequality f
 or $p=1$. While the naive extension of HLS to $p=1$ fails and the example 
 that breaks the endpoint inequality is given by approximations of a delta 
 measure\, there are a few options how to obtain a correct inequality in th
 e limit case. One of them\, suggested by the work of Bourgain--Brezis\, Va
 n Schaftingen\, and others\, is to exclude the delta measures by imposing 
 a linear translation and dilation invariant constraint on the functions in
  question. Another\, suggested by Maz'ja\, is based on adding certain non-
 linearity to the inequality. I will survey new results in this direction.\
 n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bogdan Raita (Scuola Normale Superiore)
DTSTART:20221220T073000Z
DTEND:20221220T083000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/8
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/8/">On Compensation Phenomena for Concentration Effects</a
 >\nby Bogdan Raita (Scuola Normale Superiore) as part of NCTS Conference o
 n Fractional Integrals and Related Phenomena in Analysis\n\nLecture held i
 n Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nWe study compensa
 tion phenomena for fields satisfying both a pointwise and a linear differ-
 \nential constraint. This effect takes the form of nonlinear elliptic esti
 mates\, where constraining the values of the field to lie in a cone compen
 sates for the lack of ellipticity of the differential operator. We give a 
 series of new examples of this phenomenon for a geometric class of cones a
 nd operators 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
  operator and the cone\, contradicting in particular a recent conjecture f
 rom arXiv:2106.03077. This extends the recent theory of compensated integr
 ability due to D. Serre. In particular\, we find a new family of integrand
 s that are Div-quasiconcave under convex constraints. Joint work with A. G
 uerra and M. Schrecker.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Jaye (Georgia Tech)
DTSTART:20221221T010000Z
DTEND:20221221T020000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/9
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/9/">Removable sets for the fractional Laplacian</a>\nby Be
 njamin Jaye (Georgia Tech) as part of NCTS Conference on Fractional Integr
 als and Related Phenomena in Analysis\n\nLecture held in Room 515 in the C
 osmology Building\, NTU.\n\nAbstract\nWe consider the following question: 
  Suppose\, for some $\\alpha\\in (0\,1)$\,  that $u$ solves the equation $
 (-\\Delta)^{\\alpha}u=0$ outside of a compact set $E$\, and that $u$ is  L
 ipschitz continuous.   How small does $E$ have to be to redefine $u$ to so
 lve $(-\\Delta)^{\\alpha}u=0$ on a neighborhood of $E$?   We shall answer 
  this question by reducing  it to a result about fractional Riesz transfor
 ms\, which was solved in joint work with F. Nazarov\, M.-C. Reguera\, and 
 X. Tolsa.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ji Li (Macquarie University)
DTSTART:20221221T023000Z
DTEND:20221221T033000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/10
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/10/">Flag Hardy space theory—a complete answer to a ques
 tion by E.M. Stein.</a>\nby Ji Li (Macquarie University) as part of NCTS C
 onference on Fractional Integrals and Related Phenomena in Analysis\n\nLec
 ture held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nIn 199
 9\, Washington University in Saint Louis hosted a conference on Harmonic A
 nalysis to celebrate the 70th birthday of G. Weiss. In his talk in flag si
 ngular integral operators\, E. M. Stein asked ``What is the Hardy space th
 eory in the flag setting?'' In our recent paper\, we characterise complete
 ly a flag Hardy space on the Heisenberg group. It is a proper subspace of 
 the classical one-parameter Hardy space of Folland and Stein that was stud
 ied by Christ and Geller. Our space is useful in several applications\, in
 cluding the endpoint boundedness for certain singular integrals associated
  with the Sub-Laplacian on Heisenberg groups\, and representations of flag
  BMO functions.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritva Hurri-Syrjanen (University of Helsinki)
DTSTART:20221221T060000Z
DTEND:20221221T070000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/11
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/11/">On Choquet integrals and Poincar\\'e-Sobolev inequali
 ties</a>\nby Ritva Hurri-Syrjanen (University of Helsinki) as part of NCTS
  Conference on Fractional Integrals and Related Phenomena in Analysis\n\nL
 ecture held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nWe d
 iscuss the Poincar\\'e and Poincar\\'e-Sobolev inequalities  in terms of C
 hoquet  integrals with respect to  the Hausdorff content.\nAlso\, we consi
 der  Trudinger's inequality in this context.\nThis talk is based on joint 
 work with Petteri Harjulehto.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Franz Gmeineder (University of Konstanz)
DTSTART:20221221T073000Z
DTEND:20221221T083000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/12
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/12/">Traces via potentials for $\\mathrm{L}^{1}$-based fun
 ction spaces</a>\nby Franz Gmeineder (University of Konstanz) 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 this talk we discuss two novel approaches to optimal Besov trace estimat
 es for $\\mathrm{L}^{1}$-based function spaces. Here the differentiability
  \nis not determined by full $k$-th order gradients\, but only certain dif
 ferential expressions belonging to $\\mathrm{L}^{1}$. The approaches discu
 ssed in this talk are equally new for the classical Uspenskii theorem on s
 harp Besov traces in the higher order gradient case. By the results discus
 sed in this talk\, we also obtain an in some sense optimal  generalization
  of Aronszajn’s classical coercive inequalities to the $\\mathrm{L}^{1}$
 -framework. \n\nJoint work with L. Diening (Bielefeld) and  J. Van Schafti
 ngen (U Louvain) & B.\nRaita (U Pisa).\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Lacey (Georgia Tech)
DTSTART:20221222T010000Z
DTEND:20221222T020000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/13
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/13/">Commutators\, Schatten Classes and Besov Spaces</a>\n
 by Michael Lacey (Georgia Tech) as part of NCTS Conference on Fractional I
 ntegrals and Related Phenomena in Analysis\n\nLecture held in Room 515 in 
 the Cosmology Building\, NTU.\n\nAbstract\nWe revisit classical results of
  Peller\, and Jansson Wolf\, concerning Schatten class estimates for commu
 tators.  The latter are quantitative compactness estimates\, characterized
  in terms of the symbol of the commutator being in Besov spaces.  At the e
 ndpoint\, there is an connection to a quantized derivative introduced by C
 onnes.  We report on recent work that has extended the classical results i
 n a number of directions\, including generalized Riesz transforms\, weight
 ed and two-weighted situations.  Joint work with several\, \nincluding Pen
 g Chen\, Zhijie Fan\, Ji Li\, Manasa N. Vempati\, and Brett Wick.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phuc Cong Nguyen (Louisiana State University)
DTSTART:20221222T023000Z
DTEND:20221222T033000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/14
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/14/">Capacitary inequalities and the spherical and Hardy-L
 ittlewood maximal functions on Choquet spaces</a>\nby Phuc Cong Nguyen (Lo
 uisiana State University) as part of NCTS Conference on Fractional Integra
 ls and Related Phenomena in Analysis\n\nLecture held in Room 515 in the Co
 smology Building\, NTU.\n\nAbstract\nWe discuss the boundedness of the sph
 erical and Hardy-Littlewood maximal functions on  $L^q$  type spaces defin
 ed via Choquet integrals associate to Sobolev capacities.  We also present
  a capacitary inequality of Maz'ya type which resolves a problem proposed 
 by D. Adams.\nThis talk is based on joint work with  Keng Hao Ooi.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Augusto Ponce (UCLouvain)
DTSTART:20221222T060000Z
DTEND:20221222T070000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/15
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/15/">The uncharted territory of \\(W^{\\alpha\, 1}\\) Sobo
 lev spaces</a>\nby Augusto Ponce (UCLouvain) 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 address pro
 perties of the fractional Sobolev space \\(W^{\\alpha\, 1}(\\mathbb{R}^{d}
 )\\) for any exponent \\(\\alpha > 0\\) which cannot be answered by classi
 cal representation formulas from Harmonic Analysis.\nThey can be handled i
 nstead in terms of a strong capacitary inequality which is based itself on
  a geometric boxing inequality that connects the Hausdorff content of dime
 nsion \\(d-\\alpha\\) and the fractional perimeter of order \\(0 < \\alpha
  < 1\\).\nThese results have been obtained in collaboration with D. Specto
 r (National Taiwan Normal University).\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brett Wick (Washington University in Saint Louis)
DTSTART:20221222T073000Z
DTEND:20221222T083000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/16
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/16/">Wavelet Representation of Singular Integral Operators
 </a>\nby Brett Wick (Washington University in Saint Louis) as part of NCTS
  Conference on Fractional Integrals and Related Phenomena in Analysis\n\nL
 ecture held in Room 515 in the Cosmology Building\, NTU.\n\nAbstract\nIn t
 his talk\, we'll discuss a novel approach to the representation of singula
 r integral operators of Calderón-Zygmund type in terms of continuous mode
 l operators. The representation is realized as a finite sum of averages of
  wavelet projections  of either cancellative or noncancellative type\, whi
 ch are themselves Calderón-Zygmund operators. Both properties are out of 
 reach for the established dyadic-probabilistic technique. Unlike their dya
 dic counterparts\, our representation reflects the additional kernel smoot
 hness of the operator being analyzed.   Our representation formulas lead
  naturally to a new family of T1 theorems on weighted Sobolev spaces whose
  smoothness index is naturally related to kernel smoothness. In the one pa
 rameter case\, we obtain the Sobolev space analogue of the $A_2$ theorem\;
  that is\, sharp dependence of the Sobolev norm of T on the weight charact
 eristic is obtained in the full range of exponents. As an additional appli
 cation\, it is possible to provide a proof of the commutator theorems of C
 alderón-Zygmund operators with BMO functions.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu Jia Zhai (Clemson University)
DTSTART:20221223T010000Z
DTEND:20221223T020000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/17
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/17/">Generic Leibniz rules and Leibniz-type estimates</a>\
 nby Yu Jia Zhai (Clemson University) as part of NCTS Conference on Fractio
 nal Integrals and Related Phenomena in Analysis\n\nLecture held in Room 51
 5 in the Cosmology Building\, NTU.\n\nAbstract\nWe will introduce a class 
 of multilinear inequalities\, in particular Leibniz rules\, that can be pe
 rceived as a generalization of the classical Leibniz rule or product rule.
  Some cases have been studied by Kato-Ponce\, Bourgain-Li\, Oh-Wu and Musc
 alu et al. We will resolve this problem and prove the generic Leibniz rule
 s and Leibniz-type estimates.\nWe will also discuss their connections with
  the multilinear singular integral operators. If time permits\, we will pr
 ovide the main idea of the proof\, which combines tools from multilinear h
 armonic analysis and the commutator estimates originally introduced by Bou
 rgain-Li. This is joint with C. Benea.\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Cruz-Uribe (University of Alabama)
DTSTART:20221223T023000Z
DTEND:20221223T033000Z
DTSTAMP:20260404T094149Z
UID:Fractional_Integrals/18
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Fract
 ional_Integrals/18/">Matrix weighted norm inequalities for fractional inte
 grals</a>\nby David Cruz-Uribe (University of Alabama) 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 the s
 calar case\, Carlos P\\'erez proved that a\nsufficient condition for the f
 ractional 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{\\al
 pha}{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 ov
 er all cubes $Q$ in $\\mathbb{R}^n$\,\n$\\|\\cdot\\|_{A\,Q}$ and $\\|\\cdo
 t\\|_{B\,Q}$ are normalized local Orlicz\nnorms\, and the Young functions 
 $A$ and $B$ satisfy certain growth\nconditions.  This was one of the first
  examples of the so-called $A_p$\nbump conditions for two-weight norm ineq
 ualities.\n\n\n\nIn this talk we discuss a generalization of this result t
 o the setting\nof matrix weights.  We showed that if $d\\times d$\, self-a
 djoint\,\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
 ^{\\frac{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 also proved the analogous result for a generalization of the\nfractio
 nal maximal operator to the matrix weighted setting.  These\nresults were 
 the first two-weight inequalities proved for matrix weights.\n\n\nWe will 
 provide some background on matrix weights and then discuss the key ideas i
 n\nthe proofs of our results.  If time permits we will also discuss very r
 ecent work on a\nmatrix weighted\, weak-type endpoint estimate for $I_\\al
 pha$. \n\n\nResults in this talk are joint work with Kabe Moen (UA)\, Josh
 \nIsralowitz (SUNY Albany)\, and Brandon\nSweeting (UA).\n
LOCATION:https://stable.researchseminars.org/talk/Fractional_Integrals/18/
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