BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Eric Sawyer (McMaster University)
DTSTART:20240603T020000Z
DTEND:20240603T025000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/1/">A probabilistic analogue of the Fourier extension con
 jecture and an implication for the Kakeya conjecture</a>\nby Eric Sawyer (
 McMaster University) as part of NCTS/NTNU Conference on Fractional Integra
 ls and Related Phenomena in Analysis\n\nLecture held in Room 515 in NCTS i
 n NTU.\n\nAbstract\nWe prove that the Fourier extension conjecture holds w
 hen averaged over all smooth Alpert\nmultipliers of +/-. We also show that
  if this average extension conjecture holds for all conjugations of the sm
 ooth Alpert multipliers by a unimodular function\, then the Kakeya conject
 ure holds. To prove the probabilistic analogue of the extension conjecture
 \, we use frames for Lp consisting of smooth compactly supported Alpert wa
 velets having a large number of vanishing moments\, along with standard es
 timates on oscillatory integrals and probabilistic interpolation of L2 and
  L4 estimates\, as part of a two weight testing strategy using pigeonholin
 g via the uncertainty principle to define various subforms. It is crucial 
 to use probability in our method to obtain L4 estimates with the correct d
 ecay when dealing with resonant subforms.  See arXiv:2311.03145v7.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanghyuk Lee (Seoul National University)
DTSTART:20240603T031000Z
DTEND:20240603T040000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/2/">Bounds on the Hermite spectral projection</a>\nby San
 ghyuk Lee (Seoul National University) as part of NCTS/NTNU Conference on F
 ractional Integrals and Related Phenomena in Analysis\n\nLecture held in R
 oom 515 in NCTS in NTU.\n\nAbstract\nIn this talk\, we are concerned with 
 $L^q$--$L^q$ bounds on the Hermite spectral projection operator $\\Pi_\\la
 mbda$ in $\\mathbb R^d$\, which is defined by the Hermite operator $\\math
 cal H=-\\Delta+ |x|^2.$ \nThe operator  $\\Pi_\\lambda$ is the projection 
 onto the eigenspace spanned by the eigenfunctions with eigenvalue $\\lambd
 a$ of $\\mathcal H$. The bounds on  $\\Pi_\\lambda$ not only give $L^p$ es
 timates for the eigenfunctions but also have applications to various relat
 ed problems such as the Bochner--Riesz problem and the unique continuation
  problem for the heat operator.  However\, the current understanding of  t
 he $L^q$--$L^q$ bounds on $\\Pi_\\lambda$  are far from being complete.  I
 n this talk\, we mainly focus on $L^2$--$L^{q}$ of  $\\Pi_\\lambda$. For $
 d\\ge 2$\,  the optimal  $L^2$--$L^{q}$ bound on $\\Pi_\\lambda$ has been 
 known except for the endpoint case $q_\\circ=2(d+3)/(d+1)$. However\, the 
 endpoint $L^2$--$L^{q_\\circ}$ bound  has been left unsettled for a long t
 ime. We prove this missing endpoint case for every $d\\ge 3$. Our result i
 s based on a new phenomenon: improvement of bounds due to asymmetric local
 ization near the sphere $\\{x: |x|^2=\\lambda\\}$. Additionally\, we discu
 ss other related recent results.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petteri Harjulehto (University of Helsinki)
DTSTART:20240603T053000Z
DTEND:20240603T062000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/3/">On Hausdorff content maximal operator and Riesz poten
 tial for non-measurable functions</a>\nby Petteri Harjulehto (University o
 f Helsinki) as part of NCTS/NTNU Conference on Fractional Integrals and Re
 lated Phenomena in Analysis\n\n\nAbstract\nProperties of the Hausdorff con
 tent and the Choquet integral with respect to the Hausdorff content are di
 scussed. Then boundedness results for the Hausdorff content fractional max
 imal operator are proved\, without assuming any measurability for the func
 tions. The Hausdorff content Riesz potential can be point-wise estimated b
 y  the Hausdorff content maximal operator\, and thus boundedness results a
 re obtain also for the Hausdorff content Riesz potential.  The primary aim
  of this presentation is to show that harmonic analysis results can be ext
 ended to Lebesgue non-measurable functions using the Choquet integral. Thi
 s talk is based on my joint work  with  Ritva Hurri-Syrj\\"anen.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phuc Nguyen (Lousiana State University)
DTSTART:20240603T063000Z
DTEND:20240603T072000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/4/">Capacitary strong type inequalities and  related func
 tion spaces</a>\nby Phuc Nguyen (Lousiana State University) as part of NCT
 S/NTNU Conference on Fractional Integrals and Related Phenomena in Analysi
 s\n\n\nAbstract\nWe answer question posed by D. R. Adams on a capacitary s
 trong type inequality that generalizes the classical capacitary strong typ
 e inequality of V. G. Maz'ya. As a result\, we characterize related functi
 on spaces as   K\\"othe duals to a class of  Sobolev multiplier type space
 s. The boundedness of the Hardy-Littlewood maximal function and the spheri
 cal maximal function on related Choquet spaces are also discussed. This ta
 lk is based on joint work with Keng H. Ooi.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:You-Wei Chen (National Taiwan University)
DTSTART:20240603T075000Z
DTEND:20240603T084000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/5/">On functions of bounded $\\beta$-dimensional mean osc
 illation and its applications</a>\nby You-Wei Chen (National Taiwan Univer
 sity) as part of NCTS/NTNU Conference on Fractional Integrals and Related 
 Phenomena in Analysis\n\n\nAbstract\nIn this talk\, we introduce the conce
 pt of $\\beta$-dimensional mean oscillation of functions $u: Q_0 \\subset 
 \\mathbb{R}^d \\to \\mathbb{R}$\, characterized by the norm\n\\[\n\\|u\\|_
 {BMO^{\\beta}(Q_0)} := \\sup_{Q \\subset Q_0} \\inf_{c \\in \\mathbb{R}} \
 \frac{1}{l(Q)^\\beta} \\int_{Q} |u - c| \\\; d\\mathcal{H}^{\\beta}_\\inft
 y < \\infty\,\n\\]\nwhere $l(Q)$ denotes the side length of the cube $Q$ a
 nd $\\mathcal{H}^{\\beta}_\\infty$ represents the Hausdorff content. Our m
 ain result asserts that for every $\\beta \\in (0\, d]$\, there exist cons
 tants \\(c\, C > 0\\) such that\n\\[\n\\mathcal{H}^{\\beta}_\\infty \\left
 (\\{x \\in Q : |u(x) - c_Q| > t\\}\\right) \\leq C l(Q)^\\beta \\exp\\left
 (-\\frac{ct}{\\|u\\|_{BMO^\\beta(Q_0)}}\\right)\n\\]\nholds for every \\(t
  > 0\\)\, \\(u \\in BMO^\\beta(Q_0)\\)\, \\(Q \\subset Q_0\\)\, and suitab
 le \\(c_Q \\in \\mathbb{R}\\). Additionally\, we demonstrate that for non-
 negative measurable functions \\(f\\)\,\n\\[\nI_\\alpha f \\in BMO(\\mathb
 b{R}^n) \\text{ if and only if } I_\\alpha f \\in BMO^\\beta(\\mathbb{R}^n
 ) \\text{ for } \\beta \\in (n - \\alpha\, n]\,\n\\]\nwhere \\(I_\\alpha\\
 ) denotes the Riesz potential of order \\(\\alpha\\).\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Perez (University of the Basque Country UPV/EHU and BCAM)
DTSTART:20240604T020000Z
DTEND:20240604T025000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/6/">A New Look at Subrepresentation Formulas:  The Influe
 nce of Harmonic and Fractional Analysis</a>\nby Carlos Perez (University o
 f the Basque Country UPV/EHU and BCAM) as part of NCTS/NTNU Conference on 
 Fractional Integrals and Related Phenomena in Analysis\n\n\nAbstract\nIn t
 his talk\, I will present recent work with Cong Hoang and Kabe Moen\, wher
 e we proved some extensions of the classical Sobolev-type subrepresentatio
 n formula\n\\[\n|f(x)|\\le c_n\\\,I_1(|\\nabla f|)(x) \n\\]\nin several wa
 ys\, where $I_1$ denotes the usual classical fractional integral operator.
  First\, I will show how to replace $I_1$ with a family of $A_1$-potential
  type operators. Second\, I will show that we can further improve the righ
 t-hand side by using fractional derivatives instead of the $|\\nabla|$ ope
 rator inserting the Bourgain-Brezis-Mironescu factor. Finally\, I will dis
 cuss how our results apply to rough singular integral operators\, as we pr
 oved in a previous work. \n\nWe will emphasize the connection with fundame
 ntal concepts such as isoperimetric inequalities and extrapolation theory.
 \n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hitoshi Tanaka (National University Corporation Tsukuba University
  of Technology)
DTSTART:20240604T031000Z
DTEND:20240604T040000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/7/">Tiling Morrey spaces -- as a dyadic toy model</a>\nby
  Hitoshi Tanaka (National University Corporation Tsukuba University of Tec
 hnology) as part of NCTS/NTNU Conference on Fractional Integrals and Relat
 ed Phenomena in Analysis\n\n\nAbstract\nA.K. Lerner introduced a class of 
 weighted Morrey spaces in [1]. The purpose of this note is to understand i
 t better by introducing tiling Morrey spaces\, as a dyadic toy model. We c
 onsider a class of function spaces that are close to Morrey spaces. We pas
 s it to the natural weighted counterpart and obtain the characterization o
 f weights for which the dyadic Hardy-Littlewood maximal operator is bounde
 d. Actually\, we introduce and investigate tiling Morrey spaces and their 
 weighted counterparts and a number of examples are given. Seemingly\, the 
 class Lerner [1] introduced do not fall within the scope of tiling Morrey 
 spaces. However\, our geometric observation shows that the class by Lerner
  is a special case of tiling Morrey spaces. \n\n0.1. The $A_{\\mathcal{M}_
 {\\lambda\,\\mathcal{T}}^p(w)}$ weight class:\n\nWe employ the terminology
  from [1]. We adapt it to our setting. Denote by $\\mathcal{Q}$ the family
  of all cubes in $\\R^n$ with sides parallel to the axes. Given a cube $Q\
 \in\\mathcal{Q}$\, denote by $c_{Q}$ and $\\ell_{Q}$ its center and its si
 de length of $Q$\, respectively. We denote by $\\mathcal{D}$ the family of
  all dyadic cubes $Q=2^{-k}(m+[0\,1)^n)$\, $k\\in\\mathbb{Z}\,\\\,m\\in\\m
 athbb{Z}^n$. For a cube $Q\\in\\mathcal{D}$\, denote by $Q^{(1)}$ its dyad
 ic parent\, the minimal dyadic cube that strictly contains $Q$. \n\nBy def
 inition\, a $\\mathit{tiling}$ is a collection of cubes $\\mathcal{T}\\sub
 set\\mathcal{Q}$ that satisfies\n\\[\n\\chi_{\\mathbb{R}^n}=\\sum_{Q\\in\\
 mathcal{T}}\\chi_{Q}\n\\]\nalmost everywhere\, where $\\chi_{E}$ denotes t
 he characteristic function of the set $E$. In this note\, we always consid
 er a $\\mathit{dyadic~tiling}$ $\\mathcal{T}\\subset\\mathcal{D}$. \n\nGiv
 en $Q\\in\\mathcal{D}$ and $\\mathcal{G}\\subset\\mathcal{D}$\, we write $
 \\mathcal{G}|_{Q}\n:=\\{R\\in\\mathcal{G}:\\\,R\\subseteq Q\\}$: the restr
 iction to $Q$ of $\\mathcal{G}$. We also write \n$\\mathcal{D}|_{\\mathcal
 {T}}:=\\bigcup_{Q\\in\\mathcal{T}}\\mathcal{D}|_{Q}$. Notice that \n$Q\\ca
 p R\\in\\{Q\,R\,\\phi\\}$ for all $Q\,R\\in\\mathcal{D}$ (this is fundamen
 tal dyadic structure).\nFrom this\, one sees that \n\\[\n\\chi_{Q}=\\sum_{
 R\\in\\mathcal{T}|_{Q}}\\chi_{R}\n\\]\nfor all $Q\\in(\\mathcal{D}\\setmin
 us\\mathcal{D}|_{\\mathcal{T}})$. \n\nWe now follow the argument in [1]. W
 e write $\\langle g \\rangle_{Q}:=\\frac{1}{|Q|}\\int_{Q}g$\, the average 
 of a locally integrable $g$ over $Q$. Recall that the (dyadic) Hardy-Littl
 ewood maximal operator $M$ with respect to $\\mathcal{D}$ is defined by\n\
 \[\nMf(x):=\\sup_{Q\\in\\mathcal{D}}\\langle |f| \\rangle_{Q}\\chi_{Q}(x)\
 ,\\quad x\\in\\mathbb{R}^n.\n\\]\nThe operator $M$ plays the role of the t
 oy model of the original Hardy-Littlewood maximal operator\nbecause we can
  easily replace $\\mathcal{D}$ by other dyadic grids an control the origin
 al Hardy-Littlewood maximal operator by collecting maximal operators gener
 ated by at most $3^n$ different grids. \n\nLet $X$ be a Banach function sp
 ace over $\\mathbb{R}^n$. We say that the $A_{X}$ condition holds\nif ther
 e exists $C>0$ such that for every locally integrable $f$ and for all $Q\\
 in\\mathcal{D}$\,\n\\[\n\\langle |f| \\rangle_{Q}\\|\\chi_{Q}\\|_{X}\\leq 
 C\\|f\\chi_{Q}\\|_{X}.\n\\]\nRecall that the associate space $X'$ is the s
 et of all measurable functions $f$ for which $f\\cdot g\\in L^1$ for all $
 g\\in X$ and that $X'$ is equipped with norm\n\\[\n\\|f\\|_{X'}=\\sup\\{\\
 |f\\cdot g\\|_{L^1}:\\\,g\\in X\, \\|g\\|_{X}=1\\}.\n\\]\nUsing the langua
 ge of the associate space $X'$\, the $A_{X}$ condition can be written in t
 he following equivalent form: There exists $C>0$ such that for all $Q\\in\
 \mathcal{D}$\,\n\\[\n\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\qua
 d\\quad\\quad\\quad\\quad\\quad\\quad\\frac{\\|\\chi_{Q}\\|_{X}\\|\\chi_{Q
 }\\|_{X'}}{|Q|}\\leq C.\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\q
 uad\\quad\\quad\\quad\\quad\\quad\\quad(0.1)\n\\]\nThe $A_{X}$ condition i
 s a trivial necessary condition for the boundedness of the maximal operato
 r $M$ on $X$. In fact\, this follows from the elementary pointwise estimat
 e $\\langle |f| \\rangle_{Q}\\chi_{Q}\\leq M(f\\chi_{Q})$. Therefore\, giv
 en a concrete space $X$\, the question of interest is whether the $A_{X}$ 
 condition is sufficient for the boundedness of $M$ on $X$. We have a simpl
 e example showing that the $A_{X}$ condition is not sufficient for the bou
 ndedness of $M$ on $X$. \n\nLet $0<\\lambda\\leq 1$\, $p>1$ and $w$ be a l
 ocally integrable non-negative function. We consider the weighted tiling M
 orrey space $\\mathcal{M}_{\\lambda\,\\mathcal{T}}^p(w)$ of Samko-type [2]
  with norm\n\\[\n\\|f\\|_{\\mathcal{M}_{\\lambda\,\\mathcal{T}}^p(w)}:=\\s
 up_{Q\\in\\mathcal{T}}\n\\left(\\frac{1}{|Q|^{\\lambda}}\\int_{Q}|f|^pw\\r
 ight)^{\\frac{1}{p}}=\n\\left\\|\n\\sum_{Q\\in\\mathcal{T}}\n\\left(\\frac
 {1}{|Q|^{\\lambda}}\\int_{Q}|f|^pw\\right)^{\\frac{1}{p}}\n\\chi_{Q}\n\\ri
 ght\\|_{L^{\\infty}(\\mathbb{R}^n)}.\n\\]\n\n0.2. Theorems\n\nWe establish
  that the $A_{\\mathcal{M}_{\\lambda\,\\mathcal{T}}^p(w)}$ condition is su
 fficient for the boundedness of the maximal operator acting on weighted ti
 ling Morrey spaces.\n\n$\\mathbf{Theorem~1.}$ Let $0<\\lambda\\leq 1$\, $p
 >1$ and $w$ be a locally integrable non-negative function. The maximal ope
 rator $M$ is bounded on $\\mathcal{M}_{\\lambda\,\\mathcal{T}}^p(w)$\, $p>
 1$\, \nif and only if the $A_{\\mathcal{M}_{\\lambda\,\\mathcal{T}}^p(w)}$
  condition holds.\n\nWe have an equivalent form of the $A_{\\mathcal{M}_{\
 \lambda\,\\mathcal{T}}^p(w)}$ condition.\n\n$\\mathbf{Theorem~2.}$ Let $0<
 \\lambda\\leq 1$\, $p>1$ and $w$ be a locally integrable non-negative func
 tion. The $A_{\\mathcal{M}_{\\lambda\,\\mathcal{T}}^p(w)}$ condition can b
 e written in the following equivalent form : There exists $C>0$ such that 
 for all $Q\\in\\mathcal{D}|_{\\cT}$\,\n\\[\n\\quad\\quad\\quad\\quad\\quad
 \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\frac{w(Q)^{\
 \frac{1}{p}}w^{1-p'}(Q)^{\\frac{1}{p'}}}{|Q|}\\leq C\,\\quad\\quad\\quad\\
 quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad(0.2
 )\n\\]\nand for all $Q\\in(\\mathcal{D}\\setminus\\mathcal{D}|_{\\mathcal{
 T}})$\,\n\\[\n\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\frac{1}{|Q
 |}\\left[\\sup_{R\\in\\mathcal{T}|_{Q}}|R|^{-\\frac{\\lambda}{p}}w(R)^{\\f
 rac{1}{p}}\\right]\\left[\\sum_{R\\in\\mathcal{T}|_{Q}}|R|^{\\frac{\\lambd
 a}{p}}w^{1-p'}(R)^{\\frac{1}{p'}}\n\\right]\\leq C.\\quad\\quad\\quad\\qua
 d\\quad\\quad\\quad\\quad(0.3)\n\\]\n\n$\\mathbf{Remark~1.}$ Let $\\mathca
 l{T}=\\{Q\\in\\mathcal{D}:\\\,\\ell_{Q}=1\\}$. Then the space $\\mathcal{M
 }_{1\,\\mathcal{T}}^p(w)$ is the (so-called) weighted local uniform $p$-in
 tegrable Lebesgue space and the $A_{\\mathcal{M}_{1\,\\mathcal{T}}^p(w)}$ 
 condition is equivalent to the following condition:\n\\[\n\\sup_{Q\\in\\ma
 thcal{D}:\\\,\\ell_{Q}\\leq 1}\\frac{w(Q)^{\\frac{1}{p}}w^{1-p'}(Q)^{\\fra
 c{1}{p'}}}{|Q|}\n+\n\\sup_{Q\\in\\mathcal{D}:\\\,\\ell_{Q}>1}\n\\frac{1}{|
 Q|}\\left[\n\\sup_{R\\in\\mathcal{T}|_{Q}}w(R)^{\\frac{1}{p}}\n\\right]\\l
 eft[\n\\sum_{R\\in\\mathcal{T}|_{Q}}w^{1-p'}(R)^{\\frac{1}{p'}}\n\\right]<
 \\infty.\n\\]\n\\[\n\\mathbf{References}\n\\]\n\n[1] A.K. Lerner\, $\\math
 it{A~note~on~the~maximal~operator~on~weighted~Morrey~spaces}$\, Analysis M
 athematica\, $\\mathbf{49}$ (2023)\, 1073--1086. arXiv:2211.07974.\n\n[2] 
 S. Nakamura\, Y. Sawano and H. Tanaka\, $\\mathit{Weighted~local~Morrey~sp
 aces}$\, Ann. Acad. Sci. Fenn. Math. $\\mathbf{45}$ (2020)\, 67--93.\n\n[3
 ] N. Samko\, $\\mathit{Weighted~Hardy~and~singular~operators~in~Morrey~spa
 ces}$\, J. Math. Anal. Appl.\, $\\mathbf{350}$ (2009)\, no.1\, 56--72.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naoya Hatano (Chuo University)
DTSTART:20240604T053000Z
DTEND:20240604T062000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/8/">Endpoint estimates for commutators with respect to th
 e fractional integral operators on Orlicz-Morrey spaces</a>\nby Naoya Hata
 no (Chuo University) as part of NCTS/NTNU Conference on Fractional Integra
 ls and Related Phenomena in Analysis\n\n\nAbstract\nIt is known that the n
 ecessary and sufficient conditions of the boundedness of commutators on Mo
 rrey spaces are given by Di Fazio\, Ragusa and Shirai.\nMoreover\, accordi
 ng to the result of Cruz-Uribe and Fiorenza in 2003\, it is given that the
  weak-type boundedness of the commutators of the fractional integral opera
 tors on the Orlicz spaces as the endpoint estimates. \n\nIn this talk\, we
  introduce the extention to the weak-type boundedness on the Orlicz-Morrey
  spaces.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Volberg (Michigan State University)
DTSTART:20240604T063000Z
DTEND:20240604T072000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/9/">Maximum principle and David-Semmes problem for fracti
 onal Riesz transforms of small codimension: the solution</a>\nby Alexander
  Volberg (Michigan State University) as part of NCTS/NTNU Conference on Fr
 actional Integrals and Related Phenomena in Analysis\n\n\nAbstract\nThe ta
 lk is based on a joint paper with Vladimir Eiderman and Fedor Nazarov. I w
 ill try to explain the solution of David-Semmes problem for Riesz transfor
 ms of small codimension.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hiroki Saito (Nihon University)
DTSTART:20240604T075000Z
DTEND:20240604T084000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/10/">Choquet integrals\, Hausdorff content and fractional
  operators</a>\nby Hiroki Saito (Nihon University) as part of NCTS/NTNU Co
 nference on Fractional Integrals and Related Phenomena in Analysis\n\n\nAb
 stract\nIt is shown that the fractional integral operator $I_{\\alpha}$\, 
 $0<\\alpha< n$\, and the fractional maximal operator $M_{\\alpha}$\, $0\\l
 eq\\alpha< n$\, are bounded on weak Choquet spaces \nwith respect to Hausd
 orff content. We also investigate these operators on Choquet-Morrey spaces
 .\nThese results are extensions of the previous works due to Adams [1]\, O
 robitg and Verdera [2]\, \nand Tang [3]. \n\nFor $0 < p < \\infty$\, the C
 hoquet spaces $L^p(H^d)$ and the weak Choquet spaces $\\mathrm{w}\\hskip-0
 .6pt{L}^p(H^d)$ consist of all functions with the following properties\,\n
 \\[\n\\|f\\|_{L^p(H^d)}:=\\left(\\int_{{\\mathbb R}^n}|f|^p\\\,{\\rm d}H^d
 \\right)^{\\frac1p}<\\infty\n\\]\nand\n\\[\n\\|f\\|_{\\mathrm{w}\\hskip-0.
 6pt{L}^p(H^d)}:=\\sup_{t>0}t\\\,H^d\\left(\\left\\{\nx\\in{\\mathbb R}^n:\
 \\,|f(x)|>t\n\\right\\}\\right)^{\\frac1p}<\\infty\,\n\\]\nrespectively. F
 or $0< q\\leq p <\\infty$\,the Choquet-Morrey spaces ${\\mathcal M}^p_q(H^
 d)$\nconsists of all functions with the following property\, \n\\[\n\\|f\\
 |_{{\\mathcal M}^p_q(H^d)}\n:=\n\\sup_{Q\\in{\\mathcal Q}}\n\\ell(Q)^{\\fr
 ac dp-\\frac dq}\n\\left(\n\\int_{Q}|f|^q\\\,{\\rm d}H^d\n\\right)^{\\frac
 1q}<\\infty.\n\\]\n\nWe establish the following theorems.\n\n$\\mathbf{The
 orem~1.}$ Let $0< d \\leq n$ and $0\\leq \\alpha < n$. Suppose that $d/n \
 \leq r < p < d/\\alpha$ and\n\\[\n\\frac{d-\\alpha r}{q}\n=\n\\frac{d-\\al
 pha p}{p}.\n\\]\nThen (i)\n$\n\\|M_{\\alpha}f\\|_{\\mathrm{w}\\hskip-0.6pt
 {L}^q(H^{d-\\alpha r})}\n\\lesssim\n\\|f\\|_{\\mathrm{w}\\hskip-0.6pt{L}^p
 (H^d)}\;\n$\n(ii)\n$\n\\|I_{\\alpha}f\\|_{\\mathrm{w}\\hskip-0.6pt{L}^q(H^
 {d-\\alpha r})}\n\\lesssim\n\\|f\\|_{\\mathrm{w}\\hskip-0.6pt{L}^p(H^d)}\n
 $\nfor $0 < d < n\,0 < \\alpha < n$ and $d/n < r$. (iii)\n$\\|I_{\\alpha}f
 \\|_{L^q(H^{d-\\alpha r})}\n\\lesssim\n\\|f\\|_{L^p(H^d)}$\nfor $0 < d < n
 \,0 < \\alpha < n$ and $d/n < r$.\n\n$\\mathbf{Theorem~2.}$ Let $0 < d \\l
 eq n$ and $0\\leq \\alpha < n$. Suppose that \n$d/n < r\\leq p < d/\\alpha
 $ and\n\\[\n\\frac{d-\\alpha r}{q}\n=\n\\frac{d-\\alpha p}{p}.\n\\]\nThen:
 (i)\n$\n\\|M_{\\alpha}f\\|_{{\\mathcal M}^q_r(H^{d-\\alpha r})}\n\\lesssim
 \n\\|f\\|_{{\\mathcal M}^p_r(H^d)}\;\n$\n(ii)\n$\n\\|I_{\\alpha}f\\|_{{\\m
 athcal M}^q_r(H^{d-\\alpha r})}\n\\lesssim\n\\|f\\|_{{\\mathcal M}^p_s(H^d
 )}\n$\nfor $0 < d < n$\, $0 < \\alpha < n$ and \n$d/n < r < s < p < d/\\al
 pha$ with\n$p-r \\leq (n/d-1)(d/\\alpha-p)$.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/10
 /
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bogdan Raita (Georgetown University)
DTSTART:20240606T020000Z
DTEND:20240606T025000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/11
DESCRIPTION:by Bogdan Raita (Georgetown University) as part of NCTS/NTNU C
 onference on Fractional Integrals and Related Phenomena in Analysis\n\nAbs
 tract: TBA\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/11
 /
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitriy Stolyarov (St. Petersburg State University)
DTSTART:20240606T031000Z
DTEND:20240606T040000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/12/">Hardy spaces of fractional order (joint work in prog
 ress with Daniel Spector)</a>\nby Dmitriy Stolyarov (St. Petersburg State 
 University) as part of NCTS/NTNU Conference on Fractional Integrals and Re
 lated Phenomena in Analysis\n\n\nAbstract\nI will introduce a new scale of
  function spaces called Hardy spaces of fractional order. These spaces bri
 dge between the space of measures of bounded total variation at one endpoi
 nt and the real Hardy class at the other. We define them in terms of atomi
 c decompositions. These spaces\, on the one hand\, contain measures of fra
 ctional dimension\, and on the other hand\, are friendly to the action of 
 the Riesz potential. The motivation comes from the field of Bourgain--Brez
 is inequalities and dimensional estimates for measures satisfying differen
 tial constraints. We believe that Hardy spaces of fractional order may sim
 plify many subtle estimates for functions and measures that involve the $L
 _1$ norm.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/12
 /
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julian Weigt (University of Warwick)
DTSTART:20240607T020000Z
DTEND:20240607T025000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/13/">Endpoint regularity of the fractional maximal functi
 on</a>\nby Julian Weigt (University of Warwick) as part of NCTS/NTNU Confe
 rence on Fractional Integrals and Related Phenomena in Analysis\n\n\nAbstr
 act\nIn 1997 Juha Kinnunen proved that for all \\(p>1\\) and \\(f\\in W^{1
 \,p}(\\mathbb R^d)\\) the Hardy-Littlewood maximal operator \\(M\\) satisf
 ies the gradient bound\n\\(\n\\|\\nabla M f\\|_{L^p(\\mathbb R^d)}\n\\leq 
 C_{d\,p}\n\\|\\nabla f\\|_{L^p(\\mathbb R^d)}\n.\n\\)\nIn 2004 Haj\\l{}asz
  and Onninen asked if it continues to hold in the endpoint \\(p=1\\)\, a c
 onjecture which remains open. \\\\\nA corresponding bound holds for the fr
 actional maximal operator \\(M_\\alpha\\) as well\, whose endpoint case is
 \n\\(\n\\|\\nabla M_\\alpha f\\|_{L^{d/(d-\\alpha)}(\\mathbb R^d)}\n\\leq 
 C\n\\|\\nabla f\\|_{L^1(\\mathbb R^d)}\n.\n\\)\nFor \\(\\alpha\\geq1\\) th
 is bound has been proven to hold by Carneiro and Madrid\, based on a point
 wise bound by Kinnunen and Saksman.\nWe present a new proof which works fo
 r all \\(\\alpha>0\\).\nWe further strengthen the result by proving the co
 rresponding operator continuity\, i.e.\\ that for any \\(f\\in W^{1\,1}(\\
 mathbb R^d) \\) and \\(\\varepsilon>0\\) there exists a \\(\\delta>0\\) su
 ch that for any \\(g\\in W^{1\,1}(\\mathbb R^d)\\) with \\(\\|f-g\\|_{W^{1
 \,1}(\\mathbb R^d)}<\\delta\\) we have \\(\\|\\nabla M_\\alpha f-\\nabla M
 _\\alpha g\\|_{L^{d/(d-\\alpha)}(\\mathbb R^d)}<\\varepsilon\\).\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/13
 /
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ji Li (Maquarie University)
DTSTART:20240607T031000Z
DTEND:20240607T040000Z
DTSTAMP:20260416T153328Z
UID:TaiwanAnalysisConference2024/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TaiwanAnalys
 isConference2024/14/">Schatten properties of Riesz transform commutator in
  the two weight setting</a>\nby Ji Li (Maquarie University) as part of NCT
 S/NTNU Conference on Fractional Integrals and Related Phenomena in Analysi
 s\n\n\nAbstract\nSchatten class estimates of the commutator of Riesz trans
 form in $\\mathbb R^n$ link to the quantised derivative of A. Connes. A ge
 neral setting for quantised calculus is a spectral triple $(\\mathcal A\,\
 \mathcal H\,D)$\, which consists of a Hilbert space $\\mathcal H$\, a pre-
 $C^*$-algebra $\\mathcal A $\, represented faithfully on $\\mathcal H$ and
  a self-adjoint operator $D$ acting on $\\mathcal H$ such that every $a\\i
 n A$ maps the domain of $D$ into itself and the commutator $[D\,a] = Da-aD
 $ extends from the domain of $D$ to a bounded linear endomorphism of $\\ma
 thcal H$. Here\, the quantised differential $\\qd a$ of $a \\in \\mathcal 
 A$ is defined to be the bounded operator ${\\rm i} [{\\rm sgn}(D)\,a]$\, $
 {\\rm i}^2=-1$. \n\nWe provide full characterisation of the Schatten prope
 rties of $[M_b\,R_j]$\,  the commutator of the $j$-th Riesz transform on $
 \\mathbb R^n$ with symbol $b$ $(M_bf(x):=b(x)f(x))$\, in the two weight se
 tting. \n\n\n\nThis talk is based on my recent work joint with Michael Lac
 ey\, Brett Wick and Liangchuan Wu.\n
LOCATION:https://researchseminars.org/talk/TaiwanAnalysisConference2024/14
 /
END:VEVENT
END:VCALENDAR