Theory of Function Spaces: On Classical Tools for Modern Spaces\nby Markus Weimar (Die Julius-Maximilians-Universität Würzburg) as par t of Function spaces\n\n\nAbstract\nIn the first part of this talk\, we di scuss basic principles of the theory of function spaces. In particular\, w e briefly recall the Fourier analytical approach towards classical smoothn ess spaces of distributions and point out their importance in the areas of approximation theory and the regularity theory of PDEs.\n\nThe main part of the talk is devoted to so-called Triebel-Lizorkin-Morrey spaces $\\math cal{E}_{u\,p\,q}^s$ of positive smoothness $s$ which attracted some attent ion in the last 15 years. This family of function spaces generalizes the b y now well-established scale of Triebel-Lizorkin spaces $F^s_{p\,q}$ which particularly contains the usual $L_p$-Sobolev spaces $H^s_p=F^s_{p\,2}$ a s special cases. Moreover\, there are strong relations to standard classes of functions like BMO and Campanato spaces which are widely used in the a nalysis of PDEs.\nWe will present new characterizations of Triebel-Lizorki n-Morrey spaces in terms of classical tools such as local oscillations (i. e.\, local polynomial bestapproximations) as well as ball means of higher order differences. Hence\, under standard assumptions on the parameters in volved\, we extend assertions due to Triebel 1992 and Yuan/Sickel/Yang 201 0 for spaces $\\mathcal{E}_{u\,p\,q}^s$ on $\\mathbb{R}^d$ and additionall y consider their restrictions to (bounded) Lipschitz domains $\\Omega\\sub seteq \\mathbb{R}^d$. \nIf time permits\, we moreover indicate possible ap plications to the regularity theory of quasi-linear elliptic PDEs. \nThe r esults to be presented are based on a recent preprint [1] in joint work wi th Marc Hovemann (Marburg).\n\n[1] M.~Hovemann and M.~Weimar. Oscillations and differences in Triebel-Lizorkin-Morrey spaces. Submitted preprint (ar Xiv:2306.15239)\, 2023.\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/88/ END:VEVENT BEGIN:VEVENT SUMMARY:Igor I. Skrypnik (Institute Applied Mathematics and Mechanics of t he NAS Ukraine) DTSTART;VALUE=DATE-TIME:20231205T130000Z DTEND;VALUE=DATE-TIME:20231205T140000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/89 DESCRIPTION:Title: Some remarks on the weak Harnack inequality for unbounded minimiz ers of elliptic functionals with generalized Orlicz growth\nby Igor I . Skrypnik (Institute Applied Mathematics and Mechanics of the NAS Ukraine ) as part of Function spaces\n\n\nAbstract\nWe prove the weak Harnack type inequalities for nonnegative unbounded minimizers of corresponding ellip tic functionals under the non-logarithmic \nZhikov's conditions\, roughly speaking we consider the following De Giorgi's classes\n$$\n\\int\\limits_ {B_{(1-\\sigma)r}(x_{0})}\\varPhi\\Big(x\,|(u-k)_{-}|\\Big)\\\,dx \\leqsla nt \\gamma \\int\\limits_{B_{r}(x_{0})}\\varPhi\\Big(x\,\\frac{(u-k)_{-}}{ \\sigma r}\\Big)\\\,dx\,\n$$\n$\\sigma$\, $r\\in(0\,1)$\, $k>0$ and $\\var Phi(x\,\\cdot)$ satisfies the so-called ($p\,q$)-growth conditions. We are interesting in the case when \n$\\gamma$ depends on $r$\, it turns out th at in this case it is impossible to use standard classical techniques. Our results cover new cases of double-phase\, degenerate double-phase functi onals\, non uniformly elliptic functionals and functionals with variable exponents.\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/89/ END:VEVENT BEGIN:VEVENT SUMMARY:Emiel Lorist (Delft University of Technology) DTSTART;VALUE=DATE-TIME:20231219T130000Z DTEND;VALUE=DATE-TIME:20231219T140000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/90 DESCRIPTION:Title: A discrete framework for the interpolation of Banach spaces\nb y Emiel Lorist (Delft University of Technology) as part of Function spaces \n\n\nAbstract\nInterpolation of bounded linear operators on Banach spaces is a widely used technique in analysis. Key roles are played by the real and complex interpolation methods\, but there is also a wealth of other in terpolation methods\, for example relevant in the study of (S)PDE. In this talk I will introduce interpolation of Banach spaces using a new\, discre te framework. I will discuss how this framework extends and unifies variou s results in the literature. Moreover\, I will discuss its applications to parabolic boundary value problems.\nThis talk is based on joint work with Nick Lindemulder (Radboud University Nijmegen).\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/90/ END:VEVENT BEGIN:VEVENT SUMMARY:Jennifer Duncan (ICMAT Madrid) DTSTART;VALUE=DATE-TIME:20231121T130000Z DTEND;VALUE=DATE-TIME:20231121T140000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/91 DESCRIPTION:Title: Brascamp-Lieb Inequalities: Their Theory and Some Applications \nby Jennifer Duncan (ICMAT Madrid) as part of Function spaces\n\n\nAbstra ct\nThe Brascamp-Lieb inequalities form a class of multilinear inequalitie s that includes a variety of well-known classical results\, such as Hölde r’s inequality\, Young’s convolution inequality\, and the Loomis-Whitn ey inequality\, for example. Their theory is surprisingly multifaceted\, i nvolving ideas from semigroup interpolation\, convex optimisation\, and ab stract algebra. In the first half of this talk\, we will discuss some of t he key aspects of this theory and some important variants on the Brascamp- Lieb framework\; in the second half\, we will talk specifically about how these inequalities arise in harmonic analysis\, in particular about their use in fourier restriction theory and in recent results on the boundedness of the helical maximal function. If time permits\, we will then talk abou t some more far-reaching connections with other areas of mathematics and t he sciences.\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/91/ END:VEVENT BEGIN:VEVENT SUMMARY:Swarnendu Sil (Indian Institute of Science Bengaluru) DTSTART;VALUE=DATE-TIME:20231107T130000Z DTEND;VALUE=DATE-TIME:20231107T140000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/92 DESCRIPTION:Title: BMO estimates for Hodge-Maxwell systems with discontinuous anisotr opic coefficients\nby Swarnendu Sil (Indian Institute of Science Benga luru) as part of Function spaces\n\n\nAbstract\nThe time-harmonic Maxwell system in a bounded domain is \n $$\n \\left\\lbrace \\begin{aligned}\n curl H &= i\\omega \\varepsilon \\left(x\\right) E + J_{e} \n && \\text{ in } \\Omega\, \\\\\n \\operatorname*{curl} E &= -i\\omega \\mu \\left(x\\right) H + J_{m} &&\\text{ in } \\Omega\, \\\\\n \\nu \\t imes E &= \\nu \\times E_{0} &&\\text{ on } \\partial\\Omega\,\n \\end{ aligned} \n \\right.\n $$\n where $E\, H$ are unknown vector fields\, $E_{0}\, J_{e}\, J_{m}$ are given vector fields and $\\varepsilon\, \\mu$ are given $3\\times 3$ matrix fields which are bounded\, measurable and un iformly elliptic. When $\\varepsilon\, \\mu$ have sufficient regularity\, e.g. Lipschitz\, then one can show that $(E\, H)$ inherits the same regula rity as $(J_{e}\, J_{m})$\, as long as $E_{0}$ is as regular. \n \n \\pa r In this talk\, we shall discuss the sharpest regularity assumptions on $ \\varepsilon\, \\mu$ under which $(E\, H)$ inherits BMO regularity from $( J_{e}\, J_{m}).$ As it turns out\, the minimal regularity assumption on $ \\varepsilon\, \\mu$ is that their components belong to a class of `small multipliers of BMO'. This class neither contains nor is contained in $C^{0 }.$ Thus our results prove the validity of BMO estimates for a class of di scontinuous coefficients. Our results are actually holds more generally\, for systems of differential $k$-forms of similar type in any dimension $n \\geq 3.$ \n \n \\par This is a joint work with my post-doctoral student Dharmendra Kumar.\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/92/ END:VEVENT BEGIN:VEVENT SUMMARY:Kim Myyryläinen (Aalto University) DTSTART;VALUE=DATE-TIME:20240220T130000Z DTEND;VALUE=DATE-TIME:20240220T140000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/93 DESCRIPTION:Title: Parabolic Muckenhoupt weights\nby Kim Myyryläinen (Aalto Uni versity) as part of Function spaces\n\n\nAbstract\nWe discuss parabolic Mu ckenhoupt weights related to a doubly nonlinear parabolic partial differen tial equation (PDE). In the natural geometry of the PDE\, the time variabl e scales to the power in the structural conditions for the PDE. Consequent ly\, the Euclidean balls and cubes are replaced by parabolic rectangles re specting this scaling in all estimates. The main challenge is that in the definition of parabolic Muckenhoupt weights one of the integral averages i s evaluated in the past and the other one in the future with a time lag be tween the averages. Another main motivation is that the parabolic theory i s a higher dimensional version of the one-sided setting and the correspond ing one-sided maximal function.\nThe main results include a characterizati on of weak and strong type weighted norm inequalities for forward in time parabolic maximal functions and parabolic versions of the Jones factorizat ion and the Coifman--Rochberg characterization. In addition to parabolic M uckenhoupt weights\, the class of parabolic $A_\\infty$ weights is discuss ed from the perspective of parabolic reverse H\\"older inequalities. We co nsider several characterizations and self-improving properties for this cl ass of weights and study their connection to parabolic Muckenhoupt conditi ons. A sufficient condition is given for the implication from parabolic re verse Holder classes to parabolic Muckenhoupt classes.\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/93/ END:VEVENT BEGIN:VEVENT SUMMARY:Lorenzo Brasco (Università degli Studi di Ferrara) DTSTART;VALUE=DATE-TIME:20240319T130000Z DTEND;VALUE=DATE-TIME:20240319T140000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/94 DESCRIPTION:Title: Around Hardy's inequality for convex sets\nby Lorenzo Brasco ( Università degli Studi di Ferrara) as part of Function spaces\n\nInteract ive livestream: https://cesnet.zoom.us/j/99825599862\n\nAbstract\nWe start by reviewing the classical Hardy inequality for convex sets.\nWe then dis cuss the counterpart of Hardy's inequality for the case of fractional Sobo lev-Slobodecki\\u{\\i} spaces\, still in the case of open convex subsets o f the Euclidean space. In particular\, we determine the sharp constant in this inequality\, by constructing explicit supersolutions based on the dis tance function.\nWe also show that this method works only for the {\\it mi ldly nonlocal} regime and it is bound to fail for the {\\it strongly nonlo cal} one. We conclude by presenting some open problems.\n\\par\nSome of th e results presented are issued from papers in collaboration with Francesca Bianchi (Ferrara \\& Parma)\, Eleonora Cinti (Bologna)\, Firoj Sk (Oldenb urg) and Anna Chiara Zagati (Ferrara \\& Parma).\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/94/ URL:https://cesnet.zoom.us/j/99825599862 END:VEVENT BEGIN:VEVENT SUMMARY:Firoj Sk (University of Oldenburg) DTSTART;VALUE=DATE-TIME:20240402T120000Z DTEND;VALUE=DATE-TIME:20240402T130000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/95 DESCRIPTION:by Firoj Sk (University of Oldenburg) as part of Function spac es\n\nInteractive livestream: https://cesnet.zoom.us/j/99825599862\nAbstra ct: TBA\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/95/ URL:https://cesnet.zoom.us/j/99825599862 END:VEVENT BEGIN:VEVENT SUMMARY:Rémy Rodiac (University of Warsaw) DTSTART;VALUE=DATE-TIME:20240305T130000Z DTEND;VALUE=DATE-TIME:20240305T140000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/96 DESCRIPTION:Title: A relaxation approach for the minimisation of the neo-Hookean ener gy\nby Rémy Rodiac (University of Warsaw) as part of Function spaces\ n\nInteractive livestream: https://cesnet.zoom.us/j/99825599862\n\nAbstrac t\nThe neo-Hookean model is a famous model for elastic materials. However it is still not known if the neo-Hookean energy admits a minimiser in an a ppropriate function space in 3D. I will explain what is the difficulty one encounters when we try to apply the direct method of calculus of variatio ns to this problem: this is the lack of compactness of the minimisation sp ace. I will also present a relaxation approach whose aim is to transform t he problem of lack of compactness into a problem of regularity for a modif ied problem. The talk will be based on joint works with M. Barchiesi\, C. Mora-Corral and D. Henao.\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/96/ URL:https://cesnet.zoom.us/j/99825599862 END:VEVENT BEGIN:VEVENT SUMMARY:Emil Airta (University of Málaga) DTSTART;VALUE=DATE-TIME:20240416T120000Z DTEND;VALUE=DATE-TIME:20240416T130000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/97 DESCRIPTION:by Emil Airta (University of Málaga) as part of Function spac es\n\nInteractive livestream: https://cesnet.zoom.us/j/99825599862\nAbstra ct: TBA\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/97/ URL:https://cesnet.zoom.us/j/99825599862 END:VEVENT BEGIN:VEVENT SUMMARY:Valentina Ciccone (University of Bonn) DTSTART;VALUE=DATE-TIME:20240430T120000Z DTEND;VALUE=DATE-TIME:20240430T130000Z DTSTAMP;VALUE=DATE-TIME:20240224T053752Z UID:FunctionSpaces/98 DESCRIPTION:by Valentina Ciccone (University of Bonn) as part of Function spaces\n\nInteractive livestream: https://cesnet.zoom.us/j/99825599862\nAb stract: TBA\n LOCATION:https://researchseminars.org/talk/FunctionSpaces/98/ URL:https://cesnet.zoom.us/j/99825599862 END:VEVENT END:VCALENDAR