Off-diagonal and two-sided commutator bounds

Singular integrals quantify regularity in partial differential equations and geometric measure theory, while commutators of these singular integrals are tied to factorizations of Hardy spaces, div-curl lemmas and to the Jacobian problem; all relevant examples in contemporary analysis. Our commutators are formed by commuting a singular integral with a function (symbol) acting as a pointwise multiplication. In this dissertation we are concerned with both the qualitative and quantitative properties of commutators. More precisely, we develop real analytic techniques that yield lower- and upper bounds for operator norms of commutators in terms of oscillatory testing conditions on the symbol of the commutator. Classically, between the commutator lower- and upper bounds, the latter has dominated the commutator scene due to the surplus of flexible tools such as the representation theorem and the sparse domination principle; while for the opposite reason, the lower bounds have been restricted to algebraically well-structured operators such as the Riesz transforms. Lately, a powerful argument to obtain commutator lower bounds was introduced by Hytönen, adding an approximation on top of the weak factorization scheme, originally due to Uchiyama. Through the approximate weak factorization argument commutator lower bounds became available under minimal a priori assumptions on the kernel of the singular integral and the symbol. The main contribution of this dissertation is the systematic development of the approximate weak factorization argument in the bi-parameter, bilinear and parabolic settings over Euclidean spaces. Moreover, we obtain commutator norm bounds in the off-diagonal setting, where functions in the domain and the range of the commutator have non-matching integrability. In the second and third articles, we study two different bi-parameter commutators in the mixed product setting and obtain several new characterizations of commutator boundedness in terms of mixed oscillatory testing conditions on the symbol; in the fourth article we study bilinear commutators in the quasi-Banach setting and obtain a close to full extension of the linear theory to the bilinear setting; in the fifth article we introduce a factorization scheme for parabolic atoms in terms of the parabolic Hilbert transform, and thus complete the characterization of the boundedness of the parabolic Hilbert commutator. Additionally, in the first article we obtain an almost full characterization of the boundedness of the iterated commutator with two different symbols.Singulaari-integraalien juuret ovat Fourier-sarjan suppenemisen kattavassa klassisessa harmonisessa analyysissä, ne kvantifioivat esimerkiksi Laplacen yhtälön ratkaisun säännöllisyyttä, ja ovat suoristuvuuden mittareita geometrisessa mittateoriassa. Singulaari-integraalien kommmutaattorit taas liittyvät muun muassa Jakobiaaniongelmaan, Hardy-avaruuksien faktorisointeihin ja kompensoituun kompaktisuuteen. Kaikki esimerkit edellä ovat oleellisia matemaattisen analyysin mailmassa. Viimeisenä vuosikymmenenä kehittynyt yleinen singulaari-integraaleja koskeva teknologia esityslauseiden ja harvadominointien muodossa on mahdollistanut kommutaattoriylärajojen nopean kehityksen. Samanlaista kehitystä ei ole kuitenkaan suotu kommutaattorialarajoille, jotka ovat olleet saatavilla lähinnä yksittäisille algebrallisesti hyvin strukturoiduille singulaari-integraaleille kuten Rieszin muunnoksille. Hiljattain professori Hytönen esitteli lisäyksen edesmenneen professori Uchiyaman klassiseen Hardy-atomien heikon hajottamisen teoriaan, näin luoden pohjaa monille uusille kommutaattorialarajoille. Tässä väitöstyössä tutkimme kommutaattorien kvalitatiivisia- ja kvantitatiivisia ominaisuuksia, erityisesti reaalianalyyttisiä metodeja, joista seuraa kommutaattoreille sekä ala- että ylärajoja. Tarkentuen, kehitämme yllä mainittua heikon hajottamisen teoriaa, jonka seurauksena todistamme kommutaattoriestimaatteja Lebesguen p-normin suhteen niin bilineaarisessa, bi-parametrisessa kuin parabolisissakin asetelmassa

Boundedness of The Bilinear Calderón-Zygmund Operator and Lp- Estimation Tools and Techniques

Tämän työn päämäärä on kehitellä Lp−estimointi tekniikoita, soveltaa niitä mallioperaattoreiden tutkimiseen, ja edelleen, siirtää näiden mallioperaattoreiden rajoittuneisuus ominaisuuksia bilineaariselle Calderón-Zygmund -operaattorille. Tämä siirtäminen tehdään olettamalla tunnetuksi Bilineaarisen Calderón-Zygmund -operaattorin esityslause satunnaistettujen mallioperaattoreiden summana. Oletamme tunnetuksi hieman interpolointi ja maksimaalifunktioiden teoriaa. Aloitamme esittelemällä hyödyllisiä peruskäsitteitä, mm. neliöfunktion, sen kautta Lp -avaruuksien karakterisaation, ja Haarin kannan. Kappaleissa kaksi ja kolme määrittelemme mallioperattorit: shiftit ja paratulot ja todistamme niitä koskevia vahvoja estimaatteja Lp -avaruuksissa, kun p > 1. Tarkoituksemme on interpoloida vahvoja estimaatteja quasi-Banach alueelle, siis kun p < 1, ja tätä varten todistamme heikkoja päätepiste estimaatteja. Neljäs kappale on omistettu interpolointitekniikan esittelemiselle, joka mahdollistaa edellämainitun interpoloinnin. Viidennessä ja viimeisessä kappaleessa määrittelemme bilineaarisen Calderó-Zygmund operaattorin, kokoamme yhteen aikaisemmin kehitetyn teorian tuloksia ja vedämme näistä tuloksista lyhyehkönä korollaarina bilineaarisen Calderón-Zygmund -operaattorin rajoittuneisuuden

Lower bound of the parabolic Hilbert commutator

Funding Information: T. Oikari was supported by the Academy of Finland project numbers 306901 and 314829 , by the Finnish Centre of Excellence in Analysis and Dynamics Research project No. 307333 , by the three-year research grant of the University of Helsinki No. 75160010 and by the Jenny and Antti Wihuri Foundation grant No. 00200253 . Publisher Copyright: © 2022 The Author(s)Answering a key point left open in the recent work of Bongers, Guo, Li and Wick [2], we provide the lower bound ‖b‖BMOγ(R2)≲‖[b,Hγ]‖Lp(R2)→Lp(R2), where Hγ is the parabolic Hilbert transform.Peer reviewe

Note on compactness of commutators

In this note we remark that classical compactness interpolation methods show that the optimal sufficient conditions for the $L^p$-to-$L^q$ compactness of commutators of Calder\'on-Zygmund operators in the ranges $q<p,$ $q = p$ and $q>p$ quickly follow from each other.Comment: v3: the proof of Theorem 3.7. in v2 was wrong; however, the claimed results are still correct and are in this v3 noted to follow quickly from classical compactnesss interpolation methods; not for publicatio

The dynamic metabolism of hyaluronan regulates the cytosolic concentration of UDP-GlcNAc

Hyaluronan, a macromolecular glycosaminoglycan, is normally synthesized by hyaluronan synthases at the plasma membrane using cytosolic UDP-GlcUA and UDP-GlcNAc substrates and extruding the elongating chain into the extracellular space. The cellular metabolism (synthesis and catabolism) of hyaluronan is dynamic. UDP-GlcNAc is also the substrate for O-GlcNAc transferase, which is central to the control of many cytosolic pathways. This Perspective outlines recent data for regulation of hyaluronan synthesis and catabolism that support a model that hyaluronan metabolism can be a rheostat for controlling an acceptable normal range of cytosolic UDP-GlcNAc concentrations in order to maintain normal cell functions

Approximation in H\"older Spaces

We introduce new vanishing subspaces of the homogeneous H\"{o}lder space $\dot{C}^{0,\omega}(X,Y),$ in the generality of a doubling modulus $\omega$ and normed spaces $X$ and $Y.$ For many couples $X,Y,$ we show these vanishing subspaces to completely characterize those H\"older functions that admit approximations, in the H\"{o}lder seminorm, by smooth, Lipschitz and boundedly supported functions. Beyond the intrinsic interest of these results, we also present connections to bi-parameter harmonic analysis on the Euclidean space and in particular to compactness of the bi-commutator.Comment: v3: minor improvements, 29 page

Fractional Bloom boundedness and compactness of commutators

Let $T$ be a non-degenerate Calder\'on-Zygmund operator and let $b:\mathbb{R}^d\to\mathbb{C}$ be locally integrable. Let $1<p\leq q<\infty$ and let $\mu^p\in A_p$ and $\lambda^q\in A_q,$ where $A_{p}$ denotes the usual class of Muckenhoupt weights. We show that \begin{align*} \|[b,T]\|_{L^p_{\mu}\to L^q_{\lambda}}\sim \|b\|_{\operatorname{BMO}_{\nu}^{\alpha}},\qquad [b,T]\in \mathcal{K}(L^p_{\mu}, L^q_{\lambda})\quad\mbox{iff}\quad b\in \operatorname{VMO}_{\nu}^{\alpha}, \end{align*} where $L^p_\mu=L^p(\mu^p)$ and $\alpha/d = 1/p-1/q,$ , the symbol $\mathcal{K}$ stands for the class of compact operators between the given spaces, and the fractional weighted $\operatorname{BMO}_{\nu}^{\alpha}$ and $\operatorname{VMO}_{\nu}^{\alpha}$ spaces are defined through the following fractional oscillation and Bloom weight \begin{align*} \mathcal{O}_{\nu}^{\alpha}(b;Q) = \nu^{-\alpha/d}(Q)\Big(\frac{1}{\nu(Q)}\int_Q |b-\langle b\rangle_Q|\Big),\qquad \nu = \big(\frac{\mu}{\lambda}\big)^{\beta},\quad \beta = (1+\alpha/d)^{-1}. \end{align*} The key novelty is dealing with the off-diagonal range $p<q$, whereas the case $p=q$ was previously studied by Lacey and Li. However, another novelty in both cases is that our approach allows complex-valued functions $b$, while other arguments based on the median of $b$ on a set are inherently real-valued.Comment: V2: 26 pages, minor revision according to referee comments, accepted for publication in Forum Mathematicum. V1:26 page