Multilinear and Nonlinear Harmonic Analysis and Applications (MUNHANAP)

Project summary

• The project falls into the realm of harmonic analysis, but it can be regarded as inter-disciplinary in the sense that our group intends to solve several open problems from diverse branches of mathematics. We will reduce those problems to the appropriate (multilinear or nonlinear) estimates and then prove those estimates using analytical techniques. The established estimates will be applied in ergodic theory (e.g., to quantify convergence of multiple ergodic averages), in additive combinatorics (e.g., to Szemerédi-type theorems in the Euclidean space), and in the harmonic/Fourier analysis itself (e.g., to the Dirac nonlinear Fourier transform).

Principal investigator

• Vjekoslav Kovač, Department of Mathematics, Faculty of Science, University of Zagreb

Team members

• Aleksandar Bulj, Department of Mathematics, Faculty of Science, University of Zagreb (team member from March 1, 2020)
• Rudi Mrazović, Department of Mathematics, Faculty of Science, University of Zagreb
• Bruno Predojević, Department of Mathematics, Faculty of Science, University of Zagreb (team member from December 20, 2021)
• Jelena Rupčić, Faculty of Transport and Traffic Sciences, University of Zagreb
• Mario Stipčić, Department of Mathematics, Faculty of Science, University of Zagreb
• Kristina Ana Škreb, Faculty of Civil Engineering, University of Zagreb

Research publications

Completed in the 1st project year

[1]	P. Durcik, V. Kovač, Boxes, extended boxes, and sets of positive upper density in the Euclidean space, Math. Proc. Cambridge Philos. Soc. 171 (2021), no. 3, 481-501.	[arXiv] [journal]
[2]	V. Kovač, D. Oliveira e Silva, A variational restriction theorem, Arch. Math. 117 (2021), 65-78.	[arXiv] [journal]
[3]	V. Kovač, Fourier restriction implies maximal and variational Fourier restriction, J. Funct. Anal. 277 (2019), no. 10, 3355-3372.	[arXiv] [journal]
[4]	V. Kovač, P. Zorin-Kranich, Variational estimates for martingale paraproducts, Electron. Commun. Probab. 24 (2019), paper no. 48, 14 pp.	[arXiv] [journal]
[5]	M. Stipčić, T(1) theorem for dyadic singular integral forms associated with hypergraphs, J. Math. Anal. Appl. 481 (2020), no. 2, article 123496, 27 pp.	[arXiv] [journal]
[6]	J. Rupčić, Convergence of lacunary SU(1,1)-valued trigonometric products, Commun. Pure Appl. Anal. 19 (2020), no. 3, 1275-1289.	[arXiv] [journal]

Completed in the 2nd project year

[7]	K. Domelevo, S. Petermichl, K. A. Škreb, Failure of the matrix weighted bilinear Carleson embedding theorem, Linear Algebra Appl. 582 (2019), 452-466.	[arXiv] [journal]
[8]	R. Han, V. Kovač, M. T. Lacey, J. Madrid, F. Yang, Improving estimates for discrete polynomial averages, J. Fourier Anal. Appl. 26 (2020), article no. 42, 11 pp.	[arXiv] [journal]
[9]	P. Durcik, V. Kovač, A Szemerédi-type theorem for subsets of the unit cube, Anal. PDE. 15 (2022), no. 2, 507-549.	[arXiv] [journal]
[10]	S. Eberhard, F. Manners, R. Mrazović, An asymptotic for the Hall-Paige conjecture, Adv. Math. 404, Part A (2022), article 108423, 73 pp.	[arXiv] [journal]
[11]	V. Kovač, M. Stipčić, Convergence of ergodic-martingale paraproducts, Statist. Probab. Lett. 164 (2020), article 108826, 6 pp.	[arXiv] [journal]

Completed in the 3rd project year

[12]	V. Kovač, Density theorems for anisotropic point configurations, Canad. J. Math. 74 (2022), no. 5, 1244-1276.	[arXiv] [journal]
[13]	K. Falconer, V. Kovač, A. Yavicoli, The density of sets containing large similar copies of finite sets, J. Anal. Math. 148 (2022), 339-359.	[arXiv] [journal]
[14]	A. Carbonaro, O. Dragičević, V. Kovač, K. A. Škreb, Trilinear embedding for divergence-form operators with complex coefficients, Adv. Math. 431 (2023), article 109239, 72 pp.	[arXiv] [journal]
[15]	V. Kovač, Popular differences for right isosceles triangles, Electron. J. Combin. 28 (2021), no. 4, article P4.27, 10 pp.	[arXiv] [journal]
[16]	R. Mrazović, N. Sandrić, S. Šebek, Capacity of the range of random walks on groups, Kyoto J. Math. 63 (2023), no. 4, 783-805.	[arXiv] [journal]

Completed in the 4th project year

[17]	V. Kovač, D. Oliveira e Silva, J. Rupčić, Asymptotically sharp discrete nonlinear Hausdorff-Young inequalities for the SU(1,1)-valued Fourier products, Q. J. Math. 73 (2022), no. 3, 1179-1188.	[arXiv] [journal]
[18]	A. Carbonaro, O. Dragičević, V. Kovač, Sharp $L^p$ estimates of powers of the complex Riesz transform, Math. Ann. 386 (2023), 1081-1125.	[arXiv] [journal]
[19]	V. Kovač, K. A. Škreb, Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients, J. Funct. Anal. 284 (2023), no. 9, article 109884, 32 pp.	[arXiv] [journal]
[20]	F. Nazarov, S. Petermichl, K. A. Škreb, S. Treil, The matrix-weighted dyadic convex body maximal operator is not bounded, Adv. Math. 410, Part A (2022), article 108711, 20 pp.	[arXiv] [journal]
[21]	A. Bulj, V. Kovač, Asymptotic behavior of $L^p$ estimates for a class of multipliers with homogeneous unimodular symbols, Trans. Amer. Math. Soc. 376 (2023), no. 7, 4539-4567.	[arXiv] [journal]

Completed in the 5th project year

[22]	P. Durcik, M. Stipčić, Quantitative bounds for product of simplices in subsets of the unit cube, Israel J. Math., accepted for publication (2023).	[arXiv] [journal]
[23]	A. Bulj, V. Kovač, Multi-parameter maximal Fourier restriction, J. Fourier Anal. Appl. 30 (2024), issue 3, article no. 26, 23 pp.	[arXiv] [journal]
[24]	S. Eberhard, F. Manners, R. Mrazović, Transversals in quasirandom latin squares, Proc. Lond. Math. Soc. 127 (2023), issue 1, 84-115.	[arXiv] [journal]
[25]	K. Domelevo, S. Petermichl, K. A. Škreb, Continuous sparse domination and dimensionless weighted estimates for the Bakry Riesz vector, preprint, submitted.	[arXiv]
[26]	P. Durcik, V. Kovač, M. Stipčić, A strong-type Furstenberg-Sárközy theorem for sets of positive measure, J. Geom. Anal. 33 (2023), issue 8, article no. 255, 16 pp.	[arXiv] [journal]

The list of all published papers associated with this project is also available in CROSBI.

Doctoral dissertations

• M. Stipčić, Multilinear singular integrals associated with hypergraphs, PMF, University of Zagreb, 2020.

Conference and seminar talks

In the 1st project year

• M. Stipčić, Entangled multilinear forms associated with hypergraphs (conference talk), Probabilistic Aspects of Harmonic Analysis, Będlewo, May 10, 2018.
• M. Stipčić, Hypergraph representation and boundedness of entangled multilinear forms (summer school presentation), Mini-courses in Mathematical Analysis, Padova, July 5, 2018.
• V. Kovač, Bressan's problem on mixing flows (conference talk), ApplMath18, Šibenik, September 18, 2018.
• K. A. Škreb, Bellman functions for paraproducts and modified martingale transforms (seminar talk), Michigan State University, East Lansing, October 31, 2018.
• V. Kovač, Neke primjene multilinearnih singularnih integrala (colloquium talk), Sveučilište u Splitu, January 18, 2019.
• K. A. Škreb, Dimensionless Lp estimates for the Riesz vector - a probabilistic approach (seminar talk), University of Nantes, January 18, 2019.

In the 2nd project year

• V. Kovač, Problemi klasične i maksimalne teorije Fourierove restrikcije (seminar talk), Seminar za diferencijalne jednadžbe i numeričku analizu, Zagreb, April 15, 2019.
• V. Kovač, Varijacijske ocjene za paraprodukte martingala (seminar talk), Seminar za teoriju vjerojatnosti, Zagreb, April 23, 2019.
• V. Kovač, On maximal and variational Fourier restriction (workshop talk), Follow-up Workshop to Trimester Program Harmonic Analysis and Partial Differential Equations, Bonn, May 9, 2019.
• V. Kovač, Variants of the Christ-Kiselev lemma and an application to the maximal Fourier restriction (seminar talk), Seminar Alpe-Jadran, Zagreb, May 18, 2019.
• K. A. Škreb, From Riesz transforms to stochastic processes (conference talk), Harmonic Analysis and PDEs (HAPDE2019), Helsinki, June 5, 2019.
• M. Stipčić, Characterization of boundedness of singular integral forms associated with hypergraphs (summer school presentation), Mini-courses in Mathematical Analysis, Padova, June 25, 2019.
• V. Kovač, Variants of the Christ-Kiselev lemma and an application to the maximal Fourier restriction (seminar talk), Analysis Seminar, Georgia Institute of Technology, Atlanta, September 25, 2019.
• V. Kovač, On maximal and variational Fourier restriction (seminar talk), Caltech/UCLA joint Analysis seminar, Pasadena, December 6, 2019.
• R. Mrazović, Asimptotika za Hall-Paigeovu hipotezu (colloquium talk), Znanstveni kolokvij HMD-a, Sveučilište u Zagrebu, January 22, 2020.
• V. Kovač, A Szemerédi-type theorem for subsets of the unit cube (seminar talk), Analysis Seminar, Georgia Institute of Technology, Atlanta, January 29, 2020.
• V. Kovač, A Szemerédi-type theorem for subsets of the unit cube (seminar talk), Analysis Seminar, University of Alabama, Tuscaloosa, February 7, 2020.
• M. Stipčić, T(1) teorem za dijadske singularne integralne forme pridružene hipergrafovima (conference talk), 4. simpozij studenata doktorskih studija PMF-a, Zagreb, February 28, 2020.

In the 3rd project year

• M. Stipčić, Multilinearni singularni integrali pridruženi hipergrafovima (seminar talks), Seminar za teoriju vjerojatnosti, Zagreb, May 13, October 27, November 3, November 10, 2020.
• V. Kovač, Improving estimates for discrete polynomial averaging operators (online workshop talk), Zagreb Workshop on Operator Theory 2020, Zagreb, June 29-30, 2020.
• M. Stipčić, Convergence and Lp estimate of generalized ergodic averages (online workshop talk), Zagreb Workshop on Operator Theory 2020, Zagreb, June 29-30, 2020.
• V. Kovač, A Szemerédi-type theorem for subsets of the unit cube (online seminar talk), Mathematical Institute, University of Bonn, July 3, 2020.
• V. Kovač, Density theorems for anisotropic point configurations (online seminar talk), Analysis and PDE Webinar, IMPA, Rio de Janeiro, July 21, 2020.
• V. Kovač, Trilinear embedding for divergence-form operators (conference talk), ApplMath20, Brijuni, September 15, 2020.
• V. Kovač, Density theorems for Euclidean point configurations (online workshop talk), LMS Harmonic Analysis and PDEs Meeting, December 11, 2020.
• M. Stipčić, Multilinearni singularni integrali pridruženi hipergrafovima (dissertation defense talk), PMF-MO, Zagreb, December 14, 2020.

In the 4th project year

• M. Stipčić, Lp estimates for dyadic singular integral forms associated with hypergraphs and for ergodic-martingale paraproducts (online seminar talk), Analysis and Geometry seminar, Chapman University, March 12, 2021.
• R. Mrazović, An asymptotic for the Hall-Paige conjecture (online seminar talk), CMSA Australasian Combinatorics Seminar, March 17, 2021.
• V. Kovač, Density theorems for Euclidean point configurations (online conference talk), Croatian-German meeting on analysis and mathematical physics, March 25, 2021.
• A. Bulj, Tenzorski produkt u rješavanju problema (online seminar talk), Seminar za funkcionalnu analizu, Zagreb, May 19, 2021.
• K. A. Škreb, The matrix-weighted Hardy-Littlewood maximal function is unbounded (online seminar talk), APDE Webinar, Bilbao, June 3, 2021.
• K. A. Škreb, Hardy-Littlewoodova maksimalna funkcija nije ograničena u prostorima s matričnim težinama (online seminar talk), Seminar za funkcionalnu analizu, Zagreb, June 16, 2021.
• R. Mrazović, Counting transversals in group multiplication tables (online conference talk), 8th European Congress of Mathematics, Portorož, June 24, 2021.
• V. Kovač, Multilinear estimates via optimal control (conference talk), Brijuni Applied Mathematics Workshop 2021, July 8, 2021.
• A. Bulj, Dimension conditions for non-geometric Brascamp-Lieb inequalities (summer school presentation), Brascamp-Lieb inequalities, summer school, Kopp, September 27, 2021.
• V. Kovač, Density theorems for anisotropic point configurations (online conference talk), 2021 Canadian Mathematical Society Winter Meeting, December 4, 2021.
• V. Kovač, Polugrupe generirane kompleksnim eliptičkim operatorima i ocjene u Orliczevim prostorima (online seminar talk), Seminar za funkcionalnu analizu, Zagreb, January 20, 2022.
• B. Predojević, Bourgainov teorem o simpleksu (online seminar talk), Seminar za funkcionalnu analizu, Zagreb, January 27, 2022.
• V. Kovač, Bilinear and trilinear estimates for semigroups generated by complex elliptic operators (online seminar talk), Virtual Harmonic Analysis Seminar, UK Harmonic Analysis group, February 9, 2022.

In the 5th project year

• K. A. Škreb, Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients (online seminar talk), Probability and Analysis Webinar (PAW), March 7, 2022.
• V. Kovač, Lower bounds for the Lp norms of some Fourier multipliers (online seminar talk), Probability and Analysis Webinar (PAW), May 2, 2022.
• M. Stipčić, Quantitative bounds for products of simplices in subsets of the unit cube (conference talk), 2022 Workshop on Operator Theory with an Eye on Linear Systems and Hypercomplex Analysis, Orange, June 2, 2022.
• A. Bulj, Asimptotsko ponašanje Lp ocjena za jednu klasu multiplikatora (seminar talk), Seminar za funkcionalnu analizu, Zagreb, June 9, 2022.
• A. Bulj, Asymptotic behavior of Lp estimates for a class of multipliers with homogeneous unimodular symbols (conference talk), 7th Croatian Mathematical Congress, Split, June 15, 2022.
• M. Stipčić, Quantitative bounds for products of simplices in subsets of the unit cube (conference talk), 7th Croatian Mathematical Congress, Split, June 15, 2022.
• J. Rupčić, Asymptotically sharp nonlinear Hausdorff-Young inequalities for continuous and discrete SU(1,1)-scattering transforms (conference talk), 7th Croatian Mathematical Congress, Split, June 16, 2022.
• R. Mrazović, Extractors in Paley graphs: a random model (conference talk), 7th Croatian Mathematical Congress, Split, June 17, 2022.
• M. Stipčić, Quantitative bounds for products of simplices in subsets of the unit cube (conference poster), A Young Researchers Symposium in Harmonic Analysis, Edinburgh, June 23-24, 2022.
• R. Mrazović, One-point concentration of the clique number of the random Cayley graph on Fn2 (workshop talk), Combinatorial Constructions Workshop, Zagreb, June 27, 2022.
• V. Kovač, Density theorems for large copies of finite configurations (peer reviewed slides), Short Communications Satellite of the International Congress of Mathematicians 2022 (SCS2022), July 6-14, 2022.
• M. Stipčić, On singular integral forms associated with hypergraphs and products of simplices in subsets of the unit cube (seminar talk), Participating Analysis Seminar, University of California, Los Angeles, September 23, 2022.
• A. Bulj, Fourierova transformacija i višeparametarske maksimalne ocjene (seminar talk), Seminar za funkcionalnu analizu, Zagreb, September 29, 2022.
• R. Mrazović, Circle method and counting transversals in group multiplication tables (conference talk), Representation Theory XVII, Dubrovnik, October 3, 2022.
• M. Stipčić, The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions (seminar talk), Nodal domains and landscape functions, summer school, Kopp, October 4, 2022.
• V. Kovač, Bilinear and trilinear embeddings for complex elliptic operators (online seminar talk), International Prague seminar on function spaces, October 20, 2022.
• A. Bulj, Asymptotic growth of Lp bounds of unimodular multipliers (seminar talk), Participating Analysis Seminar, University of California, Los Angeles, November 4, 2022.
• V. Kovač, Lower bounds for the Lp norms of some Fourier multipliers (workshop talk), Extremal Problems in Harmonic Analysis, Convexity, and Bellman Functions, ICERM, Providence, December 2, 2022.

Participation in conferences, workshops, and summer schools

In the 1st project year

• M. Stipčić, Probabilistic Aspects of Harmonic Analysis (conference), Będlewo, May 6-12, 2018.
• M. Stipčić, Mini-courses in Mathematical Analysis (summer school), Padova, July 2-6, 2018.
• J. Rupčić, Harmonic Analysis and PDE Day (workshop), Birmingham, September 11, 2018.
• V. Kovač, ApplMath18 (conference), Šibenik, September 17-20, 2018.

In the 2nd project year

• V. Kovač, Follow-up Workshop to Trimester Program "Harmonic Analysis and Partial Differential Equations" (workshop), Bonn, May 6-10, 2019.
• K. A. Škreb, Harmonic Analysis and PDEs (HAPDE2019) (conference), Helsinki, June 3-7, 2019.
• M. Stipčić, Mini-courses in Mathematical Analysis (summer school), Padova, June 24-28, 2019.

In the 3rd project year

• V. Kovač, Zagreb Workshop on Operator Theory 2020 (online workshop), Zagreb, June 29-30, 2020.
• M. Stipčić, Zagreb Workshop on Operator Theory 2020 (online workshop), Zagreb, June 29-30, 2020.
• A. Bulj, LMS Harmonic Analysis and PDEs Meeting (online workshop), December 11, 2020.
• V. Kovač, LMS Harmonic Analysis and PDEs Meeting (online workshop), December 11, 2020.

In the 4th project year

• A. Bulj, Fourier Restriction Online 2021 (online workshop), March 4-6 and 11-13, 2021.
• V. Kovač, Croatian-German meeting on analysis and mathematical physics (online conference), March 22-25, 2021.
• A. Bulj, Contemporary Analysis and Its Applications (online conference), June 16-18, 2021.
• A. Bulj, 8th European Congress of Mathematics (online conference), June 20-26, 2021.
• R. Mrazović, 8th European Congress of Mathematics (online conference), June 20-26, 2021.
• V. Kovač, Brijuni Applied Mathematics Workshop 2021 (conference), July 4-10, 2021.
• A. Bulj, Brascamp-Lieb inequalities (summer school), Kopp, September 26-October 1, 2021.
• V. Kovač, 2021 Canadian Mathematical Society Winter Meeting (online conference), December 2-7, 2021.

In the 5th project year

• M. Stipčić, 2022 Workshop on Operator Theory with an Eye on Linear Systems and Hypercomplex Analysis (conference), Orange, May 28-June 4, 2022.
• A. Bulj, 7th Croatian Mathematical Congress (conference), Split, June 15-18, 2022.
• R. Mrazović, 7th Croatian Mathematical Congress (conference), Split, June 15-18, 2022.
• J. Rupčić, 7th Croatian Mathematical Congress (conference), Split, June 15-18, 2022.
• M. Stipčić, 7th Croatian Mathematical Congress (conference), Split, June 15-18, 2022.
• M. Stipčić, A Young Researchers Symposium in Harmonic Analysis (symposium), Edinburgh, June 23-24, 2022.
• R. Mrazović, Combinatorial Constructions Workshop (workshop), Zagreb, June 27-29, 2022.
• M. Stipčić, Fourier Analysis @200 (workshop), Edinburgh, July 6-14, 2022.
• M. Stipčić, Nodal domains and landscape functions (summer school), Kopp, October 2-7, 2022.
• R. Mrazović, Representation Theory XVII (conference), Dubrovnik, October 3-8, 2022.
• V. Kovač, Extremal Problems in Harmonic Analysis (workshop), ICERM, Providence, November 28-December 2, 2022.
• M. Stipčić, Joint Mathematics Meeting 2023 (conference), Boston, January 4-7, 2023.

Organization of workshops

In the 2nd project year

• Probability, Analysis, and Applications (international workshop with 17 invited talks and 5 contributed talks), Zagreb, June 12-14, 2019.
• Analysis and Combinatorics (small workshop), Zagreb, October 21-25, 2019.

In the 4th project year

• Singular integrals in higher dimensions (small workshop), Zagreb, December 16-18, 2021.

Purchased subscriptions, equipment, and software

In the 1st project year

• Books (covered by HRZZ)
• 2 laptop computers (covered by HRZZ)
• 2 desktop computers with monitors (covered by PMF)
• 1 MATLAB university licence (covered by HRZZ)
• 1 Wolfram Mathematica university licence (covered by HRZZ)

In the 2nd project year

• Subscription to journal Communications on Pure and Applied Analysis (covered by HRZZ)
• Subscription to journal Journal d'Analyse Mathématique (covered by PMF)
• Books (covered by HRZZ)
• 2 Wolfram Mathematica university licences (covered by HRZZ)
• 5 Overleaf Collaborator licences (covered by HRZZ)
• 1 WinEdt educational licence (covered by HRZZ)
• 2 memberships for the Croatian Mathematical Society (covered by HRZZ)

In the 3rd project year

• Subscription to journal Communications on Pure and Applied Analysis (covered by HRZZ)
• Subscription to journal Journal d'Analyse Mathématique (covered by PMF)
• 2 Wolfram Mathematica university licences (covered by HRZZ)
• 5 Overleaf Collaborator licences (covered by HRZZ)
• 1 membership for the American Mathematical Society (covered by HRZZ)
• 3 memberships for the Croatian Mathematical Society (covered by HRZZ)

In the 4th project year

• Subscription to journal Communications on Pure and Applied Analysis (covered by HRZZ)
• Subscription to journal Journal d'Analyse Mathématique (covered by PMF)
• 1 laptop computer (covered by HRZZ)
• 5 Overleaf Collaborator licences (covered by HRZZ)
• 1 membership for the American Mathematical Society (covered by HRZZ)
• 4 memberships for the Croatian Mathematical Society (covered by HRZZ)

Announcements

February 28, 2023	Mario Stipčić accepted a preceptor position at Harvard University. Congratulations!
July 20, 2022	The work of Rudi Mrazović [10] was discussed on this blog.
December 20, 2021	Bruno Predojević joined the project. Welcome!
April 19, 2021	Mario Stipčić accepted a postdoctoral fellowship at Chapman University. Congratulations!
December 14, 2020	Mario Stipčić defended Ph.D. dissertation. Congratulations!
March 1, 2020	Aleksandar Bulj joined the project. Welcome!
February 28, 2020	Mario Stipčić won the prize for the best presentation at the 4th symposium of students of doctoral studies at the Faculty of science. Congratulations!
April 22, 2019	The work of Vjekoslav Kovač [3] was discussed on this blog.
April 8, 2019	Vjekoslav Kovač received the Fulbright Visiting Research award. He will use it to spend a year at the Georgia Institute of Technology.
December 3, 2018	Public presentation of the project will be held on Wednesday, December 12, 2018 at 11:15 in room A102, Department of Mathematics, Faculty of Science.
March 1, 2018	Kristina Škreb accepted a postdoctoral fellowship at the University of Toulouse. Congratulations!