About

Camil Muscalu is a professor in the Department of Mathematics at Cornell University. He holds a Ph.D. from Brown University, obtained in 2000. His research focuses on harmonic analysis and partial differential equations, particularly the study of Fourier series and their applications to understanding physical phenomena. Muscalu is interested in the process of discovery and analysis of new mathematical objects, such as iterated Fourier series, and their deep connections to other areas of mathematics, including the theory of multiple zeta functions, number theory, and physics. His work involves exploring fundamental questions like the convergence of these series almost everywhere and their relation to natural phenomena, contributing to the broader understanding of harmonic analysis and its applications.

Research topics

• Computer Science
• Mathematics
• Law
• Geometry
• Combinatorics
• Pure mathematics

Selected publications

• A new approach to the Fourier extension problem for the paraboloid

  Analysis & PDE · 2024-10-12

  articleOpen access1st authorCorresponding

  We propose a new approach to the Fourier restriction conjectures. It is based on a discretization of the Fourier extension operators in terms of quadratically modulated wave packets. Using this new point of view, and by combining natural scalar and mixed norm quantities from appropriate level sets, we prove that all the L²-based k-linear extension conjectures are true up to the endpoint for every 1≤k≤d+1 if one of the functions involved is a full tensor. We also introduce the concept of <i>weak transversality</i>, under which we show that all conjectured L²-based multilinear extension estimates are still true up to the endpoint, provided that one of the functions involved has a weaker tensor structure, and we prove that this result is sharp. Under additional tensor hypotheses, we show that one can improve the conjectured threshold of these problems in some cases. In general, the largely unknown multilinear extension theory beyond L² inputs remains open even in the bilinear case; with this new point of view, and still under the previous tensor hypothesis, we obtain the near-restriction target for the k-linear extension operator if the inputs are in a certain L^p space for p sufficiently large. The proof of this result is adapted to show that the k-fold product of linear extension operators (no transversality assumed) also “maps near restriction” if one input is a tensor. Finally, we exploit the connection between the geometric features behind the results of this paper and the theory of Brascamp–Lieb inequalities, which allows us to verify a special case of a conjecture by Bennett, Bez, Flock and Lee.

  PublisherOA PDFDOI

• Mixed‐norm estimates via the helicoidal method

  Mathematika · 2024-04-21 · 1 citations

  articleOpen accessSenior author

  Abstract We prove multiple vector‐valued and mixed‐norm estimates for multilinear operators in , more precisely for multilinear operators associated to a symbol singular along a ‐dimensional space and for multilinear variants of the Hardy‐Littlewood maximal function. When the dimension , the input functions are not necessarily in and can instead be elements of mixed‐norm spaces . Such a result has interesting consequences especially when spaces are involved. Among these, we mention mixed‐norm Loomis‐Whitney‐type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy “purely mixed‐norm estimates” and no classical estimates. Relying on previous estimates implied by the helicoidal method, we also prove (non‐mixed‐norm) estimates for generic singular Brascamp‐Lieb‐type inequalities.

  PublisherOA PDFDOI

• A New Proof of Strichartz Estimates for the Schrödinger Equation in $$2+1$$ Dimensions

  Applied and numerical harmonic analysis · 2023-01-01 · 1 citations

  book-chapter1st author
  PublisherDOI

• A new proof of Strichartz estimates for the Schrödinger equation in $2+1$ dimensions

  arXiv (Cornell University) · 2022-12-11

  preprintOpen access1st authorCorresponding

  This note presents a new proof of the well-known Strichartz estimates for the Schrödinger equation in $2+1$ dimensions, building on ideas from our recent work \cite{MO}.

  PublisherOA PDFDOI

• Five-linear singular integral estimates ofBrascamp–Lieb-type

  Analysis & PDE · 2022-09-03 · 3 citations

  preprintOpen access1st authorCorresponding

  We prove the full range of estimates for a five-linear singular integral of Brascamp-Lieb type. The study is methodology-oriented with the goal to develop a sufficiently general technique to estimate singular integral variants of Brascamp-Lieb inequalities that do not obey Hölder scaling. The invented methodology constructs localized analysis on the entire space from local information on its subspaces of lower dimensions and combines such tensor-type arguments with the generic localized analysis. A direct consequence of the boundedness of the five-linear singular integral is a Leibniz rule which captures nonlinear interactions of waves from transversal directions.

  PublisherOA PDFDOI

• Multiple vector-valued, mixed norm estimates for Littlewood-Paley square functions

  Publicacions Matemàtiques · 2022 · 6 citations

  Senior authorCorresponding
  • Mathematics
  • Pure mathematics
  • Combinatorics

  International audience

  PublisherOA PDFDOI

• Sparse domination via the helicoidal method

  Revista Matemática Iberoamericana · 2021 · 14 citations

  Senior authorCorresponding
  • Computer Science
  • Computer Science

  Using exclusively the localized estimates upon which the helicoidal method was built by the authors, we show how sparse estimates can also be obtained. This approach yields a sparse domination for scalar and multiple vector-valued extensions of operators alike. We illustrate these ideas for an n -linear Fourier multiplier whose symbol is singular along a k -dimensional subspace of \Gamma=\{\xi_1+\cdots+\xi_{n+1}=0\} , where k &lt; (n+1)/{2} , and for the variational Carleson operator.

  PublisherDOI

• Strichartz, Robert

  eCommons (Cornell University) · 2021-01-01

  other

  Also available as a printed booklet and from the Dean of Faculty website https://theuniversityfaculty.cornell.edu/

  Publisher

• A new approach to the Fourier extension problem for the paraboloid

  arXiv (Cornell University) · 2021-10-24

  preprintOpen access1st authorCorresponding

  We propose a new approach to the Fourier restriction conjectures. It is based on a discretization of the Fourier extension operators in terms of quadratically modulated wave packets. Using this new point of view, and by combining natural scalar and mixed norm quantities from appropriate level sets, we prove that all the $L^{2}$-based $k$-linear extension conjectures are true up to the endpoint for every $1 \leq k \leq d+1$ if one of the functions involved is a full tensor. We also introduce the concept of \textit{weak transversality}, under which we show that all conjectured $L^{2}$-based multilinear extension estimates are still true up to the endpoint provided that one of the functions involved has a weaker tensor structure, and we prove that this result is sharp. Under additional tensor hypotheses, we show that one can improve the conjectured threshold of these problems in some cases. In general, the largely unknown multilinear extension theory beyond $L^{2}$ inputs remains open even in the bilinear case; with this new point of view, and still under the previous tensor hypothesis, we obtain the near-restriction target for the $k$-linear extension operator if the inputs are in a certain $L^{p}$ space for $p$ sufficiently large. The proof of this result is adapted to show that the $k$-fold product of linear extension operators (no transversality assumed) also ``maps near restriction" if one input is a tensor. Finally, we exploit the connection between the geometric features behind the results of this paper and the theory of Brascamp-Lieb inequalities, which allows us to verify a special case of a conjecture by Bennett, Bez, Flock and Lee.

  PublisherDOI

• Mixed-norm estimates via the helicoidal method

  arXiv (Cornell University) · 2020-07-02 · 1 citations

  preprintOpen accessSenior author

  We prove multiple vector-valued and mixed-norm estimates for multilinear operators in $\rr R^d$, more precisely for multilinear operators $T_k$ associated to a symbol singular along a $k$-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension $d \geq 2$, the input functions are not necessarily in $L^p(\rr R^d)$ and can instead be elements of mixed-norm spaces $L^{p_1}_{x_1} \ldots L^{p_d}_{x_d}$. Such a result has interesting consequences especially when $L^\infty$ spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy ``purely mixed-norm estimates" and no classical $L^p$ estimates. Relying on previous estimates implied by the helicoidal method, we also prove (non-mixed-norm) estimates for generic singular Brascamp-Lieb-type inequalities.

  PublisherOA PDFDOI

Recent grants

• Topics in Multi-linear Harmonic Analysis

  NSF · $180k · 2007–2011

• Iterated Fourier Series and Integrals

  NSF · $240k · 2015–2019

Frequent coauthors

• Christoph Thiele

  49 shared

• Terence Tao

  University of California, Los Angeles

  48 shared

• Cristina Benea

  28 shared

• Wilhelm Schlag

  Yale University

  27 shared

• Jill Pipher

  19 shared

• Itamar Oliveira

  University of Birmingham

  7 shared

• Yen Do

  University of Virginia

  4 shared

• Yujia Zhai

  3 shared