arXiv subjects
Publications and source records attributed to Camil Muscalu.
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}.
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.
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.
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.
We prove that for any $L^Q$-valued Schwartz function $f$ defined on $\mathbb{R}^d$, one has the multiple vector-valued, mixed norm estimate $$ \| f \|_{L^P(L^Q)} \lesssim \| S f \|_{L^P(L^Q)} $$ valid for every $d$-tuple $P$ and every $n$-tuple $Q$ satisfying $0 < P, Q < \infty$ componentwise. Here $S:= S_{d_1}\otimes ... \otimes S_{d_N}$ is a tensor product of several Littlewood-Paley square functions $S_{d_j}$ defined on arbitrary Euclidean spaces $\mathbb{R}^{d_j}$ for $1\leq j\leq N$, with the property that $d_1 + ... + d_N = d$. This answers a question that came up implicitly in our recent works and completes in a natural way classical results of the Littlewood-Paley theory. The proof is based on the \emph{helicoidal method} introduced by the authors.
Using exclusively the localized estimates upon which the helicoidal method was built, we show how sparse estimates can also be obtained. This approach yields a sparse domination for multiple vector-valued extensions of operators as well. We illustrate these ideas for an $n$-linear Fourier multiplier whose symbol is singular along a $k$-dimensional subspace of $Γ=\lbrace ξ_1+\ldots+ξ_{n+1}=0 \rbrace$, where $k<\dfrac{n+1}{2}$, and for the variational Carleson operator.
This is an expository paper on the helicoidal method, a tool designed for proving multiple vector-valued inequalities for operators in harmonic analysis, which is based on stopping times and localizations. As it turns out, the local estimate can be used for proving sparse domination for the scalar operator and its multiple vector-valued extensions, and hence also weighted estimates.
We extend the helicoidal method that we previously developed to the quasi-Banach context, proving in this way multiple Banach and quasi-Banach vector-valued inequalities for paraproducts $Π$ and for the bilinear Hilbert transform $BHT$. As an immediate application, we obtain mixed norm estimates for $Π\otimes Π$ in the whole range of Lebesgue exponents. One of the novelties in the quasi-Banach framework (that is, when $0<r<1$), which we expect to be useful in other contexts as well, is the "linearization" of the operator $ \left( \sum_k | T(f_k, g_k) |^r \right)^{1/r}$ by dualizing its weak-$L^p$ quasinorms through $L^r$. Another important role is played by the sharp evaluation of the operatorial norm $\| T_{I_0}(f \cdot \mathbf{1}_F, g \cdot \mathbf{1}_G) \cdot \mathbf{1}_{H'}\|_r$, which is obtained by dualizing the weak-$L^p$ quasinorms through $L^τ$, with $τ\leq r$. In the Banach case, the linearization of the operator and the sharp estimates for the localized operatorial norm can be both achieved through the classical (generalized restricted type) $L^1$ dualization.
We develop a new method of proving vector-valued estimates in harmonic analysis, which we like to call "the helicoidal method". As a consequence of it, we are able to give affirmative answers to some questions that have been circulating for some time. In particular, we show that the tensor product $BHT \otimes Π$ between the bilinear Hilbert transform $BHT$ and a paraproduct $Π$ satisfies the same $L^p$ estimates as the $BHT$ itself, solving completely a problem introduced in a paper of Muscalu, Pipher, Tao and Thiele. Then, we prove that for "locally $L^2$ exponents" the corresponding vector valued $\overrightarrow{BHT}$ satisfies (again) the same $L^p$ estimates as the $BHT$ itself. Before the present work there was not even a single example of such exponents. Finally, we prove a bi-parameter Leibniz rule in mixed norm $L^p$ spaces, answering a question of Kenig in nonlinear dispersive PDE.
We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.
This article is the second in a series of three papers, whose scope is to give new proofs to the well known theorems of Calderón, Coifman, McIntosh and Meyer. Here we treat the case of the Cauchy integral on Lipschitz curves and some of its generalizations.
In the first part of the paper we prove a bi-parameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the Kato-Ponce inequality in nonlinear PDE, obtaining a fractional Leibnitz rule for derivatives in the $x_1$ and $x_2$ directions simultaneously. Then, we show that the double bilinear Hilbert transform does not satisfy any $L^p$ estimates.
We prove that for every integer $n\geq 4$, the $n$-linear operator whose symbol is given by a product of two generic symbols of $n$-linear Hilbert transform type, does not satisfy any $L^p$ estimates similar to those in Hölder inequality. Then, we extend this result to multi-linear operators whose symbols are given by a product of an arbitrary number of generic symbols of $n$-linear Hilbert transform kind. As a consequence, under the same assumption $n\geq 4$,these immediately imply that for any $1< p_1, ..., p_n \leq \infty$ and $0<p<\infty$ with $1/p_1 + ... + 1/p_n = 1/p$, there exist non-degenerate subspaces $Γ\subseteq \mathbb{R}^n$ of maximal dimension $n-1$, and Mikhlin symbols $m$ singular along $Γ$, for which the associated $n$-linear multiplier operators $T_m$ do not map $L^{p_1}\times ... \times L^{p_n}$ into $L^p$. These counterexamples are in sharp contrast with the bi-linear case, where similar operators are known to satisfy many such $L^p$ estimates.
This article is the first in a series of three papers, whose scope is to give new proofs to the well known theorems of Calderón, Coifman, McIntosh and Meyer. Here we treat the case of the first commutator and some of its generalizations.
This article is the last in a series of three papers, whose scope is to give new proofs to the well known theorems of Calderón, Coifman, McIntosh and Meyer. Here we extend the results of the previous two papers to the polydisc setting. In particular, we solve completely an open question of Coifman from the early eighties.
We generalize a family of variation norm estimates of Lepingle with endpoint estimates of Bourgain and Pisier-Xu to a family of variational estimates for paraproducts, both in the discrete and the continuous setting. This expands on work of Friz and Victoir, our focus being on the continuous case and an expanded range of variation exponents.
We describe the theory of "flag paraproducts" and their relationship with the field of differential equations.
In this article we prove that the $n$-linear operator whose symbol is the characteristic function of the simplex $Δ_n = ξ_1 < ... < ξ_n$ is bounded from $L^2 \times ... \times L^2$ into $L^{2/n}$, generalizing in this way our previous work on the "bi-est" operator (which corresponds to the case $n=3$) as well as Lacey-Thiele theorem on the bi-linear Hilbert transform (which corresponds to the case $n=2$).