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Loukas Grafakos

Publications and source records attributed to Loukas Grafakos.

53 records · Page 3Linked to original sources
arXiv2015

Multilinear Fourier Multipliers with Minimal Sobolev Regularity, II

We provide characterizations for boundedness of multilinear Fourier operators on Hardy-Lebesgue spaces with symbols locally in Sobolev spaces. Let $H^q(\mathbb R^n)$ denote the Hardy space when $0<q\le 1$ and the Lebesgue space $L^q(\mathbb R^n)$ when $1<q\le \infty$. We find optimal conditions on $m$-linear Fourier multiplier operators to be bounded from $H^{p_1}\times \cdots \times H^{p_m}$ to $L^p$ when $1/p=1/p_1+\cdots +1/p_m$ in terms of local $L^2$-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderón and Torchinsky [http://www.sciencedirect.com/science/article/pii/S0001870877800169] and of the bilinear results of Miyachi and Tomita [http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=29&iss=2&rank=4]. The extension to general $m$ is significantly more complicated both technically and combinatorially, the optimal Sobolev space smoothness required of the symbol depends on the Hardy-Lebesgue exponents and is constant on various convex simplices formed by configurations of $m2^{m-1} +1$ points in $[0,\infty)^m$.

math.AP↗
arXiv2015

Multilinear Fourier Multipliers with Minimal Sobolev Regularity, I

We find optimal conditions on $m$-linear Fourier multipliers to give rise to bounded operators from a product of Hardy spaces $H^{p_j}$, $0<p_j\le 1$, to Lebesgue spaces $L^p$. The conditions we obtain are necessary and sufficient for boundedness and are expressed in terms of $L^2$-based Sobolev spaces. Our results extend those obtained in the linear case ($m=1 $) by Calderón and Torchinsky [http://www.sciencedirect.com/science/article/pii/S0001870877800169] and in the bilinear case ($m=2$) by Miyachi and Tomita [http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=29&iss=2&rank=4]. We also prove a coordinate-type Hörmander integral condition which we use to obtain certain extreme cases.

math.AP↗
arXiv2015

Certain Multi(sub)linear square functions

Let $d\ge 1, \ell\in\Z^d$, $m\in \mathbb Z^+$ and $θ_i$, $i=1,\dots,m $ are fixed, distinct and nonzero real numbers. We show that the $m$-(sub)linear version below of the Ratnakumar and Shrivastava \cite{RS1} Littlewood-Paley square function $$T(f_1,\dots , f_m)(x)=\Big(\sum\limits_{\ell\in\Z^d}|\int_{\mathbb{R}^d}f_1(x-θ_1 y)\cdots f_m(x-θ_m y)e^{2πi \ell \cdot y}K (y)dy|^2\Big)^{1/2} $$ is bounded from $L^{p_1}(\mathbb{R}^d) \times\cdots\times L^{p_m}(\mathbb{R}^d) $ to $L^p(\mathbb{R}^d) $ when $2\le p_i<\infty$ satisfy $1/p=1/p_1+\cdots+1/p_m$ and $1\le p<\infty$. Our proof is based on a modification of an inequality of Guliyev and Nazirova \cite{GN} concerning multilinear convolutions.

math.CA↗
arXiv2013

A remark on an endpoint Kato-Ponce inequality

This note introduces bilinear estimates intended as a step towards an $L^\infty$-endpoint Kato-Ponce inequality. In particular, a bilinear version of the classical Gagliardo-Nirenberg interpolation inequalities for a product of functions is proved.

math.AP↗
arXiv2013

Multilinear generalized Radon transforms and point configurations

We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving $(k+1)$-point configurations in geometric measure theory, with $k \ge 2$, including the distribution of simplices, volumes and angles determined by the points of fractal subsets $E \subset {\Bbb R}^d$, $d \ge 2$. If $T_k(E)$ denotes the set of noncongruent $(k+1)$-point configurations determined by $E$, we show that if the Hausdorff dimension of $E$ is greater than $d-\frac{d-1}{2k}$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. This compliments previous work on the Falconer conjecture (\cite{Erd05} and the references there), as well as work on finite point configurations \cite{EHI11,GI10}. We also give applications to Erdös-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in \cite{EIT11}.

math.CA↗
arXiv2013

The Fourier transform of multiradial functions

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form $Φ(x)=ϕ(|x_1|, \dots, |x_m|)$, $x_i\in \mathbf R^{n_i}$, in terms of the Fourier transform of the function $ϕ$ on $\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}$, where $r_i$ is either 1 or 2.

math.CA↗
arXiv2013

The bilinear Bochner-Riesz problem

Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|ξ|^2-|η|^2)^δ_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\times L^2 $ into $L^1$ with minimal smoothness, i.e., any $δ>0$, and we obtain estimates for other pairs of spaces for larger values of $δ$. Our study is broad enough to encompass general bilinear multipliers $m(ξ,η)$ radial in $ξ$ and $η$ with minimal smoothness, measured in Sobolev space norms. Our results are based on a variety of techniques, that include Fourier series expansions, orthogonality, and bilinear restriction and extension theorems.

math.CA↗
arXiv2013

The Kato-Ponce Inequality

In this article we develop a simplistic approach to revisit the classical Kato-Ponce inequality, which is also known as 'fractional Leibniz rule.' As a consequence, we derive the validity of this inequality even in quasi-Banach spaces $L^p$ for $p<1$ with a certain restrictions on the indices. Also, we display the sharpness of this restriction by means of a counter-example. Finally, we also prove a multi-parameter variant of the inequality, which allows for partial fractional derivatives on $R^n$.

math.AP↗
arXiv2013

On Fourier transforms of radial functions and distributions

We find a formula that relates the Fourier transform of a radial function on $\mathbf{R}^n$ with the Fourier transform of the same function defined on $\mathbf{R}^{n+2}$. This formula enables one to explicitly calculate the Fourier transform of any radial function $f(r)$ in any dimension, provided one knows the Fourier transform of the one-dimensional function $t\to f(|t|)$ and the two-dimensional function $(x_1,x_2)\to f(|(x_1,x_2)|)$. We prove analogous results for radial tempered distributions.

math.CA↗
arXiv2011

The multilinear strong maximal function

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Certain bilinear interpolation results between distributional estimates, such as that obtained for the multivariable strong maximal function, are also proved.

math.CA↗
arXiv2005

On maximal functions for Mikhlin-Hoermander multipliers

Given Mikhlin-Hörmander multipliers $m_i$, $i=1,..., N$, with uniform estimates we prove an optimal $\sqrt{\log(N+1)}$ bound in $L^p$ for the maximal function $\sup_i|\cF^{-1}[m_i\hat f]|$ and related bounds for maximal functions generated by dilations.

math.CA↗
arXiv2002

L^p bounds for a maximal dyadic sum operator

We prove $L^p$ bounds in the range $1<p<\infty$ for a maximal dyadic sum operator on $\rn$. This maximal operator provides a discrete multidimensional model of Carleson's operator. Its boundedness is obtained by a simple twist of the proof of Carleson's theorem given by Lacey and Thiele, adapted in higher dimensions by Pramanik and Terwilleger. In dimension one, the $\lp$ boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunt's extension of Carleson's theorem on almost everywhere convergence of Fourier integrals.

math.CA↗
arXiv2001

Multilinear interpolation between adjoint operators

Multilinear interpolation is a powerful tool used in obtaining strong type boundedness for a variety of operators assuming only a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak $L^q$ estimate for a single index $q$ (which may be less than one) and that all the adjoints of the multilinear operator are of similar nature, and thus they also satisfy the same weak $L^q$ estimate. Under this assumption, in this expository note we give a general multilinear interpolation theorem which allows one to obtain strong type boundedness for the operator (and all of its adjoints) for a large set of exponents. The key point in the applications we discuss is that the interpolation theorem can handle the case $q \leq 1$. When $q > 1$, weak $L^q$ has a predual, and such strong type boundedness can be easily obtained by duality and multilinear interpolation.

math.FA↗
arXiv2000

The Marcinkiewicz multiplier condition for bilinear operators

This article is concerned with the question of whether Marcinkiewicz multipliers on $\mathbb R^{2n}$ give rise to bilinear multipliers on $\mathbb R^n\times \mathbb R^n$. We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.

math.FA↗
arXiv1997

$L^p$ bounds for singular integrals and maximal singular integrals with rough kernels

Convolution type Calderón-Zygmund singular integral operators with rough kernels $\pv \Om(x)/|x|^n$ are studied. A condition on $\Om$ implying that the corresponding singular integrals and maximal singular integrals map $L^p \to L^p$ for $1<p<\nf$ is obtained. This condition is shown to be different from the condition $\Om\in H^1(\sn)$.

math.FA↗