We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some $n$-linear analogues for $n<d$.
We give an essentially self-contained proof of Guth's recent endpoint multilinear Kakeya theorem which avoids the use of somewhat sophisticated algebraic topology, and which instead appeals to the Borsuk-Ulam theorem.
Bennett, Carbery, and Tao formulated an n-linear analogue of the Kakeya conjecture in R^n. They proved the conjecture except for the endpoint case. We prove the endpoint case.
We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form $R^\varepsilon$ (or $\log^{O(1)} R$) for some scale factor $R$. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long a…
Suppose that $(X,d,μ)$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f \colon X \to \mathbb R$, $\operatorname{Lip}(f,\cdot)$ is dominated by every upper gradient of $f$. We show that $X$ is a Lipschitz differentiability space, and the differentiable structure of $X$ has dimension at most $\dim_{\mathrm{H}} X$. Since our assumptions are satisfied whenever $X$ is doubling and satisfies a Poincaré inequality, we thus obtain a new proof of Cheeger's generalisation of Rademacher's theorem. Our approach uses Guth's multilinear Kakeya inequality for neighbour…
The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.
We show that a certain conjectured optimal reverse Littlewood- Paley inequality would, if true, imply sharp results for the Kakeya maximal function, the Bochner-Riesz means and the Fourier restriction operator.
Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the $(L^p,L^p)$ spherical restriction theorem of Wolff from $p > 42/11$ to $p > 34/9$, and also obtain a sharp $(L^p,L^q)$ spherical restriction theorem for $q > 4 - 5/27$.
We provide an alternative and self contained proof of the main result of Bennett, Carbery, Tao regarding the multilinear restriction estimate. The approach is inspired by the recent result of Guth about the Kakeya version of multilinear restriction estimate. At lower levels of multilinearity we provide a refined estimate in the context of small support for one of the terms involved.
In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp $L^2$ estimate on the Schrödinger maximal function in $\Bbb R^n$, $n \geq 2$. In this paper, we show that the Du-Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in $\Bbb R^n$ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in $\Bbb R^2$.
Here we show some results related with Kakeya conjecture which says that for any integer $n\geq 2$, a set containing line segments in every dimension in $\mathbb{R}^n$ has full Hausdorff dimension as well as box dimension. We proved here that the Kakeya books, which are Kakeya sets with some restrictions on positions of line segments have full box dimension. We also prove here a relation between the projection property of Kakeya sets and the Kakeya conjecture. If for any Kakeya set $K\subset\mathbb{R}^n$, the Hausdorff dimension of orthogonal projections on $k\leq n$ subspaces is independent o…
In the finite field setting, we show that the restriction conjecture associated to any one of a large family of $d=2n+1$ dimensional quadratic surfaces implies the $n+1$ dimensional Kakeya conjecture (Dvir's theorem). This includes the case of the paraboloid over finite fields in which $-1$ \emph{is} a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein-Tomas exponent in this setting.
The first result in this paper provides a very general $ε$-removal argument for the multilinear restriction estimate. The second result provides a refinement of the multilinear restriction estimate in the case when some terms have appropriate localization properties; this generalizes a prior result of the author.
We investigate analogues for curves of the Kakeya problem for straight lines. These arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. Using some of the geometric and arithmetic techniques developed for the straight line case by Bourgain, Wolff, Katz and Tao, we are able to prove positive results for families of parabolas whose coefficients satisfy certain algebraic conditions.
In this paper we establish the optimal multilinear restriction estimate for n-1 hypersurfaces with some curvature, where $n$ is the dimension of the underlying space. The result is sharp up to the endpoint and the role of curvature is made precise in terms of the shape operator.
The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these problems on finite fields. The restriction problem then becomes a question about general bounds for certain types of exponential sums, while the Kakeya problem has an algebraic geometry flavor, asking for the extent to which lines in different directions can overlap. In many cases the Euclidean arguments carry over easily to the finite setting (and are in fact some…
We apply the Bennett-Carbery-Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L^p estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for Hormander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for Hormander-type oscillatory integral operators in even dimensions. These last oscillatory estimates…
Bennett, Carbery and Tao considered the $k$-linear restriction estimate in $\mathbb{R}^{n+1}$ and established the near optimal $L^\frac2{k-1}$ estimate under transversality assumptions only. We have shown that the trilinear restriction estimate improves its range of exponents under some curvature assumptions. In this paper we establish almost sharp multilinear estimates for a class of hypersurfaces with curvature for $4 \leq k \leq n$. Together with previous results in the literature, this shows that curvature improves the range of exponents in the multilinear restriction estimate at all level…
We improve the $L^{p}\rightarrow L^p$ restriction estimate in $\mathbb{R}^3$ to the range $p>3+3/14$, based on some Kakeya type incidence estimates and the refined decoupling theorem.
Christ, Li, Tao, and Thiele have established multilinear oscillatory integral operator inequalities under severe dimensional restrictions. These restrictions are relaxed substantially in the present paper, but are by no means eliminated. The simple method used here has the advantage of yielding power type bounds, but cannot be expected to extend to the general case.
This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in $\mathbb{R}^3$, what we expect to hold due to the maximal Kakeya operator conjecture, as well as…
The restriction problem is better understood for hypersurfaces and recent progresses have been made by bilinear and multilinear approaches and most recently polynomial partitioning method which is combined with those estimates. However, for surfaces with codimension bigger than 1, bilinear and multilinear generalization of restriction estimates are more involved and effectiveness of these multilinear estimates is not so well understood yet. Regarding the restriction problem for the surfaces with codimensions bigger than 1, the current state of the art is still at the level of $TT^*$ method whi…
We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
We prove that recent breaking by Zahl of the $\frac32$ barrier in Wolff's estimate on the Kakeya maximal operator in $\mathbb R^4$ leads to improving the $\frac{14}{5}$ threshold for the restriction problem for the paraboloid in $\mathbb R^4$. One of the ingredients is a new trilinear estimate. The proofs are deliberately presented in a nontechnical and concise format, so as to make the arguments more readable and focus attention on the key tools.
We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.
This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction and geometric averaging operators. It is shown that, under very general circumstances, the boundedness of such functionals is equivalent to a geometric inequality for measures which has recently appeared in work by D. Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and Seeger (J. Aust. Math. Soc., 85, 2008).
The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and other hypersurfaces. As the field is quite large and has so many applications, it will be impossible to completely survey the field, but we will try to at least give the main ideas and developments in this area. The restriction problem are connected to many other conjectures, notably the Kakeya and Bochner-Riesz conjectures, as well as PDE conjectures such as t…
In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an application, we prove that at most $δ^{-2-\varepsilon}$ direction-separated $δ$-tubes can be contained in the $δ$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$. This result leads to improved bounds on the restriction and Kakeya problems in $\mathbb{R}^4$. Combined with previous work of…
In [C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6(1):1-15, 1999], the following type of multilinear fractional integral \[ \int_{\mathbb{R}^{mn}} \frac{f_1(l_1(x_1,\ldots,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\ldots,x_m,x))}{(|x_1|+\ldots+|x_m|)^λ} dx_1\ldots dx_m \] was studied, where $l_i$ are linear maps from $\mathbb{R}^{(m+1)n}$ to $\mathbb{R}^n$ satisfying certain conditions. They proved the boundedness of such multilinear fractional integral from $L^{p_1}\times \ldots \times L^{p_{m+1}}$ to $L^q$ when the indices satisfy the homogeneity c…
In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere $$\big\|\widehat{f σ}\big\|_{L^{p}(\mathbb{R}^{d})} \lesssim \|f\|_{L^{q}(\mathbb{S}^{d-1},σ)}$$ in the cases $(d,p,q) = (d,2k, q)$ with $d,k \in \mathbb{N}$ and $q\in \mathbb{R}^+ \cup \{\infty\}$ satisfying: (a) $k = 2$, $q \geq 2$ and $3 \leq d\leq 7$; (b) $k = 2$, $q \geq 4$ and $d \geq 8$; (c) $k \geq 3$, $q \geq 2k$ and $d \geq 2$. We also prove a sharp multilinear weighted restriction inequality, with weight related to the $k$-fold convoluti…
We use the method of induction-on-scales to prove certain diffeomorphism invariant nonlinear Brascamp--Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru--Herr--Tataru and Bennett--Carbery--Wright.
In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain's level set result on Strichartz estimates associated with Schrödinger equations on torus. Some sharp estimates on $L^{\frac{2(d+2)}{d}}$ norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on $L^2(\mathbb Z)$.
Consider the Fourier restriction operator associated to a curve in $R^d$, $d\ge 3$. We prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve. An essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector-valued functions.
Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition be…
We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which (up to $N^ε$ losses) implies the full range of expected $L^p_{x,t}$ Strichartz estimates for both classical and irrational tori. Another one is an improvement in the range for the discrete restriction theory for lattice points on the sphere. Various applications in Additive Combinatorics, Incidence Geometry and Number Theory are also disc…
