Inspired by a question of Lie, we study boundedness in subspaces of $L^1(\mathbb{R})$ of oscillatory maximal functions. In particular, we construct functions in $L^1(\mathbb{R})$ which are never integrable under action of our class of maximal functions. On the other hand, we prove that these maximal functions map certain classes of spaces resembling Sobolev spaces into $L^1(\mathbb{R})$ continuously under mild curvature assumptions on the phase $γ$.
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…
We establish endpoint bounds on a Hardy space $H^1$ for a natural class of multiparameter singular integral operators which do not decay away from the support of rectangular atoms. Hence the usual argument via a Journé-type covering lemma to deduce bounds on product $H^1$ is not valid. We consider the class of multiparameter oscillatory singular integral operators given by convolution with the classical multiple Hilbert transform kernel modulated by a general polynomial oscillation. Various characterisations are known which give $L^2$ (or more generally $L^p, 1<p<\infty$) bounds. Here we initi…
The one-dimensional oscillatory integral operator associated to a real analytic phase $S$ is given by $$ T_λf(x) =\int_{-\infty}^\infty e^{iλS(x,y)} χ(x,y) f(y) dy. $$ In this paper, we obtain a complete characterization for the mapping properties of $T_λ$ on $L^p(\mathbb R)$ spaces, namely we prove that $\|T_λ\|_p \lesssim |λ|^{-α}\|f\|_p$ for some $α>0$ if and only if the point $(\frac 1 {αp} , \frac 1 {αp'})$ lies in the reduced Newton polygon of $S$, and this estimate is sharp if and only if it lies on the reduced Newton diagram.
We consider the maximal function of oscillatory integrals and prove a global estimate for radial test functions which is almost sharp with respect to the Sobolev regularity.
For a function $f\in L^p(\Bbb R^d)$, $d\ge 2$, let $A_t f(x)$ be the mean of $f$ over the sphere of radius $t$ centered at $x$. Given a set $E\subset (0,\infty)$ of dilations we prove endpoint bounds for the maximal operator $M_E$ defined by $M_E f(x)=\sup_{t\in E} |A_t f(x)|$.
We expand on counterxamples of Bourgain showing how Nikodym maximal estimates and oscillatory integral estimates can break down in the non-Euclidean case. Our examples show the role that the non-existence of totally geodesic submanifolds can play in these problems.
Let Mf denote the strong maximal function of f on R^n, that is the maximal average of f with respect to n-dimensional rectangles with sides parallel to the coordinate axes. For any dimension n>1 we prove the natural endpoint Fefferman-Stein inequality for M and any strong Muckenhoupt weight w: w({x \in R^n: M f (x) > t}) \lesssim_{w,n} \int_{R^n} |f|/t [1 + (log^+ |f|/t)^{n-1}] Mw. This extends the corresponding two-dimensional result of T. Mitsis.
We study the boundedness problem for maximal operators in 3-dimensional Euclidean space associated to hypersurfaces given as the graph of $c+f$, where $f$ is a mixed homogeneous function which is smooth away from the origin and $c$ is a constant. Our result generalizes a corresponding theorem on mixed homogeneous polynomial functions by A. Iosevich and E. Sawyer.
In this paper we study some questions about the continuity of classical and fractional maximal operators in the Sobolev space $W^{1,1}$, in both continuous and discrete setting, giving a positive answer to two questions posed recently, one of them regarding the continuity of the map $f \mapsto \big(\widetilde M_βf\big)'$ from $W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$, for $q=\frac{1}{1-β}$. Here $\widetilde M_β$ denotes the non-centered fractional maximal operator on $\mathbb{R}$ with $β\in(0,1)$. The second one regarding the continuity of the discrete centered maximal operator in the space o…
In this paper, we shall prove the $L^{p}$ endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases $S$ in $\mathbb{R} \times \mathbb{R}$. As a consequence, sharp $L^{p}$ decay estimates are also obtained when polynomial phases have the form $S(x^{m_{1}},y^{m_{2}})$ with $m_1$ and $m_2$ being positive integers.
In this paper we prove sharp $L^\infty$-$L^\infty$-$L^\infty$ decay for certain trilinear oscillatory integral forms of convolution type on $\mathbb R^2$. These estimates imply earlier $L^2$-$L^2$-$L^2$ results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ and Christ, Li, Tao, and Thiele. New connections to the multilinear results of Phong, Stein, and Sturm are also considered.
We establish continuity mapping properties of the non-centered fractional maximal operator $M_β$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-β)$ the map $f \mapsto |\nabla M_βf|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$ for $ 0 < β< 1$ if $f$ is radial and for $1 \leq β< d$ for general $f$. The results for $1\leq β< d$ extend to the centered counterpart $M_β^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_β^c$ implies i…
We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim \frac{1}ε \int_{\mathbb{R}^n} |f(x)| M_{L(\log L)^ε} (w)(x)dx$, for $w\geq 0, 0<ε\leq 1.$ This follows from the sharp $L^p$ estimate $ \|T^*f \|_{ L^{p}(w) } \lesssim p' (\frac{1}δ)^{1/p'} \|f \|_{L^{p}(M_{ L(\log L)^{p-1+δ}} (w))}$, for $1<p<\infty, w\geq 0, 0<δ\leq 1. $ As as a consequence we deduce that $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim [w]_{A_{1}} \log(e+ [w]_{A_{\infty}}) \int_{\mathbb{R}^n} |f| w dx, $ extending the endpo…
For each $ d \geq 2$, the Hilbert transform with a polynomial oscillation as below satisfies a $ (1, r )$ sparse bound, for all $ r>1$ $$ H _{ \ast } f (x) = \sup _{ε} \Bigl\lvert \int_{|y| > ε} f (x-y) \frac { e ^{2 πi y ^d }} y\; dy \Bigr\rvert. $$ This quickly implies weak-type inequalities for the maximal truncations, which hold for $A_1$ weights, but are new even in the case of Lebesgue measure. The unweighted weak-type estimate \emph{without maximal truncations} but with arbitrary polynomials, is due to Chanillo and Christ (1987).
We introduce a maximal operator for the natural analogue of the disc multiplier in the framework of Jacobi analysis and determine the mapping properties of said maximal operator, including weak type endpoint results. In particular we obtain sharp information on the almost everywhere convergence of the inverse Jacobi transform.
We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in $\Bbb R^d$, $d\ge 3$, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in $\Bbb R^d$ we obtain sharp uniform $L^p\to L^q$ bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall.
Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates
University of Alabama（阿拉巴马大学）；Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires（布宜诺斯艾利斯大学理学院）；IMAS-CONICET（阿根廷国家科学技术研究委员会数学研究所）；University of the Basque Country（巴斯克大学）；Ikerbasque（伊克拉巴斯科基金会）；BCAM（巴斯克数学中心）；IMAS- CONICET-UBA（阿根廷国家科学技术研究委员会数学研究所-布宜诺斯艾利斯大学）
We provide off-diagonal estimates for a maximal operator arising from a geometric problem of estimating the size of certain geometric configuration of k- skeletons in $\mathbb{R}^n$. This is achieved by interpolating a weak-type endpoint estimate with the known diagonal bounds. The endpoint estimate is proved by combining a geometric result about k-skeletons and adapting an argument used to prove off-diagonal estimates for the circular maximal function in the plane.
For $α>1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-Δ)^{α/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\in(2+4/(d+1),\infty)$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$.
We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $L^p$ bounds for maximal modulations of the associated operators. Our methods consist of identifying where to use effectively the polynomial Carleson theorem, and where we can take advantage of the presence of oscillation to obtain decay through the $TT^*$ method.
We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein's estimate on oscillatory integrals of Carleson-Sjölin-Hörmander type and some spectral projection operators on compact manifolds, and for classes of oscillatory integral operators with one-sided fold singularities.
In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the l^p(Z^d)-boundedness results for discrete k- spherical maximal functions to a restricted weak-type result at the endpoint. Moreover we introduce a novel Approximation Formula for a single average; this allows us to improve our bounds for discrete k-spherical maximal functions along sparse subsequences of radii by exploiting recent progress of Wooley on the Vinogradov mean value conjectures. In particular we have improved …
Given a discrete function $f:\Z^d \to \R$ we consider the maximal operator $$Mf(\vec{n}) = \sup_{r\geq0} \frac{1}{N(r)} \sum_{\vec{m} \in \barΩ_r} \big|f(\vec{n} + \vec{m})\big|,$$ where $\big\{\barΩ_r\big\}_{r \geq 0}$ are dilations of a convex set $Ω$ (open, bounded and with Lipschitz boudary) containing the origin and $N(r)$ is the number of lattice points inside $\barΩ_r$. We prove here that the operator $f \mapsto \nabla M f$ is bounded and continuous from $l^1(\Z^d)$ to $l^1(\Z^d)$. We also prove the same result for the non-centered version of this discrete maximal operator.
In this paper we investigate some questions related to the continuity of maximal operators in $W^{1,1}$ and $BV$ spaces, complementing some well-known boundedness results. Letting $\widetilde M$ be the one-dimensional uncentered Hardy-Littlewood maximal operator, we prove that the map $f \mapsto \big(\widetilde Mf\big)'$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$. In the discrete setting, we prove that $\widetilde M: BV(\mathbb{Z}) \to BV(\mathbb{Z})$ is also continuous. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexample…
We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on $L^{p}$ radial functions. This is a consequence of endpoint bounds for a corresponding square function for Hankel multipliers. We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel multipliers and $L^p$ bounds of maximal operators generated by Hankel multipliers as corollaries. The proof is built on techniques developed by Garrigós and Seeger for characterizations of Hankel multipliers.
Let $0<α<d$ and $1\leq p<d/α$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_αf$ are weakly differentiable and $ \|\nabla\mathcal M_αf\|_{p^*} \leq C_{d,α,p} \|\nabla f\|_p , $ where $ p^* = (p^{-1}-α/d)^{-1} . $ In particular it covers the endpoint case $p=1$ for $0<α<1$ where the bound was previously unknown. For $p=1$ we can replace $W^{1,1}(\mathbb{R}^d)$ by $\mathrm{BV}(\mathbb{R}^d)$. The ingredients used are a pointwise estimate for the gradient of the fractional maximal function…
Given an ergodic dynamical system $(X, \mathcal{B}, μ, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \log L)(X, μ)$, the ergodic averages \[ \frac{1}{π(N)} \sum_{p \in \mathbb{P}_N} f\big(T^p x\big), \] converge for $μ$-almost all $x \in X$, where $\mathbb{P}_N$ is the set of prime numbers not larger that $N$ and $π(N) = \# \mathbb{P}_N$.
One of the purposes of this paper is to prove that if G is a noncompact connected semisimple Lie group of real rank one with finite center, then L^{2,1}(G)\ast L^{2,1}({G})\subseteq L^{2,\infty}({G}). Let {K} be a maximal compact subgroup of {G} and X=G/K a symmetric space of real rank one. We will also prove that the noncentered maximal operator {\cal M}_2f(z) = \sup_{z\in B} \frac{1}{|B|} \int_{B}|f(z')|\,dz' is bounded from L^{2,1}(X) to L^{2,\infty}(X) and from L^p(X) to L^p(X) in the sharp range of exponents p\in(2,\infty]. The supremum in the definition of {\cal M}_2f(z) is taken over al…
We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$ norm inequalities. The multipliers involved are related to those of Coifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form $e^{i|ξ|^α}$ for both $α$ positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more no…
In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of $L^2 \mapsto L^2$ type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in $\mathbb{R}^3$.
We prove sparse bounds for maximal oscillatory rough singular integral operator $$T^{P}_{Ω,*}f(x):=\sup_{ε>0} \left|\int_{|x-y|>ε}e^{ιP(x,y)}\frac{Ω\big((x-y)/|x-y|\big)}{|x-y|^{n}}f(y)dy\right|,$$ where $P(x,y)$ is a real-valued polynomial on $\mathbb{R}^{n}\times \mathbb{R}^{n}$ and $Ω\in L^{\infty}(\mathbb{S}^{n-1})$ is a homogeneous function of degree zero with $\int_{\mathbb{S}^{n-1}}Ω(θ)~dθ=0$. This allows us to conclude weighted $L^p-$estimates for the operator $T^{P}_{Ω,*}$. Moreover, the norm $\|T^P_{Ω,*}\|_{L^p\rightarrow L^p}$ depends only on the total degree of the polynomial $P(x,…
We obtain two-weighted $L^2$ norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type. The conditions imposed on the weights involve geometrically-defined maximal functions, and the inequalities are best-possible in the sense that they imply the full $L^p(\mathbb{R})\rightarrow L^q(\mathbb{R})$ mapping properties of the oscillatory integrals. Our results build on work of Carbery, Soria, Vargas and the first author.
We prove necessary and sufficient conditions for the weak-$L^p$ boundedness, for $p \in (1,\infty)$, of a maximal operator on the infinite-dimensional torus. In the endpoint case $p=1$ we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.
We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^{p}$ into a first order Sobolev space in dimensions $n \geq 5$.
In dimensions $n\ge 2$ we obtain $L^{p_1}(\mathbb R^n) \times\dots\times L^{p_m}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results. Moreover, we obtain weak type and Lorentz space estimates as well as counterexamples in the endpoint cases.
We show that $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.
We begin with an overview on square functions for spherical and Bochner-Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner-Riesz operator, and for more general classes of radial Fourier multipliers.
In this paper, we establish some bilinear endpoint estimates of Calderón commutator $\mathcal{C}[\nabla A,f](x)$ with a homogeneous kernel when $Ω\in L\log^+L(\mathbf{S}^{d-1})$. More precisely, we prove that $\mathcal{C}[\nabla A,f]$ maps $L^q(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ if $q>d$ which improves previous result essentially. If $q=d$, we show that Calderón commutator maps $L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ which is new even if the kernel is smooth. The novelty in the paper is that we prove a new endpoint est…
We study the regularity properties of the centered fractional maximal function $M_β$. More precisely, we prove that the map $f \mapsto |\nabla M_βf|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-β)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.
Let $Ω\subset \mathbb{R}^{n}$ be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to $Ω$ maps $L^{p}(Ω) \to W^{1,p}(Ω)$ for all $p > 1$, when the smoothness index $α\geq 1$. In particular, the results are valid in the range $p \in (1, n/(n-1)]$ that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.
