We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cos\theta)\cos\phi,\,(t+s\cos\theta)\sin\phi,\, s\sin\theta ): \theta, \phi \in …
We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cos\theta)\cos\phi,\,(t+s\cos\theta)\sin\phi,\, s\sin\theta ): \theta, \phi \in [0,2\pi) \}$ with $c_0t>s>0$ for some $c_0\in (0,1)$. We prove that the associated (two-parameter) maximal function is bounded on $L^p$ if and only if $p>2$. We also obtain $L^p$--$L^q$ estimates for the local maximal operator on a sharp range of $p,q$. Furthermore, the sharp smoothing estimates are proved including the sharp local smoothing estimates for the operators $f\to \mathcal A_t^s f$ and $f\to \mathcal A_t^{c_0t} f$. For the purpose, we make use of Bourgain--Demeter's decoupling inequality for the cone and Guth--Wang--Zhang's local smoothing estimates for the $2$ dimensional wave operator.
We prove that for a finite type curve in â3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp …
We prove that for a finite type curve in â3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp â Lp1/p regularity result for the averaging operators for large p. The proofs make use of a deep result of Thomas Wolff about decompositions of cone multipliers.
We prove that for a finite type curve in $\mathbb R^3$ the maximal operator generated by dilations is bounded on $L^p$ for sufficiently large $p$. We also show the endpoint …
We prove that for a finite type curve in $\mathbb R^3$ the maximal operator generated by dilations is bounded on $L^p$ for sufficiently large $p$. We also show the endpoint $L^p \to L^{p}_{1/p}$ regularity result for the averaging operators for large $p$. The proofs make use of a deep result of Thomas Wolff about decompositions of cone multipliers.
We prove that for a finite type curve in $\mathbb R^3$ the maximal operator generated by dilations is bounded on $L^p$ for sufficiently large $p$. We also show the endpoint …
We prove that for a finite type curve in $\mathbb R^3$ the maximal operator generated by dilations is bounded on $L^p$ for sufficiently large $p$. We also show the endpoint $L^p \to L^{p}_{1/p}$ regularity result for the averaging operators for large $p$. The proofs make use of a deep result of Thomas Wolff about decompositions of cone multipliers.
We study $L^p$ boundedness of the maximal average over dilations of a smooth hypersurface $S$. When the decay rate of the Fourier transform of a measure on $S$ is $1/2$, …
We study $L^p$ boundedness of the maximal average over dilations of a smooth hypersurface $S$. When the decay rate of the Fourier transform of a measure on $S$ is $1/2$, we establish the optimal maximal bound, which settles the conjecture raised by Stein. Additionally, when $S$ is not flat, we verify that the maximal average is bounded on $L^p$ for some finite $p$, which generalizes the result by Sogge and Stein.
We study problems related to smooth nondegenerate curves $\gamma$ in $\mathbb R^d$. First we show the maximal operator $M$ defined by the averaging operator $\mathcal A_t$ over the dilations $t\gamma$ …
We study problems related to smooth nondegenerate curves $\gamma$ in $\mathbb R^d$. First we show the maximal operator $M$ defined by the averaging operator $\mathcal A_t$ over the dilations $t\gamma$ is bounded on $L^p$ for $p>2(d-1)$ when $d\ge 4$. The result is a consequence of the new sharp local smoothing estimate for $\mathcal A_t$. Secondly, we prove the sharp $L^p$ Sobolev regularity estimates for $\mathcal A_t$ with a fixed $t\neq 0$ when $d\ge5$. This settles the problem of sharp $L^p$ regularity estimate except the endpoint cases.
Let $M$ be the maximal operator associated to a smooth curve in $\mathbb R^3$ which has nonvanishing curvature and torsion. We prove that $M$ is bounded on $L^p$ if and …
Let $M$ be the maximal operator associated to a smooth curve in $\mathbb R^3$ which has nonvanishing curvature and torsion. We prove that $M$ is bounded on $L^p$ if and only if $p>3$.
Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 …
Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p is greater than 2, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Muller on this subject
Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 …
Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p is greater than 2, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Muller on this subject
Averages over smooth measures on smooth compact hypersurfaces in Rn are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for …
Averages over smooth measures on smooth compact hypersurfaces in Rn are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for example Lp estimates of multiparameter maximal functions over compact hypersurfaces.
Extending the methods developed in the authorâs recent paper and using some techniques from a paper by Sogge and Stein in conjunction with various facts about adapted coordinate systems in …
Extending the methods developed in the authorâs recent paper and using some techniques from a paper by Sogge and Stein in conjunction with various facts about adapted coordinate systems in two variables, an $L^p$ boundedness theorem is proven for maximal operators over hypersurfaces in $\mathbb {R}^3$ when $p > 2.$ When the best possible $p$ is greater than $2$, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Müller (2010).
We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let v be a vector field defined on the …
We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let v be a vector field defined on the unit square such that v(x,y) = (1,u(x)) for some measurable u from [0,1] to [0,1]. Fix a small parameter delta and let Z be the collection of rectangles R of a fixed width such that delta much of the vector field inside R is pointed in (approximately) the same direction as R. We show that the maximal averaging operator associated to the collection Z is bounded on L^p for p>1 with constants comparable to (delta)^(-1) .
The aim of this paper is to propose weak assumptions to prove maximal L^q regularity for Cauchy problem: du/dt - Lu(t)=f(t). Mainly we only require "off-diagonal" estimates on the real …
The aim of this paper is to propose weak assumptions to prove maximal L^q regularity for Cauchy problem: du/dt - Lu(t)=f(t). Mainly we only require "off-diagonal" estimates on the real semigroup (e^{tL})_{t>0} to obtain maximal L^q regularity. The main idea is to use a one kind of Hardy space H^1 adapted to this problem and then use interpolation results. These techniques permit us to prove weighted maximal regularity too.
where dσr is the normalized surface measure on r S1. It is easy to see that M is not bounded on L2 (see Example 1.1 below). A well-known result of …
where dσr is the normalized surface measure on r S1. It is easy to see that M is not bounded on L2 (see Example 1.1 below). A well-known result of Bourgain [1] asserts that M is bounded on Lp for 2 < p ≤ ∞. We will consider the question of boundedness of M and Mδ from Lp to Lq. Unless stated to the contrary, we will be dealing only with functions defined on R2. Absolute constants will be denoted by C, and the notation ??? will mean = up to a constant.
The $L^1$ Fourier transform The Schwartz space Fourier inversion and the Plancherel theorem Some specifics, and $L^p$ for $p<2$ The uncertainty principle The stationary phase method The restriction problem Hausdorff …
The $L^1$ Fourier transform The Schwartz space Fourier inversion and the Plancherel theorem Some specifics, and $L^p$ for $p<2$ The uncertainty principle The stationary phase method The restriction problem Hausdorff measures Sets with maximal Fourier dimension and distance sets The Kakeya problem Recent work connected with the Kakeya problem Bibliography for Chapter 11 Historical notes Bibliography.
PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular Integral Operators: Fourier Transform228VIIPseudo-Differential and Singular Integral Operators: Almost Orthogonality269VIIIOscillatory Integrals of the First Kind329IXOscillatory …
PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular Integral Operators: Fourier Transform228VIIPseudo-Differential and Singular Integral Operators: Almost Orthogonality269VIIIOscillatory Integrals of the First Kind329IXOscillatory Integrals of the Second Kind375XMaximal Operators: Some Examples433XIMaximal Averages and Oscillatory Integrals467XIIIntroduction to the Heisenberg Group527XIIIMore About the Heisenberg Group587Bibliography645Author Index679Subject Index685
Let [unk](f)(x) denote the supremum of the averages of f taken over all (surfaces of) spheres centered at x. Then f --> [unk](f) is bounded on L(p)(R(n)), whenever p > …
Let [unk](f)(x) denote the supremum of the averages of f taken over all (surfaces of) spheres centered at x. Then f --> [unk](f) is bounded on L(p)(R(n)), whenever p > n/(n - 1), and n >/= 3.
We prove L p -boundedness for a class of singular integral operators and maximal operators associated with a general k-parameter family of dilations on R n . This class includes …
We prove L p -boundedness for a class of singular integral operators and maximal operators associated with a general k-parameter family of dilations on R n . This class includes homogeneous operators defined by kernels supported on homogeneous manifolds. For singular integrals, only certain "minimal" cancellation is required of the kernels, depending on the given set of dilations.
We prove that for a finite type curve in â3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp …
We prove that for a finite type curve in â3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp â Lp1/p regularity result for the averaging operators for large p. The proofs make use of a deep result of Thomas Wolff about decompositions of cone multipliers.
We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we …
We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we prove that if no affine tangent plane to S passes through the origin and S is analytic, then the associated maximal operator is bounded on $ {L^p}\left( {{\mathbb{R}^3}} \right) $ if and only if p > h(S), where h(S) denotes the so-called height of the surface S (defined in terms of certain Newton diagrams). For non-analytic S we obtain the same statement with the exception of the exponent p = h(S). Our notion of height h(S) is closely related to A. N. Varchenko's notion of height h(ϕ) for functions ϕ such that S can be locally represented as the graph of ϕ after a rotation of coordinates. Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on S and the Lp-boundedness of the associated maximal operator $ \mathcal{M} $, and a conjecture by Iosevich and Sawyer which relates the Lp-boundedness of $ \mathcal{M} $ to an integrability condition on S for the distance to tangential hyperplanes, in dimension 3. In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on S, thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnold in our context. As an immediate application of this, we obtain an $ {L^p}\left( {{\mathbb{R}^3}} \right) - {L^2}(S) $ Fourier restriction theorem for S.
Bilinear restriction estimates have 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 …
Bilinear restriction estimates have 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L Superscript p Baseline comma upper L Superscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(L^p,L^p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spherical restriction theorem of Wolff from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 42 slash 11"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>42</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>11</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 42/11</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 34 slash 9"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>34</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>9</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 34/9</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and also obtain a sharp <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L Superscript p Baseline comma upper L Superscript q Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(L^p,L^q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spherical restriction theorem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than 4 minus five twenty-sevenths"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mfrac> <mml:mn>5</mml:mn> <mml:mn>27</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">q> 4 - \frac {5}{27}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
We prove that the elliptic maximal function maps the Sobolev space W_{4,\eta}(\mathbb{R}^2) into L^4(\mathbb{R}^2) for all \eta>1/6 . The main ingredients of the proof are an analysis of the intersection …
We prove that the elliptic maximal function maps the Sobolev space W_{4,\eta}(\mathbb{R}^2) into L^4(\mathbb{R}^2) for all \eta>1/6 . The main ingredients of the proof are an analysis of the intersection properties of elliptic annuli and a combinatorial method of Kolasa and Wolff.
Abstract Let A t f ( x ) denote the mean of f over a sphere of radius t and center x . We prove sharp estimates for the maximal …
Abstract Let A t f ( x ) denote the mean of f over a sphere of radius t and center x . We prove sharp estimates for the maximal function M E f ( X ) = sup t ∈ E |A tf (x)| where E is a fixed set in IR + and f is a radial function ∈ L p (IR d ). Let P d = d/ ( d− 1) (the critical exponent for Stein's maximal function). For the cases (i) p < p d , d ⩾ 2, and (ii) p = p d , d ⩽ 3, and for p ⩽ q ⩽ ∞ we prove necessary and sufficient conditions on E for M E to map radial functions in L p to the Lorentz space L P,q .
The purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of Bourgain's circular maximal theorem [1]. We use …
The purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of Bourgain's circular maximal theorem [1]. We use easy wave front analysis along with techniques previously used in proofs of the Carleson-Sj6lin theorem (see [3],[5],[7]) and in the proof of sharp regularity properties of Fourier integral operators [13]. The circular means operators are defined by
is bounded on L(R) if p > n/(n − 1). He also showed that no such result can hold for p ≤ n/(n − 1) if n ≥ 2. Thus, …
is bounded on L(R) if p > n/(n − 1). He also showed that no such result can hold for p ≤ n/(n − 1) if n ≥ 2. Thus, the 2-dimensional case is more complicated since the circular maximal operator corresponding to n = 2 is not bounded on L. Some 10 years passed before Bourgain [2] finally showed that the circular maximal function is bounded on L(R) for every 2 p depending on p > 2. In either case, though, it seems certain that the techniques in [2] or [9] will not give sharp estimates of this type. Recently, though, Schlag [11] obtained bounds which are of the best possible nature. Specifically, if we set
Extending the methods developed in the authorâs recent paper and using some techniques from a paper by Sogge and Stein in conjunction with various facts about adapted coordinate systems in …
Extending the methods developed in the authorâs recent paper and using some techniques from a paper by Sogge and Stein in conjunction with various facts about adapted coordinate systems in two variables, an $L^p$ boundedness theorem is proven for maximal operators over hypersurfaces in $\mathbb {R}^3$ when $p > 2.$ When the best possible $p$ is greater than $2$, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Müller (2010).
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 …
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 implies the full range of expected L px,t Strichartz estimates for both the rational and (up to N ε losses) the irrational torus.Another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere.Various applications to Additive Combinatorics, Incidence Geometry and Number Theory are also discussed.Our argument relies on the interplay between linear and multilinear restriction theory.
In this article, we continue the study of the problem of $L^p$-boundedness of the maximal operator $\mathcal {M}$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ …
In this article, we continue the study of the problem of $L^p$-boundedness of the maximal operator $\mathcal {M}$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ of finite type in 3-dimensional Euclidean space. An essentially complete answer to this problem was given about eight years ago by the third and fourth authors in joint work with M. Kempe [Acta Math 204 (2010), pp. 151â271] for the case where the height $h$ of the given surface is at least two. In the present article, we turn to the case $h<2.$ More precisely, in this Part I, we study the case where $h<2,$ assuming that $S$ is contained in a sufficiently small neighborhood of a given point $x^0\in S$ at which both principal curvatures of $S$ vanish. Under these assumptions and a natural transversality assumption, we show that, as in the case $h\ge 2,$ the critical Lebesgue exponent for the boundedness of $\mathcal {M}$ remains to be $p_c=h,$ even though the proof of this result turns out to require new methods, some of which are inspired by the more recent work by the third and fourth authors on Fourier restriction to $S.$ Results on the case where $h<2$ and exactly one principal curvature of $S$ does not vanish at $x^0$ will appear elsewhere.
We prove a sharp square function estimate for the cone in R 3 and consequently the local smoothing conjecture for the wave equation in 2 + 1 dimensions.2 To be …
We prove a sharp square function estimate for the cone in R 3 and consequently the local smoothing conjecture for the wave equation in 2 + 1 dimensions.2 To be more specific, Sogge originally made the conjecture for α in the range α > 1 2 -2 p and Wolff confirmed Sogge's conjecture for p ≥ 74 and α in this range.Later in the work [15] of Heo, Nazarov and Seeger it was conjectured further that when p > 4 the conjecture should hold for α ≥ 1 2 -2 p . 3 Such kind of "locally constant" heuristic will be used a few times in the current paper.To justify this intuition one can use Corollary 4.3 in [3].See also Lemma 6.1 and Lemma 6.2 in Section 6 of the current paper.4 This definition works best if τ is honestly tiled by θ.In general we abuse the notation a bit: Throughout this paper, by writing "summing over θ ⊂ τ ", we really mean "summing over all θ ∈ A(τ )" where the collection A(τ ) is chosen as follows: Each A(τ ) only contains those θ's who intersect τ , and all A(τ ) form a disjoint union {θ} = τ A(τ ).
Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma $ in $\mathbb R^d$ , $d\ge 3$ . …
Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma $ in $\mathbb R^d$ , $d\ge 3$ . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every $d\ge 3$ . As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $\gamma $ for $d\ge 4$ .