In this paper, for general plane curves $γ$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,γ}$ along the variable plane curves $(t,U(x_1, x_2)γ(t))$ and the $L^p(\mathbb{R}^2)$-boundedness of the corresponding maximal functions $M^\infty_{U,γ}$, where $p>2$ and $U$ is a measurable function. The range on $p$ is sharp. Furthermore, for $1<p\leq 2$, under the additional conditions that $U$ is Lipschitz and making a $\varepsilon_0$-truncation with $γ(2 \varepsilon_0)\leq 1/4\|U\|_{\textrm{Lip}}$, we also obtain similar boundedness for these two operators $H^{\varepsilon_0}_{U,γ}$ and $M^{\varepsilon_0}_{U,γ}$.
We prove the $L^2$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves.
We prove the $L^p (p > 3/2)$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves.
We show sufficient conditions on matrix weights $U$ and $V$ for the martingale transforms to be uniformly bounded from $L^2(V)$ to $L^2(U)$. We also show that these conditions imply the uniform boundedness of the dyadic shifts as well as the boundedness of the Hilbert transform between these spaces.
We are proving $L^2(\R)\times L^2(\R)\,\rightarrow\,L^1(\R)$ bounds for the bilinear Hilbert transform $H_Γ$ along curves $Γ=(t,-γ(t))$ with $γ$ being a smooth "non-flat" curve near zero and infinity.
In this paper, the $L^2$ boundedness of the Hilbert transform along variable flat curve $(t,P(x_1)γ(t))$ $$H_{P,γ}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x_1-t,x_2-P(x_1)γ(t))\,\frac{\textrm{d}t}{t},\quad \forall\, (x_1,x_2)\in\mathbb{R}^2,$$ is studied, where $P$ is a real polynomial on $\mathbb{R}$. A new sufficient condition on the curve $γ$ is introduced.
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.
The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result is sharp in a particular sense.
Let $p\in (1,\infty)$. In this paper, for any given measurable function $u:\ \mathbb{R}\rightarrow \mathbb{R}$ and a generalized plane curve $γ$ satisfying some conditions, the $L^p(\mathbb{R}^2)$ boundedness of the Hilbert transform along the variable plane curve $u(x_1)γ$ $$H_{u,γ}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x_1-t,x_2-u(x_1)γ(t)) \,\frac{\textrm{d}t}{t}, \quad \forall\, (x_1,x_2)\in\mathbb{R}^2, $$ is obtained. At the same time, the $L^p(\mathbb{R})$ boundedness of the corresponding Carleson operator along the general curve $γ$ $$\mathcal{C}_{u,γ}f(x):=\mathrm{p.\,v.}…
We characterize the space $BV(I)$ of functions of bounded variation on an arbitrary interval $I\subset \mathbb{R}$, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator $M_R$ from $BV(I)$ into the Sobolev space $W^{1,1}(I)$. By restriction, the corresponding characterization holds for $W^{1,1}(I)$. We also show that if $U$ is open in $\mathbb{R}^d, d >1$, then boundedness from $BV(U)$ into $W^{1,1}(U)$ fails for the local directional maximal operator $M_T^{v}$, the local strong maximal operator $M_T^S$, and the iterated local directional maximal operat…
We obtain a necessary and sufficient condition on an exponent $p(\cdot)$ for which the Hardy--Littlewood maximal operator is bounded on the variable $L^{p(\cdot)}$ space. It is formulated in terms of the Muckenhoupt-type condition $A_{p(\cdot)}$, responsible for a local control of $p(\cdot)$, and a certain integral condition on $p(\cdot)$, responsible for the behaviour of $p(\cdot)$ at infinity. Our approach is based on an earlier characterization established by L. Diening and on non-increasing rearrangements.
A long standing conjecture for the linear Schroedinger equation states that 1/4 of derivative in $L^2$, in the sense of Sobolev spaces, suffices in any dimension for the solution to that equation to converge almost everywhere to the initial datum as the time goes to 0. This is only known to be true in dimension 1 by work of Carleson. In this paper we show that the conjecture is true on spherical averages. To be more precise, we prove the $L^2$ boundedness of the associated maximal square function on the Sobolev class $H^{1/4}(R^n)$ in any dimension n.
We introduce the notion of \textit{Perron capacity} of a set of slopes $Ω\subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $Ω$ is finite then the directional maximal operator associated $M_Ω$ is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set $$Ω_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.
We establish the first pointwise ergodic theorems along thin sets of prime numbers; a set with zero density with respect to the primes. For instance we will be able to achieve this with the Piatetski-Shapiro primes. Our methods will be robust enough to solve the ternary Goldbach problem for some thin sets of primes.
We show that the bilinear Hilbert transform $H_Γ$ along curves $Γ=(t,-γ(t))$ with $γ\in\mathcal{N}\mathcal{F}^{C}$ is bounded from $L^{p}(\mathbb{R})\times L^{q}(\mathbb{R})\,\rightarrow\,L^{r}(\mathbb{R})$ where $p,\,q,\,r$ are Hölder indices, i.e. $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with $1<p<\infty$, $1<q\leq\infty$ and $1\leq r<\infty$. Here $\mathcal{N}\mathcal{F}^{C}$ stands for a wide class of smooth "non-flat" curves near zero and infinity whose precise definition is given in Section 2. This continues author's earlier work on this topic, extending the boundedness range of $H_Γ$ to a…
We present new results on the two-weighted boundedness of singular integral operators and $L^p$ boundedness of the Orlicz maximal function. Namely, we extend a theorem of Pérez regarding the necessary and sufficient conditions for the boundedness of the Orlicz maximal function as well as give a new sufficient two-weighted boundedness assumption for Calderón-Zygmund singular integrals.
We introduce and study the median maximal function \mathcal{M} f, defined in the same manner as the classical Hardy-Littlewood maximal function, only replacing integral averages of f by medians throughout the definition. This change has a qualitative impact on the mapping properties of the maximal operator: in contrast with the Hardy-Littlewood operator, which is not bounded on L^1, we prove that \mathcal{M} is bounded on L^p(w) for all 0 < p < \infty, if and only if w \in A_{\infty}. The characterization is purely qualitative and does not give the dependence on [w]_{A_{\infty}}. However, the …
Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_η} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = η}. In the endpoint case we also prove the corresponding weak type inequality. As an application we prove norm inequalities for the fractional integral operator {I_η}. Our proof for the fractional maximal operator uses the theory of dyadic cubes on spaces of homogeneous type, and even in the Euclidean setting it is simpler than existing proofs. For the fractional integral operator we extend a po…
We provide a new direct proof of the $\ell^2$-boundedness of the Discrete Spherical Maximal Function that neither relies on abstract transference theorems (and hence Stein's Spherical Maximal Function Theorem) nor on delicate asymptotics for the Fourier transform of discrete spheres.
Let $\mathcal{N}\mathcal{F}$ be the class of smooth non-flat curves near the origin and near infinity previously introduced by the second author and let $γ\in\mathcal{N}\mathcal{F}$. We show - via a unifying approach relative to the correspondent bilinear Hilbert transform $H_Γ$ - that the (sub)bilinear maximal function along curves $Γ=(t,-γ(t))$ defined as $$M_Γ(f,g)(x):=\sup\limits_{ε>0} \frac{1}{2ε} \int_{-ε}^ε |f(x-t)\,g(x+γ(t))|\,dt$$ is bounded from $L^p(\mathbb{R})\times L^{q}(\mathbb{R})\to L^r(\mathbb{R})$ for all $p, q$ and $r$ Hölder indices, that is $\frac{1}{p}+\frac{1}{q}=\frac{1…
In this paper we prove that the Hardy-Littlewood maximal operator is bounded on Morrey spaces $\mathcal{M}_{1,λ}(\rn)$, $0 \le \la < n$ for radial, decreasing functions on $\rn$
In this paper we derive the maximal subspace of positive numbers, for which the restricted maximal operator of Fejér means in this subspace is bounded from the Hardy space $H_{p}$ to the space $L_{p}$ for all $0<p\leq 1/2.$ Moreover, we prove that the result is in a sense sharp.
We study $L^{p}\times L^{q}\rightarrow L^{r}$-boundedness of (sub)bilinear maximal functions associated with degenerate hypersurfaces. First, we obtain the maximal bound on the sharp range of exponents $p,q,r$ (except some border line cases) for the bilinear maximal functions given by the model surface $\big\{(y,z)\in\mathbb{R}^{n}\times \mathbb{R}^{n}:|y|^{l_{1}}+|z|^{l_{2}}=1\big\}$, $(l_{1},l_{2})\in [1,\infty)^2$, $n\ge 2$. Our result manifests that nonvanishing Gaussian curvature is not good enough, in contrast with $L^p$-boundedness of the (sub)linear maximal operator associated to hyper…
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
We study the boundedness problem for maximal operators $\mathcal{M}$ associated to averages along families of hypersurfaces $S$ of finite type in $\mathbb{R}^n.$ In this paper, we prove that if $S$ is a finite type hypersurface which is of finite type $k$ at $x_0 \in \mathbb{R}^n$, then the associated maximal operator is bounded on $L^p(\mathbb{R}^n)$ for $p>k.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the same $L^p-$ boundedness result for $p>k.$ We also discuss the consequence of this result. In particular, we verify a conjecture by E. M. Stein a…
The aim of this paper is to study two-weight norm inequalities for fractional maximal functions and fractional Bergman operator defined on the upper-half space. Namely, we characterize those pairs of weights for which these maximal operators satisfy strong and weak type inequalities. Our characterizations are in terms of Sawyer and Békollé-Bonami type conditions. We also obtain a $Φ$-bump characterization for these maximal functions, where $Φ$ is a Orlicz function. As a consequence, we obtain two-weight norm inequalities for fractional Bergman operators. Finally, we provide some sharp weighted…
Results of P. Sjölin and F. Soria on the Schrödinger maximal operator with complex-valued time are improved by determining up to the endpoint the sharp $s \geq 0$ for which boundedness from the Sobolev space $H^s(\mathbb{R})$ into $L^2(\mathbb{R})$ occurs. Bounds are established for not only the Schrödinger maximal operator, but further for a general class of maximal operators corresponding to solution operators for certain dispersive PDEs. As a consequence of additional bounds on these maximal operators from $H^s(\mathbb{R})$ into $L^2([-1, 1])$, sharp results on the pointwise almost everywhe…
We improve the range of $\ell^p(\mathbb Z^d)$-boundedness of the integral $k$-spherical maximal functions introduced by Magyar. The previously best known bounds for the full $k$-spherical maximal function require the dimension $d$ to grow at least cubicly with the degree $k$. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we deduce improved bounds in the ergodic Waring--Goldbach problem.
We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces are derived as consequences of our main result. We also apply our boundedness results by studying both generalized neighbourhood capacities and the Lebesgue differentiation of fractional weighted Sobolev functions.
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.
Let $(\cx,\,d,\,μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ…
We study the necessity of bump conditions for the boundedness of the Hardy-Littlewood maximal operator $M$ from $L^p(v)$ into $L^p(w)$, where $1<p<\infty$. The conditions in question are obtained by replacing the average of $σ=v^{-\frac{1}{p-1}}$ in the Muckenhoupt $A_p$-condition by an average with respect to certain Banach function space, and are known to be sufficient for the two-weighted maximal inequality. We show that these conditions are in general not necessary for the boundedness of $M$ from $L^p(v)$ into $L^p(w)$.
We study weighted boundedness of Hardy-Littlewood-type maximal function involving Orlicz functions. We also obtain some sufficient conditions for the weighted boundedness of the Hardy-Littlewood maximal function of the upper-half plane.
Bilinear spherical maximal function with fractal dilations
National Center for Theoretical Sciences National Taiwan University（台湾大学国家理论科学中心）；Department of Mathematics National Taiwan University（台湾大学数学系）；Department of Mathematics Indian Institute of Science Education and Research Bhopal（印度博帕尔科学教育研究 institute 数学系）
We prove the boundedness of a class of tri-linear operators consisting of a quasi piece of bilinear Hilbert transform whose scale equals to or dominates the scale of its linear counter part. Such type of operators is motivated by the tri-linear Hilbert transform and its curved versions.
We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's maximal operator are also discussed.
Let $M_{Ω,α}$ and $T_{Ω,α}$ be the fractional maximal and integral operators with rough kernels, where $0<α<n$. In this paper, we shall study the continuity properties of $M_{Ω,α}$ and $T_{Ω,α}$ on the weighted Morrey spaces $L^{p,κ}(w)$. The boundedness of their commutators with BMO functions is also obtained.
We prove the $L^p$ boundedness of a maximal operator associated with a dyadic frequency decomposition of a Fourier multiplier, under a weak regularity assumption.
Characterization of the boundedness of the segment multiplier on rearrangement-invariant spaces
Universidad Complutense de Madrid（马德里康普顿斯大学）；Brock University（布鲁克大学）；The Ohio State University（俄亥俄州立大学）；Czech Technical University in Prague（布拉格捷克理工大学）；ICMAT（数学分析与应用研究所）
We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces. Given $p \in (1,\infty)$ and a metric measure space $\mathfrak{X}$ we let $Ω^p_{\rm HL}(\mathfrak{X}) \subset [0,1]^2$ be the set of all pairs $(\frac{1}{q},\frac{1}{r})$ such that $\mathcal{M}$ is bounded from $L^{p,q}(\mathfrak{X})$ to $L^{p,r}(\mathfrak{X})$. For each fixed $p$ all possible shapes of $Ω^p_{\rm HL}(\mathfrak{X})$ are characterized. Namely, we show that the boundary of $Ω^p_{\rm HL}(\mathfrak{X})$ either is empty or takes the form $$\{ δ\} \times [0, \lim_{u …
In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^γ$ along a variable plane curve $(t,u(x_1, x_2)γ(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $γ$ is a general curve satisfying some suitable smoothness and curvature conditions.
We study the bilinear Hilbert transform and bilinear maximal functions associated to polynomial curves and obtain uniform $L^r$ estimates for $r>\frac{d-1}{d}$ and this index is sharp up to the end point.
