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 operator $M_T^{d}\circ ...\circ M_T^{1}$. Nevertheless, if $U$ satisfies a cone condition, then $M_T^S:BV(U)\to L^1(U)$ boundedly, and the same happens with $M_T^{v}$, $M_T^{d} \circ ...\circ M_T^{1}$, and $M_R$.
We show that for any infinite set of unit vectors $U$ in $\ZR^2$ the maximal operator defined by $$ H_Uf(x)=\sup_{u\in U}\bigg|\pv\int_{-\infty}^\infty \frac{f(x-tu)}{t}dt\bigg|,\quad x\in \ZR^2, $$ is not bounded in $L^2(\ZR^2)$.
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.
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\|_{\te…
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 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…
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…
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…
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.
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 reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators which can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other singular bilinear multipliers operators. The unboundedness on product of Lebesgue spaces but the boundedness on spaces of smooth functions (which is the exotic behavior referred to) of such operators is obtained. In addition, by introducing a new way…
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.
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 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.
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 $L^p(w)$.
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）
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…
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.
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 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 …
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.
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 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.
This paper will be devoted to study the two-weight norm inequalities of the multilinear fractional maximal operator $\mathcal{M}_α$ and the multilinear fractional integral operator $\mathcal{I}_α$. The entropy conditions in the multilinear setting will be introduced and the entropy bounds for $\mathcal{M}_α$ and $\mathcal{I}_α$ will be given.
$L^p$ boundedness of the circular maximal function $\mathcal M_{\mathbb{H}^1}$ on the Heisenberg group $\mathbb{H}^1$ has received considerable attentions. While the problem still remains open, $L^p$ boundedness of $\mathcal M_{\mathbb{H}^1}$ on Heisenberg radial functions was recently shown for $p>2$ by Beltran, Guo, Hickman, and Seeger [2]. In this paper we extend their result considering the local maximal operator $M_{\mathbb{H}^1}$ which is defined by taking supremum over $1<t<2$. We prove $L^p-L^q$ estimates for $M_{\mathbb{H}^1}$ on Heisenberg radial functions on the optimal range of $p,…
In this paper we analyze the convergence of the following type of series $$ T_N^αf(x,t)=\sum_{j=N_1}^{N_2} v_j(P_{a_{j+1}}^αf(x,t)-P_{a_j}^αf(x,t)),\quad (x,t)\in \mathbb R^{n+1}, \ N=(N_1, N_2)\in \mathbb Z^2,\ α>0, $$ where $\{P_τ^α\}_{τ>0}$ is the fractional Poisson-type operators generated by the parabolic operator $L=\partial_t-Δ$ with $Δ$ being the classical Laplacian, $\{v_j\}_{j\in \mathbb Z}$ a bounded real sequences and $\{a_j\}_{j\in \mathbb Z}$ an increasing real sequence. Our analysis will consist {of} the boundedness, in $L^p(\mathbb{R}^n)$ and in $BMO(\mathbb{R}^n)$, of the oper…
Sharp estimates for the Fourier transform of surface-carried measures and maximal operators associated with hypersurfaces in $\mathbb{R}^4$ with vanishing Gaussian curvature
V. I. Romanovsky Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I. 罗曼诺夫斯基数学研究所）；Institut für Analysis und Numerik, Fachbereich Mathematik und Informatik der Universität Münster（明斯特大学数学与信息学院分析与数值研究所）
Let $γ_{-1}$ be the absolutely continuous measure on $\mathbb{R}^n$ whose density is the reciprocal of a Gaussian and consider the natural weighted Laplacian $\mathcal{A}$ on $L^2(γ_{-1})$. In this paper, we prove boundedness and unboundedness results for the purely imaginary powers and the first order Riesz transforms associated with the translated operators $\mathcal{A}+λI$, $λ\geq0$, from certain new Hardy-type spaces adapted to $γ_{-1}$ to $L^1(γ_{-1})$. We also investigate the weak type $(1,1)$ of these operators.
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 $\mathcal{L}=-Δ+V$ be a Schrödinger operator, where the nonnegative potential $V$ belongs to the reverse Hölder class $B_{q}$. By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, $\{e^{-t\mathcal{L}^α}\}_{t>0},$ associated with $\mathcal{L}$. As an application, we obtain the $BMO^γ_{\mathcal{L}}$-boundedness of the maximal function, and the Littlewood-Paley $g$-functions associated with $\mathcal{L}$ via $T1$ theorem, respectively.
We explore some variants of the Boman covering lemma, and their relationship to the boundedness properties of the maximal operator. Let $1 < p < \infty$ and let $q$ be its conjugate exponent. We prove that the strong type $(q,q)$ of the uncentered maximal operator, by itself, implies certain generalizations of the Boman covering lemma for the exponent $p$, and in turn, these generalizations entail the weak type $(q,q)$ of the centered maximal operator. We show by example that it is possible for the uncentered maximal operator to be unbounded for all $1 < s < \infty$, while the conclusion of th…
In this paper we analyze the convergence of the following type of series \begin{equation*} T_N f(x)=\sum_{j=N_1}^{N_2} v_j\Big(e^{-a_{j+1}(-Δ)^α} f(x)-e^{-a_{j}(-Δ)^α} f(x)\Big),\quad x\in \mathbb R^n, \end{equation*} where $\{e^{-t(-Δ)^α} \}_{t>0}$ is the heat semigroup of the fractional Laplacian $(-Δ)^α,$ $N=(N_1, N_2)\in \mathbb Z^2$ with $N_1<N_2,$ $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequences and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence. Our analysis will consist in the boundedness, in $L^p(\mathbb{R}^n)$ and in $BMO(\mathbb{R}^n)$, of the operators $T_N$ and it…
In this article we introduce the fractional Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree and study its weighted boundedness. We also provide examples of weights for which the fractional Hardy-Littlewood maximal function satisfies strong type $(p, q)$ estimates on the infinite rooted $k$-ary tree.
