Amiran Gogatishvili, Tengiz Kopaliani
In this paper we study some estimates of norms in variable exponent Lebesgue spaces for maximal multiplier operators.We will consider the case when multiplier is the Fourier transform of a compactly supported Borel measure
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Spectral multipliers for maximally subelliptic operators
本文研究紧流形上非负自伴极大次椭圆算子的谱乘子，证明 Mihlin–Hörmander 型条件下其为奇异积分算子，并建立相关非各向同性函数空间等价及 $L^p$ Sobolev 有界性条件。
In this article, we establish a characterization of the set $M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ of all pointwise multipliers of Besov spaces $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ with only logarithmic smoothness $b\in\mathbb{R}$ in the special cases $p=1$ and $p=\infty$. As applications of these two characterizations, we clarify whether or not the three concrete examples, namely characteristic functions of open sets, continuous functions defined by differences, and the functions $e^{ik\cdot x}$ with $k\in\mathbb{Z}^n$ and $x\in\mathbb{R}^n$, are pointwise multipliers of $B^{0,b}_{1,\infty}(\mat…
Maximal functions, conjugations and multipliers between Toeplitz kernels
本文研究了Toeplitz核通过Riemann-Hilbert问题、最大函数和乘子的多种定义方式及其关系，重点探讨了符号因子分解在构造乘子中的作用以及最大函数与共轭运算的联系。
We investigate the class $\mathcal{B}^{loc}(\mathbb{R}^{n})$ of exponents $p(\cdot)$ for with local Hardy-Littlewood maximal operator is bounded in $L^{p(\cdot)}(\mathbb{R}^{n})$ space. Littlewood-Paley square-function characterization of $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces with the above class of exponent are also obtained.
Maximal functions and multipliers for subkernels of Toeplitz operators
本文研究了Toeplitz算子子核中极值函数与乘子的关系，通过模型空间表示方法揭示了重要Toeplitz核的结构特性。
We prove that the existence of a Mihlin-Hörmander functional calculus for an operator $L$ implies the boundedness on $L^p$ of both the maximal operators and the continuous square functions build on spectral multipliers of $L.$ The considered multiplier functions are finitely smooth and satisfy an integral condition at infinity. In particular multipliers of compact support are admitted.
Characterization of weights for the variable fractional maximal operator and weighted inequalities for variable fractional rough operators
本文研究了可变分数极大算子在可变Lebesgue空间上的权重刻画，扩展了已有结果，并引入了满足新变量Hörmander条件的核类，证明了Coifman-Fefferman不等式和加权不等式。
We establish a platform to transfer $L_p$-completely bounded maps on tensor products of von Neumann algebras to $L_p$-completely bounded maps on the corresponding amalgamated free products. As a consequence, we obtain a Hörmander-Mikhlin multiplier theory for free products of groups. Let $\mathbb{F}_\infty$ be a free group on infinite generators $\{g_1, g_2,\cdots\}$. Given $d\ge1$ and a bounded symbol $m$ on $\mathbb{Z}^d$ satisfying the classical Hörmander-Mikhlin condition, the linear map $M_m:\mathbb{C}[\mathbb{F}_\infty]\to \mathbb{C}[\mathbb{F}_\infty]$ defined by $λ(g)\mapsto m(k_1,\cdo…
A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type
本文证明在 Borel 半正则且反向加倍的齐型空间上，变指数下确界大于 1 是 Hardy-Littlewood 极大算子有界性的必要条件。
Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.
In this paper, Mikhlin and Marcinkiewicz--Lizorkin type operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces are studied. By using this results embedding theorems in Sobolev-Lions type spaces is obtained. Moreover, maximal regularity properties of abstract elliptic and parabolic equations are derived
Atomic Characterization and Its Applications of Matrix-Weighted Variable Hardy Spaces
本文通过矩阵加权极大函数引入变指数Hardy空间，利用凸体值极大函数建立其等价刻画，并通过Whitney分解和凸体缩减算子获得原子刻画，进而给出对偶空间和Calderón-Zygmund算子的有界性。
Multipliers between kernels of Toeplitz operators are characterised in terms of test functions (so-called maximal vectors for the kernels); these maximal vectors may easily be parametrised in terms of inner and outer factorizations. Immediate applications to model spaces are derived. The case of surjective multipliers is also analysed. These ideas are applied to describing equivalences between two Toeplitz kernels.
Consider the multidimensional Bessel operator $$B f(x) = -\sum_{j=1}^N \left(\partial_j^2 f(x) +\frac{α_j}{x_j} \partial_j f(x)\right), \quad x\in(0,\infty)^N. $$ Let $d = \sum_{j=1}^N \max(1,α_j+1)$ be the homogeneous dimension of the space $(0,\infty)^N$ equipped with the measure $x_1^{α_1}... x_N^{α_N} dx_1...dx_N$. In the general case $α_1,...,α_N >-1$ we prove multiplier theorems for spectral multipliers $m(B)$ on $L^{1,\infty}$ and the Hardy space $H^1$. We assume that $m$ satisfies the classical Hörmander condition $$\sup_{t>0} \left||η(\cdot) m(t\cdot)\right||_{W^{2,β}(\mathbb{R})}<\in…
Characterizations of Lorentz Type Sobolev Multiplier Spaces and Their Preduals
本文研究了洛伦兹型Sobolev乘子空间及其预对偶空间的特征化，并探讨了局部Hardy-Littlewood极大函数在这些空间上的有界性问题。
New variable weighted conditions for fractional maximal operators over spaces of homogeneous type
本文在齐型空间上建立新的变多重权与变测试条件，系统给出分数次极大算子及多线性分数次极大算子的定量变权强、弱型估计，且常数与权无关。
We prove a general type description result for the multipliers acting between two periodic Bessel potential spaces, defined on the $n$--dimensional torus, in a case when their smoothness indices are of different signs. This is done through the detailed examination of a periodic analogue of the linear operator $J_s$, which is employed in the definition of the scale of the Bessel potential space defined on the whole space $\mathbb{R}^n$. Our method of defining this periodic analogue of $J_s$ uses the results about an asymptotic behaviour of the generalized Fourier coefficients and existence of a…
Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition and $L$ a non-negative self-adjoint operator on $L^2(\mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estimates. Let $H_L^{p(\cdot)}(\mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels $\{e^{-t^2L}\}_{t\in (0,\infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(\cdot)}(\mathbb R^n)$; using this, the authors then obtain its non-tangential ma…
We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimensio…
In the present work, we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calderón-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $σ(ξ_1,\dots , ξ_m)$ and their derivatives vanish on the hyperplane $ξ_1+\cdots+ξ_m=0$.
Classical discrete operators on variable $\ell^{p(\cdot)}(\mathbb{Z})$ spaces
本研究利用离散加权范数不等式与Rubio de Francia算法，证明离散Hardy-Littlewood极大算子在对偶空间有界时，离散Hilbert变换和Riesz位势在可变ℓ^{p(·)}(ℤ)空间有界，还得到离散分数次极大算子的向量值不等式。
Generic norm growth of powers of homogeneous unimodular Fourier multipliers
该研究针对齐次幺模傅里叶乘子算子族，证明通用相位函数下其范数随参数t达最大增长阶，说明此前反例中的最大增长是普遍现象。
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the anisotropic variable Hardy-Lorentz space $H_A^{p(\cdot),q}(\mathbb R^n)$ associated with $A$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain characterizations of $H_A^{p(\cdot),q}(\mathbb R^n)$, respectively, in terms of the atom and th…
A Remark on Hörmander Multipliers on Dunkl Hardy Spaces
本文在 Dunkl Hardy 空间上证明了 Hörmander 乘子定理，给出了乘子有界性的充分条件，推广了经典结果。
The Molecular Characterizations of Variable Triebel-Lizorkin Spaces Associated with the Hermite Operator and Its Applications
本文引入了与Hermite算子相关的非齐次变指数Triebel-Lizorkin空间，建立了其分子刻画，并将其应用于分数阶Hermite方程的正则性研究与谱乘子的有界性分析。
We study a class of spectral multipliers ϕ(L) for the Ornstein--Uhlenbeck operator L arising from the Gaussian measure on R^n and find a sufficient condition for integrability of ϕ(L)f in terms of the admissible conical square function and a maximal function.
Discrete Hilbert Transform And Discrete Mikhlin Multiplier On Discrete Variable Lebesgue Space
本文在变量离散Lebesgue空间上证明了离散Hilbert变换的有界性，并给出了离散Mikhlin乘子在该空间上有界的充分条件。
We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The main goal of the paper is to show that the boundedness of these operators in the spaces $\mathcal{L}^{p(\cdot), q(\cdot)}$ is possible without the local log-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents $p(s)$ and $q(s)$ should only satisfy decay conditions of log-type as $s\to 0$ and $s\to\infty$. To prove…
Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres
本文证明欧氏球全离散 Hardy-Littlewood 极大算子对所有 $p>1$ 及球面极大算子对 $d\geq5, p\geq2$ 满足无维数界，通过离散高斯乘子逼近解决 Stein 问题并加强 MSW 定理。
In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in $\mathcal L(L^p(\mathbb R^d; X))$ for $1<p<\infty$ and for a UMD Banach space $X$ in terms of the range of the corresponding symbol. For example, if the range contains $a_1,\ldots,a_N \in \mathbb C$, then the norm of the multiplier exceeds $\|a_1R_1^2 + \cdots + a_NR_N^2\|_{\mathcal L(L^p(\mathbb R^N; X))}$, where $R_n$ is the corresponding Riesz transform. We also provide sharp upper bounds of norms of Bañuelos-Bogdan type multipliers in terms of the range of the functions involved. The main tools th…
Let $A$ be a generator of an analytic semigroup having a H{ö}rmander functional calculus on $X = L^p(Ω,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for H{ö}rmander spectral multipliers decaying sufficiently fast at $\infty$, there holds a maximal estimate $\| \sup_{t \geq 0} |m(tA)f|\, \|_{L^p(Ω,Y)} \lesssim \|f\|_{L^p(Ω,Y)}$. We also show square function estimates $\left\| \left( \sum_k \sup _{t \geq 0} |m_k(tA)f_k|^2 \right)^{\frac12} \right\|_{L^p(Ω,Y)} \lesssim \left\| \left( \sum _k |f_k|^2 \right)^{\frac12}…
In this article, the authors first introduce the Triebel-Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ with variable exponents, and establish its $φ$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defined. The smooth molecular and the smooth atomic characterizations of $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$. The authors also characterize the space $F_{p(\cdot),q(\c…
We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$ and a maximally modulated Calderón-Zygmund singular integral operator $T^Φ$ is of weak type $(r,r)$ for all $r\in(1,\infty)$, then $T^Φ$ extends to a bounded operator on $X(\mathbb{R}^n)$. This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ under natural assumptions on the variable exponent $p:\mathbb{R}\to(1,\infty)$. Applications of the above r…
We consider local "complementary" generalized Morrey spaces ${\dual \cal M}_{\{x_0\}}^{p(\cdot),\om}(\Om)$ in which the $p$-means of function are controlled over $\Om\backslash B(x_0,r)$ instead of $B(x_0,r)$, where $\Om \subset \Rn$ is a bounded open set, $p(x)$ is a variable exponent, and no monotonicity type conditio is imposed onto the function $\om(r)$ defining the "complementary" Morrey-type norm. In the case where $\om$ is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal ope…
In this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel-Lizorkin-type spaces with variable exponents $B^{\mathrm{\boldsymbolω}, ϕ}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and $F^{\mathrm{\boldsymbolω}, ϕ}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$. Of big importance in general, and an essential tool here, are the characterizations of the spaces via maximal functions and local means, that we also present. These spaces were recently introduced by Wu at al. and cover not only variable 2-microlocal Besov and Triebel-Lizorkin spaces $B^{\mathrm{\boldsymbolω}}_{p(\cdot),q…
The maximal operator on variable Lebesgue spaces: an ${\mathcal A}_{\infty}$-characterization
本文通过可变指数$A_{\infty}$条件，给出了最大算子在可变指数空间上有界的新的判据。
Fractional maximal operators on weighted variable Lebesgue spaces over the spaces of homogeneous type
本文在齐型空间上引入分数次型变指标权，建立分数次极大算子在加权变指标Lebesgue空间中的强型与弱型刻画，推广了多个已有经典结果。
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on $\mathbb R^n$, $W\!H^{p(\cdot)}(\mathbb R^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W\!H^{p(\cdot)}(\mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_λ^\ast…
We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in $L^2(\mathbb{R}^d)$, we obtain norm estimates of Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, and for multidimensional Cantor iterates we derive explicit upper bound…
Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $φ:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{φ,\,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the spac…
It is shown that, under some natural additional conditions, an operator which intertwines one cyclic singular unitary operator with one dimensional perturbation of another cyclic singular unitary operator is the operator of multiplication by a multiplier between model spaces. Using this result, it is shown that if $T$ is one dimensional perturbation of a unitary operator, $T$ is a quasiaffine transform of a singular unitary operator, and $T$ is power bounded, then $T$ is similar to a unitary operator, and $\sup_{n\geq 0}\|T^{-n}\|\leq(2(\sup_{n\geq 0}\|T^n\|)^2+1)\cdot(\sup_{n\geq 0}\|T^n\|)^5…
We consider the space $\mathcal{H}(ces_p)$ of all Dirichlet series whose coefficients belong to the Cesàro sequence space $ces_p$, consisting of all complex sequences whose absolute Cesàro means are in $\ell^p$, for $1<p<\infty$. It is a Banach space of analytic functions, for which we study the maximal domain of analyticity and the boundedness of point evaluations. We identify the algebra of analytic multipliers on $\mathcal{H}(ces_p)$ as the Wiener algebra of Dirichlet series shifted to the vertical half-plane $\mathbb{C}_{1/q}:=\{s\in\mathbb{C}:\Re s>1/q\}$, where $1/p+1/q=1$.
In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator $S(\cdot + p)+T(\cdot)$, where $p\in X$ and $S$ and $T$ are maximal monotone operators on the reflexive Banach space $X$. Then, this is used to obtain sufficient conditions for the surjectivity of $S+T$ and for the situation when $0$ belongs to the range of $S+T$. Several special cases are discussed, some of the…
We give a new characterisation of inner product spaces amongst normed vector spaces in terms of the maximal cirumradius of spheres. It turns out that a normed vector space $(X,\norm{\cdot})$ with $\dim X\geq 2$ is an inner product space if and only if all spheres are not degenerate, i.e. the maximal circumradius of points on the sphere equals the radius of the sphere.
The asymptotic behavior for entropy numbers of general Fourier multiplier operators of multiple series with respect to an abstract complete orthonormal system $\{ϕ_{\textbf{m}}\}_{\textbf{m}\in \mathbb{N}^d_0}$ on a probability space and bounded in $L^{\infty}$, is studied. The orthonormal system can be of the type $ϕ_{\textbf{m}}(\textbf{x}) = ϕ_{m_1}^{(1)}(x_1)\cdots ϕ_{m_d}^{(d)}(x_d)$, where each $\left\{ϕ_{l}^{(j)}\right\}_{l\in \mathbb{N}_0}$ is an orthonormal system, that can be different for each $j$, for example, it can be a Vilenkin system, a Walsh system on a real sphere or the trig…
Given a locally convex space $(\mathcal{A},τ)$ with a Hausdorff locally convex topology $τ$ such that the following maps are continuous; $u \mapsto u^*$ for all $u \in \mathcal{A}$, $x \mapsto x\cdot y$ and $x \mapsto z\cdot x$ for every left and right multipliers of $\mathcal{A}$. In this paper we re-characterized the locally convex partial $*$-algebra $(\mathcal{A}, Γ,\cdot,*,τ)$ arising from these continuous maps in terms of convolution algebra of a Lie groupoid $Γ\rightrightarrows \mathcal{A}$. This is advantageous because the pathologies of the underlying spaces owing to their quantum mec…
The invariant subspaces of periodic Fourier multipliers with application to abstract evolution equations
本文用调和分析识别周期傅里叶乘子不变的大类Banach空间，并应用于抽象二阶积分微分方程，刻画其适定性与最大正则性条件，推广了现有理论。
We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of t…
We study $L^2$-maximal regularity for non-autonomous evolution equations of the form \begin{equation}\label{Abstract equation} \dot u(t)+\mathcal A(t)u(t)=f(t)\ \ t\in[0,T],\ \ u(0)=u_0. \end{equation} where $\mathcal A(t),\ t\in [0,T]$ arise from a non-autonomous sesquilinear forms $a(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ $L^2-$maximal regularity result is proved recently in \cite{Ar-Mo15} when $a$ is Hölder continuous of type $α>1/2.$ In this paper we recover the same results by an approximation method developed in \cite{ELLA13}, \cite{LASA14} and \cite{EL…
Let $1< α<2$ and $A$ be the generator of an $α$-times resolvent family $\{S_α(t)\}_{t \ge 0}$ on a Banach space $X$. It is shown that the fractional Cauchy problem ${\bf D}_t^αu(t) = Au(t)+f(t)$, $t \in [0,r]$; $u(0), u'(0) \in D(A)$ has maximal regularity on $C([0,r];X)$ if and only if $S_α(\cdot)$ is of bounded semivariation on $[0,r]$.
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