Let $S_{n}f$ denote the $n$th partial sum of the Vilenkin-Fourier series of a function $f \in L^{1}(G)$. For $1 < p_{-} \leq p_{+} < \infty$, we characterize all exponents $p(\cdot)$ for which the convergence of $S_{n}f$ to $f$ in $L^{p(\cdot)}(G)$ holds whenever $f \in L^{p(\cdot)}(G)$.
The main aim of this paper is to investigate Paley type and Hardy-Littlewood type inequalities and strong convergence theorem of partial sums of Vilenkin-Fourier series.
The main aim of this paper is to investigate (H_{p},L_{p})-type inequalities for maximal operators of logarithmic means of one-dimensional Vilenkin-Fourier series.
TThe main aim of this paper is to prove that there exist a martingale $f\in H_{1/2}$ such that Fejér means of Vilenkin-Fourier series of the martingale $f$ is not uniformly bounded in the space $L_{1/2}.$
The main aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. We also use our results to prove approximation and strong convergence theorems on the martingale Hardy spaces $H_{p},$ when $0<p\leq 1.$
The main aim of this paper is to investigate $\left( H_{p},L_{p}\right) $ and $\left( H_{p},L_{p,\infty }\right) $ type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin-Fourier series.
In this short article we introduce so-called anisotropic (weight) Grand Lebesgue Spaces (more exactly, Grand Lebesgue-Riesz Spaces), which are generalization of the classical Lebesgue-Riesz Spaces and ordinary Grand Lebesgue Spaces, and investigate the boundedness of weight Riesz potential and weight Fourier transforms in this spaces.
The main aim of this paper is to prove that the maximal operator $\overset{% \sim }{σ}^{*}f:=\underset{n\in P}{\sup }\frac{\left| σ_{n}f\right| }{\log ^{2}\left( n+1\right) }$ is bounded from the Hardy space $H_{1/2}$ to the space $L_{1/2}$, where $σ_{n}f$ is Fejér means of bounded Vilenkin-Fourier series.
We consider multiplication properties of elements in weighted Fourier Lebesgue and modulation spaces. Especially we extend some results by Pilipovic, Teofanov and Toft (2010).
We derive in this article the exact norm in the Grand Lebesgue Spaces (GLS) estimates for Fourier transform acting on the functions defined in the infinite local compact Abelian (LCA) group, compact or discrete.
In this article we investigate the Fourier series and transforms for the functions defined on the [-pi, pi]^ d or on the R^d and belonging to the (Bilateral) Grand Lebesgue Spaces. As a particular case we obtain some results about Fourier's transform in the so-called exponential Orlicz spaces. We construct also several examples to show the exactness of offered estimations.
Let ${\mathcal L} A(H)$ be the Lebesgue-Fourier space of a hypergroup $H$ considered as a Banach space on $H$. In addition ${\mathcal L}A(H)$ is a Banach algebra with the multiplication inherited from $L^1(H)$. Moreover If $H$ is a regular Fourier hypergroup, ${\mathcal L}A(H)$ is a Banach algebra with pointwise multiplication. We study the amenability and character amenability of these two Banach algebras.
We define and study wave-front sets for weighted Fourier-Lebesgue spaces when the weights are moderate with respect to the associated functions for general sequences $\{ M_p\} $ which satisfy Komatsu's conditions $(M.1) - (M.3)'$. In particular, when $\{ M_p\} $ is the Gevrey sequence ($M_p = p!^s$, $s>1$) we recover some previously observed results. Furthermore, we consider wave-front sets for modulation spaces in the same setting, and prove the invariance property related to the Fourier-Lebesgue type wave-front sets.
We consider different types of (local) products $f_1 f_2$ in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wave-front properties. We also consider semi-linear equations of the form $$ \qquad P(x,D)f = G(x,J_k f), $$ with appropriate polynomials $P $ and $G$. If the solution locally belongs to appropriate weighted Fourier Lebesgue space ${\mathscr F}L^q_{(ω)} (\rr d)$ and $P$ is non-characteristic at $(x_0,ξ_0),$ then we prove that $(x_0,ξ_0)\not \in WF_{{\mathscr F}L^q_{(\widetilde ω)}} (f)$, where $\widetildeω$ dep…
We derive bilateral estimates for the constants appearing in the Fourier transform restricted theorems on the Euclidean sphere for the ordinary and especially radial functions belonging to the Lebesgue-Riesz spaces as well as belonging to the Grand Lebesgue Spaces. We obtain an exact estimate for the norm of the restriction Fourier transform operator acting on the radial functions.
We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by H. Mascarenhas, P. Santos and M. Seidel.
A new general Hormander type condition involving anisotropies and mixed norms is introduced, and boundedness results for Fourier multi- pliers on anisotropic Besov and Triebel-Lizorkin spaces of distributions with mixed Lebesgue norms are obtained. As an application, the continuity of such operators is established on mixed Sobolev and Lebesgue spaces too. Some lifting properties and equivalent norms are obtained as well.
This is an attempt of a comprehensive survey of the results in which estimates of the norms of linear means of multiple Fourier series, the Lebesgue constants, are obtained by means of estimating the Fourier transform of a function generating such a method. Only few proofs are given in order to illustrate a general idea of techniques applied. Among the results are well known elsewhere as well as less known or published in an unacceptable journals and several new unpublished results.
In this paper we generalize Bochkariev's theorem, which states that for any uniformly bounded orthonormal system $Φ$, there exists a Lebesgue integrable function such that the Fourier series of it with respect to system $Φ$ diverge on the set of positive measure. We characterize the class of variable exponent Lebesgue spaces $L^{p(\cdot)}[0;1]$, $1<p(x)<\infty$ a.e. on [0;1], such that above mentioned Bochkarev's theorem is valid.
We carry on the study of Fourier integral operators of H{ö}rmander's type acting on the spaces $(\mathcal{F}L^p)_{comp}$, $1\leq p\leq\infty$, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank $r$ of the Hessian of the phase $Φ(x,η)$ with respect to the space variables $x$. Indeed, we show that operators of order $m=-r|1/2-1/p|$ are bounded on $(\mathcal{F}L^p)_{comp}$, if the mapping $x\longmapsto\nabla_xΦ(x,η)$ is constant on the fibers, of codimension …
Let $ω,ω_0$ be appropriate weight functions and $q\in [1,\infty ]$. We introduce the wave-front set, $\WF_{\mathscr FL^q_{(ω)}}(f)$ of $f\in \mathscr S'$ with respect to weighted Fourier Lebesgue space $\mathscr FL^q_{(ω)}$. We prove that usual mapping properties for pseudo-differential operators $\op (a)$ with symbols $a$ in $S^{(ω_0)}_{ρ, 0}$ hold for such wave-front sets. Especially we prove \WF_{\mathscr FL^q_{(ω/ω_0)}}(\op (a)f)\subseteq \WF_{\mathscr FL^q_{(ω)}}(f) \subseteq \WF_{\mathscr FL^q_{(ω/ω_0)}}(\op (a)f)\ttbigcup \Char (a). %% Here $\Char (a)$ is the set of characteristic point…
We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. Especially we recover some known results.
We obtain discrete characterizations of wave front sets of Fourier-Lebesgue and quasianalytic type. It is shown that the microlocal properties of an ultradistribution can be obtained by sampling the Fourier transforms of its localizations over a lattice in $\mathbb{R}^{d}$. In particular, we prove the following discrete characterization of the analytic wave front set of a distribution $f\in\mathcal{D}'(Ω)$. Let $Λ$ be a lattice in $\mathbb{R}^{d}$ and let $U$ be an open convex neighborhood of the origin such that $U\capΛ^{*}=\{0\}$. The analytic wave front set $WF_{A}(f)$ coincides with the co…
In this article we investigate an action of some operators (not necessary to be linear or sublinear) in the so-called (Bilateral) Grand Lebesgue Spaces (GLS), in particular, double weight Fourier operators, maximal operators, imbedding operators etc. We intend to calculate an exact or at least weak exact values for correspondent imbedding constant. We obtain also interpolation theorems for GLS spaces.We construct several examples to show the exactness of offered estimations. In two last sections we introduce anisotropic Grand Lebesgue Spaces, obtain some estimates for Fourier two-weight inequa…
We study divergence properties of Fourier series on Cantor-type fractal measures, also called mock Fourier series. We show that in some cases the $L^1$-norm of the corresponding Dirichlet kernel grows exponentially fast, and therefore the Fourier series are not even pointwise convergent. We apply these results to the Lebesgue measure to show that a certain rearrangement of the exponential functions, which we call scrambled Fourier series, have a corresponding Dirichlet kernel whose $L^1$-norm grows exponentially fast, which is much worse than the known logarithmic bound. The divergence propert…
Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. Suppose $a$ is a Fourier multiplier on the space $X(\mathbb{R})$. We show that the Fourier convolution operator $W^0(a)$ with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$. This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.
We estimate the truncated double trigonometric series $\sum_{n=0}^{N}\sum_{m=0}^{M}a_{mn} {e}^{2π\imath \left(m x+n y\right)}$, $a_{mn} \in\mathbb{C}$, in Lebesgue spaces with mixed norms in terms of the $p^{th}-q^{th}$ power finite double sums of its coefficients. We obtain these estimates for all possible values of the exponents involved then we provide examples of matrices in ${\mathbb{C}}^{M\times N}$ that maximize some of them up to a constant independent of $M$ and $N$.
We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets; unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum, that is, by a subset of the group; the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2; the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.
We discuss continuity of the twisted convolution on (weighted) Fourier modulation spaces. We use these results to establish continuity results for the twisted convolution on Lebesgue spaces. For example we prove that if $ω$ is an appropriate weight and $1\le p\le 2$, then $L^p_{(ω)}$ is an algebra under the twisted convolution.
For $μ>β>0$, the generalized Stieltjes operators $$ \mathcal{S}_{β,μ} f(t):={t^{μ-β}}\int_0^\infty {s^{β-1}\over (s+t)^μ}f(s)ds, \qquad t>0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(α)}(t^α)$ (where $α\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < β- \pp < μ$, then operators $\mathcal{S}_{β,μ}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Cesáro operator on $\mathcal{T}_p^{(α)}(t^α)$ . We calculate and represent explicitly their spectrum…
Last years there was increasing an interest to the so called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. …
We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.
We study inequalities of the form \begin{equation*} ρ( \lvert \hat{f} \rvert) \leq C σ(f) < \infty, \end{equation*} with $f \in L_{1}(\mathbb{R}^n)$, the Lebesgue-integrable functions on $\mathbb{R}^n$ and \begin{equation*} \hat{f}(ξ) := \int_{\mathbb{R}^n} f(x) \, e^{- 2 πi ξ\cdot x} dx, \ \ \ ξ\in \mathbb{R}^n. \end{equation*} The functionals $ρ$ and $σ$ are so-called rearrangement-invariant (r.i.) norms on $M_{+}(\mathbb{R}^n)$, the nonnegative measurable functions on $\mathbb{R}^n$. Results first proved in the general context of r.i. spaces are then both specialized and expanded on in the …
We study the cyclicity of vectors $u$ in $\ell^p(\mathbb{Z})$. It is known that a vector $u$ is cyclic in $\ell^2(\mathbb{Z})$ if and only if the zero set, $\mathcal{Z}(\widehat{u})$, of its Fourier transform, $\widehat{u}$, has Lebesgue measure zero and $\log |\widehat{u}| \not \in L^1(\mathbb{T})$, where $\mathbb{T}$ is the unit circle. Here we show that, unlike $\ell^2(\mathbb{Z})$, there is no characterization of the cyclicity of $u$ in $\ell^p(\mathbb{Z})$, $1<p<2$, in terms of $\mathcal{Z}(\widehat{u})$ and the divergence of the integral $\int\_\mathbb{T} \log |\widehat{u}| $. Moreover w…
In this paper we investigate the foundations for analysis in infinitely-many (independent) variables. We give a topological approach to the construction of the regular $\s$-finite Kirtadze-Pantsulaia measure on $\R^\iy$ (the usual completion of the Yamasaki-Kharazishvili measure), which is an infinite dimensional version of the classical method of constructing Lebesgue measure on $\R^n$ (see \cite{YA1}, \cite{KH} and \cite{KP2}). First we show that von Neumann's theory of infinite tensor product Hilbert spaces already implies that a natural version of Lebesgue measure must exist on $\R^{\iy}$.…
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
