Related papers: Note on Fourier inequalities

The Hausdorff-Young inequality for Euclidean space, in its sharp form due to Beckner, gives an upper bound for the Fourier transform in terms of Lebesgue space norms, with an optimal constant. The extremizers have been identified by Lieb to…

We give a~new proof of the known criteria for the inequality \begin{equation*} \left(\int_{0}^{\infty}\left(\int_{0}^{t}f\right)^{q}w(t)\,dt\right)^{\frac{1}{q}} \leq C \left(\int_{0}^{\infty}f^{p}v\right)^{\frac{1}{p}}. \end{equation*} The…

We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality $$ \|e^{-\tau\ell(\cdot)} u^{\frac 1q} f\|_q\leq c_\tau\| e^{-\tau\ell(\cdot)} v^{\frac 1p}\, \nabla f\|_p, \quad f\in C^\infty_0( R^n).…

Let $p(\cdot):\ \mathbb{R}^n\to(0,1]$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable…

We consider a family of jointly Gaussian random vectors $\xi_j \in \mathbb{R}^{k_j}$, each standard normal but possibly correlated, and investigate when\[ \mathbb{E}\, F\!\Bigl(B\bigl(|T_{z_1} f_1(\xi_1)|,\dots,|T_{z_n}…

We strengthen Han's Fourier entropy-influence inequality $$ H[\widehat{f}] \leq C_{1}I(f) + C_{2}\sum_{i\in [n]}I_{i}(f)\ln\frac{1}{I_{i}(f)} $$ originally proved for $\{-1,1\}$-valued Boolean functions with $C_{1}=3+2\ln 2$ and $C_{2}=1$.…

In this note, we present two arguments showing that the classical \textit{linear adjoint cone restriction conjecture} holds for the class of functions supported on the cone and invariant under the spatial rotation in all dimensions. The…

In this paper, we prove the following inequality \begin{equation*} \|\big(\int_{\mathbb{R}^n}\frac{|f(\cdot+y)-f(\cdot)|^q}{|y|^{n+sq}}dy\big)^{\frac{1}{q}}\|_{L^{p,\infty}(\mathbb{R}^n)}\lesssim\|f\|_{\dot{L}^p_s(\mathbb{R}^n)},…

The method of using rearrangements to give sufficient conditions for Fourier inequalities between weighted Lebesgue spaces is revisited. New results in the case q < p are established and a comparison between two known sufficient conditions…

We prove that in variable exponent spaces $L^{p(\cdot)}(\Omega)$, where $p(\cdot)$ satisfies the log-condition and $\Omega$ is a bounded domain in $\mathbf R^n$ with the property that $\mathbf R^n \backslash \bar{\Omega}$ has the cone…

We work in a discrete model of the nonlinear Fourier transform (following the terminology of Tao and Thiele), which appears in the study of orthogonal polynomials on the unit circle. The corresponding nonlinear variant of the…

We derive a number of equivalent criterions for the variable exponent Hardy type inequality |\frac{1}{x}\int_{0}^{x}f(t)dt|_{L^{p(.)}(0,1)}\leq C|f|_{L^{p(.)}(0,1)}; f\geq 0. to hold, whenever the exponent $p:(0,1)\to (1,\infty)$ is…

In this paper we characterize the validity of the Hardy-type inequality \begin{equation*} \left\|\left\|\int_s^{\infty}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,w,\infty}\leq c \,\|h\|_{1,v,\infty} \end{equation*} where $0<p< \infty$, $0<q\leq…

We study norm inequalities for the Fourier transform, namely, \begin{equation}\label{introduction} \|\widehat f\|_{X_{p,q}^\lambda} \lesssim \|f\|_{Y}, \end{equation} where $X$ is either a Morrey or Campanato space and $Y$ is an appropriate…

In this paper, we obtain non-symmetric and symmetric versions of the classical Heisenberg-Pauli-Weyl uncertainty principle in Lebesgue spaces with power weights.

In this work we define a Fourier transform for each $f\in L^{p(\cdot)}(\mathbb{R})$, for a large class of exponent functions $p(\cdot)$, as the distributional derivative of a H\"older continuous function. A norm is defined in the space of…

If $\phi$ is an analytic selfmap of the disk (not an elliptic automorphism) the Denjoy-Wolff Theorem predicts the existence of a point $p$ with $|p|\leq 1$ such that the iterates $\phi_{n}$ converge to $p$ uniformly on compact subsets of…

We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. Especially we recover some known results.

Let p and q be conjugate exponents, with p in [1,2]. It is shown that the Laplace transform acts boundedly between the Lp space with unit weight on the positive real semiaxis and the Lq space weighted by a well-projected measure (a term…

We refine the classical Cauchy--Schwartz inequality $\|X\|_{1} \leq \|X\|_{2}$ by demonstrating that for any $p$ and $q$ with $q>p>2$, there exists a constant $C=C(p,q)$ such that $\|X\|_1 \leq 1 - C \Big{(}\|X\|_p^p -…