Related papers: Morrey spaces over the unit circle cannot be renor…

Fix $b\in (0,\infty)$ and $p\in (1,\infty)$. Let $\phi$ be a positive measurable function on $I_b:=(0,b)$. Define the Lorentz Gamma norm, $\r_{p,\phi}$, at the measurable function $f:\R+\to\R+$ by…

The following theorem is the main result of this note. Theorem 1. Let $(E, \|\cdot\|_E) $ be a rearrangement invariant Banach function space on the interval $[0, 1]$. If $E$ is isometric to $\L_p [0, 1]$ for some $1\le p<\infty$, then $E$…

We prove that if $X$ is a real rearrangement-invariant function space on $[0,1]$, which is not isometrically isomorphic to $L_2,$ then every surjective isometry $T:X\to X$ is of the form $Tf(s)=a(s)f(\sigma(s))$ for a Borel function $a$ and…

Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces $m_{u,p}=m_{u,p}(\mathbb{Z}^d)$,…

Rearrangement-invariance in function spaces can be viewed as a kind of generalization of 1-symmetry for Schauder bases. We define subrearrangement-invariance in function spaces as an analogous generalization of 1-subsymmetry. It is then…

We study inequalities of the form \begin{equation*} \rho ( \lvert \hat{f} \rvert) \leq C \sigma(f) < \infty, \end{equation*} with $f \in L_{1}(\mathbb{R}^n)$, the Lebesgue-integrable functions on $\mathbb{R}^n$ and \begin{equation*}…

In this note, we establish several interpolation inequalities in $\mathbb R^n$ in the Lebesgue spaces and Morrey spaces. By using the classical Calderon--Zygmund decomposition, we will reprove that $L^{p}(\mathbb…

Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be probability spaces in each of which there is at least one measurable set whose…

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, the author establishes some interpolation results between Lorentz, Morrey and BMO spaces. Let $1<p<\infty$ and $p\leq r\leq\infty$. It is proved that the space $L^{p,r}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)$ is…

Given $p\in [1,\infty)$ and $\lambda\in (0,n)$, we study Morrey space ${\rm L}^{p,\lambda}({\mathbb R}^n)$ of all locally integrable complex-valued functions $f$ on ${\mathbb R}^n$ such that for every open Euclidean ball $B\subset{\mathbb…

Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces can not handle. Morrey spaces which we have been handling are called classical Morrey spaces. However, classical Morrey spaces are not totally enough…

We prove that the weak Morrey space $WM^{p}_{q}$ is contained in the Morrey space $M^{p}_{q_{1}}$ for $1\leq q_{1}< q\leq p<\infty$. As applications, we show that if the commutator $[b,T]$ is bounded from $L^p$ to $L^{p,\infty}$ for some…

We investigate a rearrangement inequality for pairs of n-square matrices: Let |A\|_p denote the C^p trace norm of an n-square matrix A. Consider the quantity |A+B|_p^p + |A-B|_p^p. Under certain positivity conditions, we show that this is…

We consider classical Tsirelson-type norms of $T[A_r,\theta]$ and their modified versions on $\ell_p$ spaces. We show that for any $1<p<\infty$ there is a constant $\lambda_p$ such that considered Tsirelson-type norms do not…

Let $f_1, f_2, ..., f_n$ be a family of independent copies of a given random variable f in a probability space $(\Omega, \mathcal{F}, \mu)$. Then, the following equivalence of norms holds whenever $1 \le q \le p < \infty$…

We give a characterization of $L^{p}(\sigma)$ for uniformly rectifiable measures $\sigma$ using Tolsa's $\alpha$-numbers, by showing, for $1<p<\infty$ and $f\in L^{p}(\sigma)$, that \[ \lVert f\rVert_{L^{p}(\sigma)}\sim…

Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces…

We construct a Schwartz function $\varphi$ such that for every exponentially small perturbation of integers $\Lambda$, the set of translates $\{\varphi(t-\lambda), \lambda\in\Lambda\}$ spans the space $L^p(R)$, for every $p > 1$. This…

In their 1976 paper, Nagel and Rudin characterize the closed unitarily and M\"obius invariant spaces of continuous and $L^p$-functions on a sphere, for $1\leq p<\infty$. In this paper we provide an analogous characterization for the…