Related papers: Some interpolation inequalities in Lorentz, Morrey…

In this manuscript, we examine the continuity properties of the Riemann-Liouville fractional integral for order $\alpha = 1/p$, where $p > 1$, mapping from $L^p(t_0, t_1; X)$ to the Banach space $BMO(t_0, t_1; X)\cap K_{(p-1)/p}(t_0, t_1;…

Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the sub-Laplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\"older…

In this note, we consider the metric Ramsey problem for the normed spaces l_p. Namely, given some 1<=p<=infinity and alpha>=1, and an integer n, we ask for the largest m such that every n-point metric space contains an m-point subspace…

Fix $p>2$. We prove that the Euclidean distortion of every $n$-point subset of $L_p$ is $p^3(\log n)^{\frac12+o(1)}$, thus, in particular, demonstrating that all $n$-point subsets of $L_p$ exhibit an asymptotic improvement over the $O(\log…

In this paper, we analyze carefully the behaviour in $L^\infty(\mathbb R)$ of the square functions $S$ and $S_\mathcal I$'s, originating from ergodic theory. Firstly, we show that we can find some function $f\in L^\infty(\mathbb{R})$, such…

Kahn and Kim (J. Comput. Sci., 1995) have shown that for a finite poset $P$, the entropy of the incomparability graph of $P$ (normalized by multiplying by the order of $P$) and the base-$2$ logarithm of the number of linear extensions of…

We propose results on interior Morrey, BMO and H\"older regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey spaces.

The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the Hardy-Littlewood maximal operator $M$, and thus, we consider the boundedness of $M$ in the…

We consider two questions on the geometry of Lipschitz free $p$-spaces $\mathcal F_p$, where $0<p\leq 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal M, \rho)$ is an infinite…

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…

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,\mu)$. The embedding of the Newton-Morrey-Sobolev space into the H\"older space is obtained if $\mathscr{X}$ supports a weak…

Let $(\Omega,\Sigma,\mu)$ be a measure space and $1< p < +\infty$. In this paper we show that, under quite general conditions, the set $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$ is maximal spaceable, that is, it contains…

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following:…

Let $\mathcal{M}$ be a von Neumann algebra equipped with a normal semifinite faithful trace, $(\mathbb{X},\,d,\,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and…

We consider a family of Gagliardo-Nirenberg-Sobolev interpolation inequalities which interpolate between Sobolev's inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the…

It is shown that every Banach space either contains $\ell ^1$ or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, $L^p(\lambda )$, $1<p<\infty $, is a quotient of a H.I Banach space.

In this paper the continuity of the set valued map $p\rightarrow B_{\Omega,\mathcal{X},p}(r),$ $p\in (1,+\infty),$ is proved where $B_{\Omega,\mathcal{X},p}(r)$ is the closed ball of the space $L_{p}\left(\Omega,\Sigma,\mu;…

We propose and study a general quasi-interpolation framework for stochastic function approximation, which stems and draws motivation from convolution-type solutions for certain practical weighted variational problems. We obtain our…

We show that the norm in the Hardy space $H^p$ satisfies \begin{equation}\label{absteq} \|f\|_{H^p}^p\asymp\int_0^1M_q^p(r,f')(1-r)^{p\left(1-\frac1q\right)}\,dr+|f(0)|^p\tag{\dag} \end{equation} for all univalent functions provided that…

The Lipschitz extension modulus $e(M)$ of a metric space $M$ is the infimum over $L\ge 1$ such that for any Banach space $Z$ and any $C\subset M$, any 1-Lipschitz function $f:C\to Z$ can be extended to an $L$-Lipschitz function $F:M\to Z$.…