Related papers: Duality for outer $L^{p,\infty}$ spaces
This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this…
An alternative characterization of Minkowski--Lyapunov functions is derived. The derived characterization enables a computationally efficient utilization of Minkowski--Lyapunov functions in arbitrary finite dimensions. Due to intrinsic…
We introduce a new norm, called $N^{p}$-norm $(1\leq{p}<\infty)$ on a space $N^{p}(V,W)$ where $V$ and $W$ are abstract operator spaces. By proving some fundamental properties of the space $N^{p}(V,W)$, we also obtain that if $W$ is…
We continue the analysis of reproducing pairs of weakly measurable functions, which generalize continuous frames. More precisely, we examine the case where the defining measurable functions take their values in a partial inner product space…
This paper studies the $H^0$ norm and $H^1$ seminorm of quadratic functions. The (semi)norms are expressed explicitly in terms of the coefficients of the quadratic function under consideration when the underlying domain is an $l_p$-ball (1…
We study a concept of inner function suited to Dirichlet-type spaces. We characterize Dirichlet-inner functions as those for which both the space and multiplier norms are equal to 1.
We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces, and the so-called $\mathcal{L}$, $\mathcal{R}$, $\mathcal{LL}$, $\mathcal{RL}$, $\mathcal{RL}$, and $\mathcal{RR}$ spaces. The quasi-norms for a function $f$ in…
We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of…
Recently, Brezis, Van Schaftingen and the second author established a new formula for the $\dot{W}^{1,p}$ norm of a function in $C^{\infty}_c(\mathbb{R}^N)$. The formula was obtained by replacing the $L^p(\mathbb{R}^{2N})$ norm in the…
In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not determined by the restriction of the…
Interpolation inequalities for $C^m$ functions allow to bound derivatives of intermediate order $0 < j<m$ by bounds for the derivatives of order $0$ and $m$. We review various interpolation inequalities for $L^p$-norms ($1 \le p \le…
We characterize the situation of having many normal measures on a measurable cardinal. We show the plausibility of having many normal measures on each compact cardinal.
We characterise $L_p$-norms on the space of integrable step functions, defined on a probabilistic space, via H\"older's type inequality with an optimality condition.
By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local…
We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le \infty$, for which the standard functional $$ ||f||_{p,s}=(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t})^{1/s} $$ is only a quasi-norm. We find the optimal constant in the…
After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well…
In this paper, we obtain an alternative expression for the distance of a function in $BLO$ from the subspace $L^\infty$. The distance is the one induced by choosing a new "norm" on $BLO$, equivalent to the usual one and that has the…
Two non-commutative versions of the classical L^q(L^p) norm on the algebra of (mn)x(mn) matrices are compared. The first norm was defined recently by Carlen and Lieb, as a byproduct of their analysis of certain convex functions on matrix…
Let $p>n$ and let $L^1_p(R^n)$ be a homogeneous Sobolev space. For an arbitrary Borel measure $\mu$ on $R^n$ we give a constructive characterization of the space $L^1_p(R^n)+L_p(R^n;\mu)$. We express the norm in this space in terms of…
We demonstrate that the set $L^\infty(X, [-1,1])$ of all measurable functions over a Borel measure space $(X, \mathcal B, \mu )$ with values in the unit interval is typically non-polyhedric when interpreted as a subset of a dual space. Our…