Related papers: Note on order-isomorphic isometric embeddings of s…

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$, and $\theta$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space…

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 $E$…

In this paper we prove some results related to the problem of isomorphically classifying the complemented subspaces of $X_{p}$. We characterize the complemented subspaces of $X_{p}$ which are isomorphic to $X_{p}$ by showing that such a…

Let $p$ be a prime number. Motivated by the local lifting problem for $(\mathbb{Z}/p\mathbb{Z})^n$ with $n>1$, we prove several new results on certain $\mathbb{F}_p$-vector spaces of logarithmic differential forms on the projective line in…

We introduce a notion of elliptic differential graded Lie algebra. The class of elliptic algebras contains such examples as the algebra of differential forms with values in endomorphisms of a flat vector bundle over a compact manifold, etc.…

Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a…

This is an exposition of the known techniques for constructing $\Cal L_p$-spaces for $p\in (1,\infty)\setminus \{2\}$, including some unpublished work of Alspach. Isomorphic and complemented embedding relations between these spaces are also…

We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\infty$ and…

We introduce Besov spaces with variable smoothness and integrability by using the continuous version of Calder\`on reproducing formula. We show that our space is well-defined, i.e., independent of the choice of basis functions. We…

We give conditions on the exponent function $p(\cdot)$ that imply the existence of embeddings between grand, small and variable Lebesgue spaces. We construct examples to show that our results are close to optimal. Our work extends recent…

Using a recent result of Batson, Spielman and Srivastava, We obtain a tight estimate on the dimension of $\ell_p^n$, $p$ an even integer, needed to almost isometrically contain all $k$-dimensional subspaces of $L_p$.

In this paper we connect Calder\'on and Zygmund's notion of $L^p$\- -differentiability with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu. We show how the…

We prove that if $X$ is a subspace of $L_p$ $(2<p<\infty)$, then either $X$ embeds isomorphically into $\ell_p \oplus \ell_2$ or $X$ contains a subspace $Y,$ which is isomorphic to $\ell_p(\ell_2)$. We also give an intrinsic…

We give a sharp sufficient condition on the distribution function, $|\{x\in \Omega :\,p(x)\leq 1+\lambda\}|$, $\lambda>0$, of the exponent function $p(\cdot): \Omega \to [1,\infty)$ that implies the embedding of the variable Lebesgue space…

We show that for all odd primes $p$, there exist ordinary elliptic curves over $\bar{\mathbb{F}}_p(x)$ with arbitrarily high rank and constant $j$-invariant. This shows in particular that there are elliptic curves with arbitrarily high rank…

Based on the characterization of surjective $L^p$-isometries of unitary groups in finite factors, we describe all surjective $L^p$-isometries between Grassmann spaces of projections with the same trace value in semifinite factors.

This paper studies absolute integrability for functions with values in semi- normed spaces and in locally convex topological vector spaces (LCTVS). We introduce an \emph{upper-integral} approach (based on a $\rho$-variational measure…

The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in $\mathbb R^n$. An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all…

In this article, we introduce a new object, a virtual quadratic space, and its group of isometries. They are presented as natural generalizations of quadratic spaces and orthogonal groups. It is then shown that by replacing quadratic spaces…

We introduce the mixed-norm amalgam spaces $(L^{\vec{p}},L^{\vec{s}})(\mathbb{R}^n)$ and $(L^{\vec{p}},L^{\vec{s}})^{\alpha}(\mathbb{R}^n)$, and show their some basic properties. In addition, we find the predual…