Related papers: The Distorion Problem

We consider norms on a complex separable Hilbert space such that $\langle a\xi,\xi\rangle\leq\|\xi\|^2\leq\langle b\xi,\xi\rangle$ for positive invertible operators $a$ and $b$ that differ by an operator in the Schatten class. We prove that…

Let H be a Hilbert space and let F be the family of all countable subsets of an orthonormal basis of H. We show that if F is infinite then F is equipollent with every linear basis of the vector space H. In doing so we also present a short…

Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a…

We show that if the dyadic Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the Hilbert transform, with a linear relation of the bounds. This result is the counterpart of [arXiv:2212.00090] where the opposite…

Given $k\in \mathbb{R},$ $v,$ $D>0,$ and $n\in \mathbb{N},$ let $\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }$ be a Gromov-Hausdorff convergent sequence of Riemannian $n$--manifolds with sectional curvature $\geq k,$ volume $>v,$ and…

For a connected $n$-dimensional compact smooth hypersurface $M$ without boundary embedded in $\mathbb{R}^{n+1}$, a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a…

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $\kappa$. For any point $p\in M$ and any number $\lambda\in\mathbb{R}$ with $\lambda^2+\kappa>0$, we show existence of a $C^2$ spherical surface…

We show that, given a metric space $(Y,d)$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $\mu$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,\mu)$ is…

We prove some existence results for the Webster scalar curvature problem on the Heisenberg group and on the unit sphere of ${\mathbb C}^{n+1}$, under the assumption of some natural symmetries of the prescribed curvatures. We use variational…

We prove oscillation stability for the Banach space $\ell_\infty$: every weak-* Borel, uniformily continuous map from the unit sphere of this space to a compact metric space can be made arbitrarily close to a constant map when restricted to…

The Banach isometric conjecture asserts that a normed space with all of its $k$-dimensional subspaces isometric, where $k\geq 2$, is Euclidean. The first case of $k=2$ is classical, established by Auerbach, Mazur and Ulam using an elegant…

We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being…

A linear operator $U$ acting boundedly on an infinite-dimensional separable complex Hilbert space $H$ is universal if every linear bounded operator acting on $H$ is similar to a scalar multiple of a restriction of $U$ to one of its…

A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved…

In this paper we show how some metric properties of the unit sphere of a normed space can help to approach a solution to Tingley's problem. In our main result we show that if an onto isometry between the spheres of strictly convex spaces is…

We construct a bounded and symmetric convex body in $\ell_2(\Gamma)$ (for certain cardinals $\Gamma$) whose translates yield a tiling of $\ell_2(\Gamma)$. This answers a question due to Fonf and Lindenstrauss. As a consequence, we obtain…

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we…

Following results of Bourgain and Gorelik we show that the spaces $\ell_p$, $1<p<\infty$, as well as some related spaces have the following uniqueness property: If $X$ is a Banach space uniformly homeomorphic to one of these spaces then it…

We show that if there exists a Lipschitz homeomorphism $T$ between the nets in the Banach spaces $C(X)$ and $C(Y)$ of continuous real valued functions on compact spaces $X$ and $Y$, then the spaces $X$ and $Y$ are homeomorphic provided…

Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575--582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap $\mathbb{L}_2$-discrepancy to give the distance integral of the…