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A coarse embedding of a metric space X into a metric space Y is a map f: X-->Y satisfying for every x, y in X: \phi_1(d(x,y)) \leq d(f(x),f(y)) \leq \phi_2(d(x,y)) where \phi_1 and \phi_2 are nondecreasing functions on [0,\infty) with…
This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces $\mcH_1$ and $\mcH_2$, such that $\mcH_1$ is a continuous dense…
A companion of Ostrowski like inequality for mappings whose second derivatives belong to $L^{\infty}$ spaces is established. Applications to composite quadrature rules, and to probability density functions are also given.
We prove that the spaces $\ell_p$, $1<p<\infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach-Mazur rotation…
This paper is devoted to the description of the lack of compactness of $H^1_{rad}(\R^2)$ in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P. -L. Lions. The approach that we adopt to…
Stochastic parabolic integro-differential problem is considered in the whole space. By verifying H\"ormander condition, the existence and uniqueness is proved in Lp-spaces of functions whose regularity is defined by a scalable Levy measure.…
We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that…
Our goal in this paper is to continue the study initiated by the authors in [Lipschitz free $p$-spaces for $0<p<1$; arXiv:1811.01265 [math.FA]] of the geometry of the Lipschitz free $p$-spaces over quasimetric spaces for $0<p\le1$, denoted…
We consider Sobolev spaces with values in Banach spaces as they are frequently useful in applied problems. Given two Banach spaces $X\neq\{0\}$ and $Y$, each Lipschitz continuous mapping $F:X\rightarrow Y$ gives rise to a mapping $u\mapsto…
In this paper, we study spaceability of subsets of generalized Orlicz and Lebesgue spaces associated to Banach function space. Also, we give some sufficient conditions for spaceability of subsets of a general Banach space which improves an…
Let $(\varphi_i)_{i=1}^n$ be mutually orthogonal functions on a probability space such that $\|\varphi_i\|_\infty \leq 1 $ for all $i \in [n]$. Let $\alpha > 0$. Let $\Phi(u) = u^2 \log^{\alpha}(u)$ for $u \geq u_{0}$, and $\Phi(u) =…
In a companion paper (Studia Math., 2023), we proved for every $\lambda\in(1,2]$ the existence of a $(\lambda^+)$-injective renorming of $\ell_\infty$ that is not $\lambda$-injective, thereby establishing a~forgotten theorem of…
We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and…
We consider the non-linear eigenvalue equations characterizing $L^p$ into $L^q$ Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the…
It is shown that for every $p\in (2,\infty)$ there exists a doubling subset of $L_p$ that does not admit a bi-Lipschitz embedding into $\R^k$ for any $k\in \N$.
We consider a generalization of the classic linear regression problem to the case when the loss is an Orlicz norm. An Orlicz norm is parameterized by a non-negative convex function $G:\mathbb{R}_+\rightarrow\mathbb{R}_+$ with $G(0)=0$: the…
We find conditions on a function space $\bf{L}$ that ensure that it behaves as an $L_p$-space in the sense that any unconditional basis of a complemented subspace of $\bf{L}$ either is equivalent to the unit vector system of $\ell_2$ or has…
In the dual $L_{\Phi^*}$ of a $\Delta_2$-Orlicz space $L_\Phi$, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology…
New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the…
In this paper, {we extend the affine dual curvature measures to the $L_p$ setting and solve the existence part of the corresponding Minkowski problem for non-symmetric discrete measures when $p>1$ and for symmetric measures when $p\geq0$.}…