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This paper introduces new parameterizations of equilibrium neural networks, i.e. networks defined by implicit equations. This model class includes standard multilayer and residual networks as special cases. The new parameterization admits a…

We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms and we show that they always admit a maximal extension which preserves the same invariance. A similar…

A basilar property and a useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space $W^{k,p}(\R^n)$ (i.e. the functions with weak derivatives of orders $0$ to $k$ in $L^p$). On Riemannian…

We introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for…

A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…

In this paper, we consider the characterizations of the Lipschitz spaces and homogeneous Lipschitz spaces associated to the biharmonic operator $\Delta^2.$ With this characterizations, we prove the boundedness of the Bessel potentials,…

A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to…

The paper deals with a nontrivial density result for $C^m(\overline{\Omega})$ functions, with $m\in{\mathbb N}\cup\{\infty\}$, in the space $$W^{k,\ell,p}(\Omega;\Gamma)= \left\{u\in W^{k,p}(\Omega): u_{|\Gamma}\in…

We investigate relations between symmetrizations of quasi-Banach function spaces and constructions such as Calderon-Lozanovskii spaces, pointwise product spaces and pointwise multipliers. We show that under reasonable assumptions the…

The well-known von Bahr--Esseen bound on the absolute $p$th moments of martingales with $p\in(1,2]$ is extended to a large class of moment functions, and now with a best possible constant factor (which depends on the moment function). This…

We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric-Sobolev space $H^{1,p}(X,\mathsf{d},\mathfrak{m})$ associated with a positive and finite Borel measure $\mathfrak{m}$ in a…

We prove that for every $n\in \mathbb{N}$ there exists a metric space $(X,d_X)$, an $n$-point subset $S\subseteq X$, a Banach space $(Z,\|\cdot\|_Z)$ and a $1$-Lipschitz function $f:S\to Z$ such that the Lipschitz constant of every function…

We aim to contribute to the folklore of function spaces on Lipschitz domains. We prove the boundedness of the trace operator for homogeneous Sobolev and Besov spaces on a special Lipschitz domain with sharp regularity. To achieve this, we…

We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional…

It is hereby established that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable…

Wigner functions are broadly used to probe non-classical effects in the macroscopic world. Here we develop an orbital-free functional framework to compute the 1-body Wigner quasi-probability for both fermionic and bosonic systems. Since the…

We consider convex monotone $C_0$-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a $\sigma$-Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and…

Many elliptic boundary value problems exhibit an interior regularity property, which can be exploited to construct local approximation spaces that converge exponentially within function spaces satisfying this property. These spaces can be…

It is shown here that if $(Y,\|\cdot\|_Y)$ is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists $c=c(Y)\in (0,\infty)$ with the following property. For every $n\in \mathbb{N}$ and…

We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite $p$th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global…