This paper deals with regulated functions having values in a Banach space. In particular, families of equiregulated functions are considered and criteria for relative compactness in the space of regulated functions are given.

In this paper, we introduce a new class of variational inequality with its weak and split forms to obtain an $LU$-optimal solution to the multi-dimensional interval-valued variational problem, which is a wider class of interval-valued programming problem in operations research. Using the concept of (strict) $LU$-convexity over the involved interval-valued functionals, we establish equivalence relationships between the solutions of variational inequalities and the (strong) $LU$-optimal solutions of the multi-dimensional...

We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_{\alpha }$ maps weak weighted Orlicz$-\phi$ spaces into appropriate weighted versions of the spaces $BM{O}_{\psi }$, where $\psi \left(t\right)=t^{\alpha /n}{\phi }^{-1}(1/t)$. This generalizes known results about boundedness of $I_{\alpha }$ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha$ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_{\alpha }$ acting on weak$-L_{\phi }$ for $\phi$ of lower type equal or greater than $n/\alpha$, is the same as the one solving the problem for weak...

We prove here that the Poincaré-Sobolev pointwise inequalities for the relative rearrangement can be considered as the root of a great number of inequalities in various sets not necessarily vector spaces. In particular, new interpolation inequalities can be derived.

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean...

We study properties of the functional$$\begin{array}{c}\hfill {ℱ}_{\mathrm{loc}}(u,\Omega ):=\underset{\left(u_j\right)}{inf}\left\{\underset{j\to \infty }{lim\; inf}{\int }_{\Omega }f\left(\nabla u_j\right)x\left|\begin{array}{cc}& \left(u_j\right)\subset W_{\mathrm{loc}}^{1,r}\left(\Omega ,\right)\hfill \\ & u_{j}u\mathrm{in}\left(\Omega ,\right)\hfill \end{array}\right.\right\},\end{array}$$F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , whereu ∈ BV(Ω;RN), and f:RN × n → R is continuous and satisfies 0 ≤ f(ξ) ≤ L(1 + | ξ | r). For r ∈ [1,2), assuming f has linear growth in certain rank-one directions, we combine a result of [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737–756] with a new technique involving mollification to prove an upper bound for Floc. Then, for $r\in [1,\frac{n}{n-1})$ r ∈ [ 1 , n n − 1 ) , we prove that...