Compactness in the space $L^p(0,T;B)$, $B$ being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961, 1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting...

Let $L^{\varphi }\left(X\right)$ be an Orlicz-Bochner space defined by an Orlicz function $\varphi$ taking only finite values (not necessarily convex) over a $\sigma$-finite atomless measure space. It is proved that the topological dual $L^{\varphi }{\left(X\right)}^{*}$ of $L^{\varphi }\left(X\right)$ can be represented in the form: $L^{\varphi }{\left(X\right)}^{*}=L^{\varphi }{\left(X\right)}_n^{\sim }\oplus L^{\varphi }{\left(X\right)}_s^{\sim }$, where $L^{\varphi }{\left(X\right)}_n^{\sim }$ and $L^{\varphi }{\left(X\right)}_s^{\sim }$ denote the order continuous dual and the singular dual of $L^{\varphi }\left(X\right)$ respectively. The spaces $L^{\varphi }{\left(X\right)}^{*}$, $L^{\varphi }{\left(X\right)}_n^{\sim }$ and $L^{\varphi }{\left(X\right)}_s^{\sim }$ are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory...

We enlarge the amount of embeddings of the group G of invertible transformations of [0,1] into spaces of bounded linear operators on Orlicz spaces. We show the equality of the inherited coarse topologies.

rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p),\theta }(X,\mu )$ to $L^{q),q\theta /p}(X,\nu )$ (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace inequalities...

Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions...

Nous étudions d’abord la transformation de Fourier sur les espaces ${\ell }^p(L^{p^{\text{'}}})$ qui sont formés de fonctions appartenant localement à $L^{p^{\text{'}}}$ et se comportant à l’infini comme des éléments de ${\ell }^p$. Si $1\le p,p^{\text{'}}\le 2$, les transformées de Fourier des éléments de ${\ell }^p(L^{p^{\text{'}}})$ appartiennent à ${\ell }^{q^{\text{'}}}(L^q)$. Dans les autres cas, nous donnons quelques résultats partiels.Nous montrons ensuite que ${\ell }^2(L^1)$ est le plus grand espace vectoriel solide de fonctions mesurables sur lequel la transformation de Fourier puisse se définir par prolongement par continuité.

We study convolution operators bounded on the non-normable Lorentz spaces $L^{1,q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1,q}$. In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals,...

Our aim in this paper is to establish Trudinger’s inequality on Musielak-Orlicz-Morrey spaces $L^{\Phi ,\kappa }\left(G\right)$ under conditions on $\Phi$ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger’s inequality for double phase functionals $\Phi (x,t)=t^{p\left(x\right)}+a\left(x\right)t^{q\left(x\right)}$, where $p(·)$ and $q(·)$ satisfy log-Hölder conditions and $a(·)$ is nonnegative, bounded and Hölder continuous.

R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on $L^2$ maps $L^{\infty }$ into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by $C\left|R{|}^s\right|R{|}^{-s}$, for some $s\&gt;0$. We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided $s\&gt;s_0\left(E\right)$. We also show that the Calderon-Coifman bicommutators,...