Composition of maximal operators

Consider the Hardy-Littlewood maximal operator $$Mf(x)=\sup_{Q\owns x}\frac{1}{|Q|}\int_Q |f(y)|\,dy.$$ It is known that $M$ applied to $f$ twice is pointwise comparable to the maximal operator $M_{L\log L}f$, defined by replacing the mean value of $|f|$ over the cube $Q$ by the $L\log L$-mean, namely $$M_{L\log L}f(x)=\sup_{x\in Q} \frac{1}{|Q|}\int_Q|f(y)| \log\left(e+\frac{|f|}{|f|_Q}\right)(y)\,dy,$$ where $|f|_Q=\frac{1}{|Q|}\int_Q|f|$ (see \cite{L}, \cite{LN}, \cite{P}). In this paper we prove that, more generally, if $\Phi(t)$ and $\Psi(t)$ are two Young functions, there exists a third function $\Theta(t)$, whose explicit form is given as a function of $\Phi(t)$ and $\Psi(t)$, such that the composition $M_\Psi\circ M_\Phi$ is pointwise comparable to $M_{\Theta}$. Through the paper, given an Orlicz function $A(t)$, by $M_A f$ we mean $$M_{A}f(x)=\sup_{Q\owns x}||f||_{A, Q}$$ where ||f||_{A, Q}=\inf \left\{\lambda > 0:\frac{1}{|Q|}\int_{Q} A\left(\frac{|f|}{\lambda}\right)(x)\, dx\le 1\right\}

Risultati di regolarità dei minimi locali di funzionali con coefficienti discontinui

We give an overview on recent regularity results of local vectorial minimizers of under two main features: the energy density is uniformly convex with respect to the gradient variable only at infinity and it depends on the spatial variable through a possibly discontinuous coefficient. More precisely, the results that we present tell that a suitable weak differentiability property of the integrand as function of the spatial variable implies the higher differentiability and the higher integrability of the gradient of the local minimizers. We also discuss the regularity of the local solutions of nonlinear elliptic equations under a fractional Sobolev assumption.Presentiamo alcuni recenti risultati di regolarità dei minimi locali vettoriali di funzionali integrali le cui caratteristiche principali sono che le densità di energia sono uniformemente convesse solo all’ infinito e che, come funzioni della variabile spaziale possono essere discontinue. Tali risultati possono essere sintetizzati come segue: una opportuna differenziabilità debole dell’ integrando rispetto alla variabile spaziale implica la maggiore differenziabilità e maggiore integrabilità del gradiente del minimo. Discutiamo anche la regolarità delle soluzioni locali di equazioni non lineari ellittiche sotto ipotesi di differenziabilità frazionaria

Bi-Sobolev mappings with differential matrices in Orlicz Zygmund classes

AbstractBi-Sobolev mappings f:Ω⊂R2→ontoΩ′⊂R2 have been defined as those homeomorphisms such that f and f−1 belong to Wloc1,1. We deduce regularity properties of the distortion of f from the regularity of the differential matrix Df−1 and conversely

On very weak solutions of a class of nonlinear elliptic systems

summary:In this paper we prove a regularity result for very weak solutions of equations of the type $- \operatorname{div} A(x,u,Du)=B(x, u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x, u, Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$ \cases -\operatorname{div} A(x,u,Du)\,=B(x, u,Du) \quad & \text{in } \Omega , \ u-u_o\in W^{1,r}(\Omega,\Bbb R^m). \endcases $

Regularity results for a class of obstacle problems with p,q−p,q-growth conditions

In this paper we prove the local boundedness as well as the local Lipschitz continuity for solutions to a class of obstacle problems of the type $$\min\left\{\int_\Omega {F(x, Dz)}: z\in \mathcal{K}_{\psi}(\Omega)\right\}.$$ Here $\mathcal{K}_{\psi}(\Omega)$ is set of admissible functions $z \in W^{1,p}(\Omega)$ such that $z \ge \psi$ a.e. in $\Omega$, $\psi$ being the obstacle and $\Omega$ being an open bounded set of $\mathbb{R}^n$, $n \ge 2$. The main novelty here is that we are assuming $F(x, Dz)$ satisfying $(p,q)$-growth conditions {and less restrictive assumptions on the obstacle with respect to the existing regularity results}

A new partial regularity result for non-autonomous convex integrals with non standard growth conditions

We establish C 1,γ -partial regularity of minimizers of non autonomous convex integral functionals of the type: F(u; Ω) :=´Ω f (x, Du) dx, with non standard growth conditions into the gradient for a couple of exponents p, q such that and α-Hölder continuous dependence with respect to the x variable. The significant point here is that the distance between the exponents p and q is independent of α. Moreover this bound on the gap between the growth and the coercitivity exponents improves previous results in this setting