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\}

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 $

Regolarita' in problemi variazionali: equazioni e sistemi

Dottorato di ricerca in matematica. 7. cicloConsiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome; Biblioteca Nazionale Centrale - P.za Cavalleggeri, 1, Florence / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

with wide range of anisotropy

of minimizers of variational integrals with wide range of anisotropy b