Weak inequalities for maximal functions in Orlicz-Lorentz spaces and applications

F. E. LEVIS

Ubeda, España

Conferencia; IX International Meeting on Approximation of the University of Jaén; 2008

Departamento de Matemática de la Universidad de Jaén.

Let $mathcal{M}_0$ be the class of all real extended $mu$-measurable functions on  $[0,alpha)$, $0<alpha le infty$, where $mu$ is the Lebesgue measure. As usual, for $f in mathcal{M}_0$ we denote its distribution function by $mu_f$  and its decreasing rearrangement by $f^*$. Let  $phi: mathbf{R}_+  to mathbf{R}_+$ be a differentiable and convex function, $phi(0)=0$, $phi(t)>0$ for $t>0$ and let $w: (0,alpha)  to (0,infty)$ be a weight function, non-increasing and locally integrable. If $alpha=infty$, we also assume $limlimits_{t to infty}w(t)=0$ and $int_0^{infty}wdmu=infty$. For $f in mathcal{M}_0$, let $Psi_{w,phi}(f)=int_0^{mu_f(0)}phi(f^*)wdmu.$ From 1990, several authors studied geometric properties of the Orlicz-Lorentz space ${f in mathcal{M}_0 : Psi_{w,phi}(lambda f)< infty ;;mbox{for some}; lambda > 0 }$. We denote by $Lambda_{w,phi}$  the subspace ${f in mathcal{M}_0 : Psi_{w,phi}(lambda f)< infty ;mbox{for all};; lambda > 0 },$ and by $Lambda_{w,phi´}$ the space analogously defined, where $phi´$ is the derivative of the function $phi$. Let $A subset [0,alpha)$ be a finite measure set and let $chi_A$ be its characteristic function. If $C(f,A)$ is the set of all constants $c$ minimizing the expression$Psi_{w,phi}((f-c)chi_A),$ we denote by $T_A$ the mapping which assigns to each $f  in Lambda_{w,phi}$ the set $C(f,A)$. In this work $T_A$ is extended from an Orlicz-Lorentz space $Lambda_{w,phi}$ to the space $Lambda_{w,phi´}$. Moreover, monotony of $T_A$, in the Larders and Rogge sense, is established. Let ${B(x,epsilon)}_{epsilon}$, $epsilon>0$, denote a net of intervals of the form $(x-epsilon,x+epsilon) subset [0,alpha)$. For $f  in Lambda_{w,phi´}$, let $f^{epsilon}(x) in T_{B(x,epsilon)}(f)$. In cite{MC1} the authors studied the pointwise convergence of $f^{epsilon}(x)$ to $f(x)$, as $epsilon to 0$ when $ f in L_{p-1}+L_{infty}$, $1 le p < infty$. Similar results in a subspace of the Orlicz space have appeared in cite{ZF}. We prove weak inequalities for the maximal function associated with the family ${f^{epsilon}(x)}_{epsilon}$, which are used in the study of pointwise convergence of $f^{epsilon}(x)$ to $f(x)$, as $epsilon to 0$. For $f  in Lambda_{w,phi´}$, we consider the maximal function defined on $(0,alpha)$, $$Mf(x)= sup left{ frac{Psi_{w,phi´}(f chi_{B(x,epsilon)})} {Psi_{w,phi´}(chi_{B(x,epsilon)})} ,: epsilon >0 text{ and }  B(x,epsilon) subset (0,alpha) ight}.$$ In cite{BMR}, weak inequalities for $Mf$ has been studied when $Lambda_{w,phi´}$ is the Lorentz space $L_{p,q}$, $1 le p,q < infty$. We extend these results to $Lambda_{w,phi´}$. As a consequence we obtain a generalization of Lebesgue´s Differentiation Theorem.  Bibliography: [BMR], J. Bastero, M. Milman, F. Ruiz, Rearrangement of Hardy-Littewood ma-ximal functions in Lorentz spaces, Proc. Amer. Math. Soc., {128}, 1 (1999), 65-74. [MC1], F. Mazzone, H. Cuenya, Maximal inequalities and Lebesgue´s Differentiation Theorem for Best Approximant by Constant over Balls, J. Approx. Theory, {110}, (2001), 171-179. [ZF], F. Zo, S. Favier, A Lebesgue type differentiation theorem for best approximations by constants in Orlicz space, Real Anal. Exchange, {30}, 1, (2004), 29-42.