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Consider the maximal operator defined for a function $f\in L^1_{loc}$: $$ Mf : x\mapsto \sup_{r>0} \frac{1}{|B(x,r)|} \int\limits_{B(x,r)} f. $$ It is well know that $M : L^1 \to L^{(1,\infty)}$ ...

I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$ \Phi_i(2t)\approx \Phi_i(t), \quad i=1,2 $$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...

I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley. And I would like to understand one question: Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...