All Questions

Question: For a smooth, bounded domain $\Omega\subset \mathbb R^d$, does there exist a function $u\in L^1(\Omega)$ such that $u\not\in L^\Phi(\Omega)$ for any Orlicz space $\Phi$? For the definition ...

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 ...

If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satisfying $1/r+1/s=1$, then we know that $fg\in L^1$. But if $g$ is a little bit more than $L^s$, say $L^s \...

I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...

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 ...