Let $S\subset {ℝ}^{n+1}$ be the graph of the function $\varphi :{[-1,1]}^n\to ℝ$ defined by $\varphi (x_1,\cdots ,x_n)={\sum }_{j=1}^n{\left|x_j\right|}^{{\beta }_j},$ with 1<${\beta }_1\le \cdots \le {\beta }_n,$ and let $\mu$ the measure on ${ℝ}^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu$ is $L^p$-$L^q$ bounded.

We deal with the integral equation $u\left(t\right)=f\left({\int }_{I}g(t,z)u\left(z\right)dz\right)$, with $t\in I=[0,1]$, $f:{𝐑}^n\to {𝐑}^n$ and $g:I\times I\to [0,+\infty [$. We prove an existence theorem for solutions $u\in L^{\infty }(I,{𝐑}^n)$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n=1$.

Let $T$ be a positive number or $+\infty$. We characterize all subsets $M$ of ${ℝ}^n\times ]0,T[$ such that $$\underset{X\in {ℝ}^n\times ]0,T[}{inf}u\left(X\right)=\underset{X\in M}{inf}u\left(X\right)i$$ for every positive parabolic function $u$ on ${ℝ}^n\times ]0,T[$ in terms of coparabolic (minimal) thinness of the set $M_{\delta }={\cup }_{(x,t)\in M}{B}^p((x,t),\delta t)$, where $\delta \in (0,1)$ and $B^p((x,t),r)$ is the “heat ball” with the “center” $(x,t)$ and radius $r$. Examples of different types of sets which can be used instead of “heat balls” are given. It is proved that (i) is equivalent to the condition ${sup}_{X\in {ℝ}^n\times {ℝ}^{+}}u\left(X\right)={sup}_{X\in M}u\left(X\right)$ for every bounded parabolic function on ${ℝ}^n\times {ℝ}^{+}$ and hence to all equivalent conditions given in the article...