Narrow Results

Let $\phi$ be a function in the complex Sobolev space $W^{\ast }(U)$, where $U$ is an open subset in ${ℂ}^k$. We show that the complement of the set of Lebesgue points of $\phi$ is pluripolar. The key ingredient in our approach is to show that $|\phi {|}^{\alpha }$ for $\alpha \in [1,2)$ is locally bounded from above by a plurisubharmonic function.

Let $\overrightarrow{p}:=(p_1,\dots ,p_n),\overrightarrow{r}:=(r_1,\dots ,r_n)\in [1,\infty {)}^n$, $L^{\overrightarrow{r}}({ℝ}^n)$ be the mixed-norm Lebesgue space, and $𝜃$ an integrable function. In this paper, via establishing the boundedness of the mixed centered Hardy–Littlewood maximal operator $M_{\overrightarrow{p}}$ from $L^{\overrightarrow{r}}({ℝ}^n)$ to itself or to the weak mixed-norm Lebesgue space $W{L}^{\overrightarrow{r}}({ℝ}^n)$ under some sharp assumptions on $\overrightarrow{p}$ and $\overrightarrow{r}$, the authors show that the $𝜃$-mean of $f\in L^{\overrightarrow{r}}({ℝ}^n)$ converges to $f$ almost everywhere over the diagonal if the Fourier transform $\widehat{𝜃}$ of $𝜃$ belongs to some mixed-norm homogeneous Herz space ${Ė}_{{\overrightarrow{p}}^{\prime }}({ℝ}^n)$ with ${\overrightarrow{p}}^{\prime }$ being the conjugate index of $\overrightarrow{p}$. Furthermore, by introducing another mixed-norm homogeneous Herz space and establishing a characterization of this Herz space, the authors then extend the above almost everywhere convergence of $𝜃$-means to the unrestricted case. Finally, the authors show that the $𝜃$-mean of $f\in L^{\overrightarrow{r}}({ℝ}^n)$ converges over the diagonal to $f$ at all its $\overrightarrow{p}$-Lebesgue points if and only if $\widehat{𝜃}$ belongs to ${Ė}_{{\overrightarrow{p}}^{\prime }}({ℝ}^n)$, and a similar conclusion also holds true for the unrestricted convergence at strong $\overrightarrow{p}$-Lebesgue points. Observe that, in all these results, those Herz spaces to which $\widehat{𝜃}$ belongs prove to be the best choice in some sense.