Recently, a multivariate multifractal analysis for pointwise regularities based on hierarchical multiresolution quantities was developed. General bounds between the Hausdorff dimension of the intersection of single fractal sets and that of the original sets were derived. Equalities were checked for some synthetic signals that include multiplicative cascades. In this paper, we focus on the setting supplied by simultaneous pointwise $(T_{u_i}^{p_i}{)}_{i=1,\dots ,L}$ regularities. The $T_u^p$ regularity for $1\le p<\mathrm{\infty }$ was first introduced in order to better study elliptic partial differential equations where the natural function space setting is $L^p$ or a Sobolev space which includes unbounded functions. We will prove that both corresponding multivariate multifractal formalism and equalities above hold Baire generically in a given product of Besov spaces $\prod _{i=1,\dots ,L}{B}_{t_i}^{s_i,r_i}\left({\mathbb{R}}^d\right)$, $s_i,t_i,r_i>0,1\le p_i\le \mathrm{\infty }$ such that $s_i-\frac{d}{t_i}>-\frac{d}{p_i}$. We therefore extend previous results where only cases ($p_i=\mathrm{\infty }$ and $s_i-\frac{d}{t_i}>0$ for all i) and ($p_i=\mathrm{\infty }$, $s_i=\frac{d}{t_i}$ and $r_i\le 1$ for all i) for simultaneous pointwise Hölder regularities have been proved.

最近，开发了一种基于分层多分辨率量的逐点规律的多元多分形分析。导出了单个分形集交集的豪斯多夫维数与原始集的一般界。检查了一些包括乘法级联的合成信号的等式。在本文中，我们专注于由同时逐点提供的设置$({吨}_{{你}_{\mathrm{一世}}}^{p_{\mathrm{一世}}}{)}_{\mathrm{一世}=1,\dots ,\mathrm{大号}}$规律性。这${吨}_{你}^p$规律性$1\le p<\mathrm{\infty }$首次引入是为了更好地研究自然函数空间设置为的椭圆偏微分方程${\mathrm{大号}}^p$或包含无界函数的 Sobolev 空间。我们将证明，在给定的 Besov 空间乘积中，相应的多元多元分形形式主义和上述等式通常都成立 Baire$\prod _{\mathrm{一世}=1,\dots ,\mathrm{大号}}{乙}_{{吨}_{\mathrm{一世}}}^{s_{\mathrm{一世}},r_{\mathrm{一世}}}\left({\mathbb{R}}^d\right)$,$s_{\mathrm{一世}},{吨}_{\mathrm{一世}},r_{\mathrm{一世}}>0,1\le p_{\mathrm{一世}}\le \mathrm{\infty }$这样$s_{\mathrm{一世}}-\frac{d}{{吨}_{\mathrm{一世}}}>-\frac{d}{p_{\mathrm{一世}}}$. 因此，我们扩展了以前的结果，其中只有案例 ($p_{\mathrm{一世}}=\mathrm{\infty }$和$s_{\mathrm{一世}}-\frac{d}{{吨}_{\mathrm{一世}}}>0$对于所有i ) 和 ($p_{\mathrm{一世}}=\mathrm{\infty }$,$s_{\mathrm{一世}}=\frac{d}{{吨}_{\mathrm{一世}}}$和$r_{\mathrm{一世}}\le 1$对于所有i ) 同时的逐点 Hölder 正则性已被证明。