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تسجيل مستخدم جديدA trace inequality for solenoidal charges
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We prove that for $alpha in (d-1,d]$, one has the trace inequality begin{align*} int_{mathbb{R}^d} |I_alpha F| ;d u leq C |F|(mathbb{R}^d)| u|_{mathcal{M}^{d-alpha}(mathbb{R}^d)} end{align*} for all solenoidal vector measures $F$, i.e., $Fin M_b(mathbb{R}^d,mathbb{R}^d)$ and $operatorname{div}F=0$. Here $I_alpha$ denotes the Riesz potential of order $alpha$ and $mathcal M^{d-alpha}(mathbb{R}^d)$ the Morrey space of $(d-alpha)$-dimensional measures on $mathbb{R}^d$.
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For a commuting $d$- tuple of operators $boldsymbol T$ defined on a complex separable Hilbert space $mathcal H$, let $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ be the $dtimes d$ block operator $big (!!big (big [ T_j^* , T_ibig ]big )!!
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