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Register a new userA 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 $big (!!big (big [ T_j^* , T_ibig ]big )!!big )$ of the commutators $[T^*_j , T_i] := T^*_j T_i - T_iT_j^*$. We define the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ by symmetrizing the products in the Laplace formula for the determinant of a scalar matrix. We prove that the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ equals the generalized commutator of the $2d$ - tuple of operators, $(T_1,T_1^*, ldots, T_d,T_d^*)$ introduced earlier by Helton and Howe. We then apply the Amitsur-Levitzki theorem to conclude that for any commuting $d$ - tuple of $d$ - normal operators, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ must be $0$. We show that if the $d$- tuple $boldsymbol T$ is cyclic, the determinant of $big [ !!big [ boldsymbol T^*, boldsymbol T big ]!!big ]$ is non-negative and the compression of a fixed set of words in $T_j^* $ and $T_i$ -- to a nested sequence of finite dimensional subspaces increasing to $mathcal H$ -- does not grow very rapidly, then the trace of the determinant of the operator $big [!! big [ boldsymbol T^* , boldsymbol Tbig ] !!big ]$ is finite. Moreover, an upper bound for this trace is given. This upper bound is shown to be sharp for a class of commuting $d$ - tuples. We make a conjecture of what might be a sharp bound in much greater generality and verify it in many examples.
In this article, the authors give a survey on the recent developments of both the John--Nirenberg space $JN_p$ and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO, $VJN_p$, and $CJN_p$ on $mathbb{R}^n$ or a given cube $Q_0subsetmathbb{R}^n$ with finite side length. In addition, some related open questions are also presented.
In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.
We improve the constant $frac{pi}{2}$ in $L^1$-Poincare inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $sqrt{frac{pi}{2}}$. For Hamming cube the sharp constant is not known, and $sqrt{frac{pi}{2}}$ gives an estimate from below for this sharp constant. On the other hand, L. Ben Efraim and F. Lust-Piquard have shown an estimate from above: $C_1le frac{pi}{2}$. There are at least two other independent proofs of the same estimate from above (we write down them in this note). Since those proofs are very different from the proof of Ben Efraim and Lust-Piquard but gave the same constant, that might have indicated that constant is sharp. But here we give a better estimate from above, showing that $C_1$ is strictly smaller than $frac{pi}{2}$. It is still not clear whether $C_1> sqrt{frac{pi}{2}}$. We discuss this circle of questions and the computer experiments.
In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Kreins formula is established. Some examples of trace compatible affine spaces of operators are given.
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