[Paper Review] On Schatten-von Neumann class properties of pseudo-differential operators. The Cordes-Kato method

Gruia Arsu|ArXiv.org|Nov 3, 2005

Spectral Theory in Mathematical Physics17 references3 citations

TL;DR

This paper extends the Cordes-Kato method to establish Schatten-von Neumann class properties of pseudo-differential operators by replacing $L^∞$-norm conditions on derivatives with $L^p$-norm conditions. It proves that if all derivatives $D_x^\alpha D_\xi^\beta a$ of order $|\alpha|,|\beta| \leq [n/2]+1$ belong to $L^p(\mathbb{R}^n \times \mathbb{R}^n)$, then the associated operator lies in the Schatten class $\mathcal{B}_p(L^2)$ for $1 \leq p < \infty$, applicable to both Weyl and Kohn-Nirenberg quantizations.

ABSTRACT

We investigate the Schatten-class properties of pseudo-differential operators with the (revisted) method of Cordes and Kato. As symbol classes we use classes similar to those of Cordes in which the $L^{\infty}$% -conditions are replaced by $L^{p}$-conditions, $1\leq p

Motivation & Objective

• To extend the Cordes-Kato method to derive Schatten-class membership for pseudo-differential operators under $L^p$-integrability conditions on derivatives of the symbol.
• To generalize the Calderón-Vaillancourt theorem to trace-class and Schatten-von Neumann class properties using $L^p$-based regularity assumptions.
• To establish trace-class properties for pseudo-differential operators under minimal differentiability and integrability conditions on the symbol.
• To demonstrate the method's applicability to both Weyl and Kohn-Nirenberg quantizations.
• To provide a unified framework for Schatten-class results via Fourier analysis and twisted convolution in symplectic harmonic analysis.

Proposed method

• Adapts the Cordes-Kato method by replacing $L^\infty$-bounds on derivatives with $L^p$-integrability conditions for $1 \leq p < \infty$.
• Employs the Weyl calculus in the Schrödinger representation formalism, using symplectic Fourier transforms and twisted convolution on a symplectic space $\mathfrak{S}$.
• Utilizes the $\ast$-algebra structure on measures and their Fourier transforms to define the composition product and operator quantization.
• Applies the representation $\mathcal{W}(\mu) = \int \mathcal{W}(\xi) \, d\mu(\xi)$ to link measures on $\mathfrak{S}$ to unitary operators on $L^2$.
• Analyzes the inverse Fourier transform of the symbol $a$ in $\mathbb{R}^{n_1} \times \mathbb{R}^{n_2}$ to derive pointwise decay estimates.
• Uses integral estimates involving $\langle x_1 \rangle^N \langle x_2 \rangle^N |\mathcal{F}^{-1}a(x_1,x_2)|$ and $L^1$-integrability conditions to prove summability of singular values.

Experimental results

Research questions

• RQ1 Under what $L^p$-regularity conditions on the derivatives of a symbol does the associated pseudo-differential operator belong to the Schatten class $\mathcal{B}_p(L^2)$?
• RQ2 Can the Cordes-Kato method be adapted to yield Schatten-class results under $L^p$-norm conditions instead of $L^\infty$-bounds?
• RQ3 Does the method extend to both Weyl and Kohn-Nirenberg quantizations?
• RQ4 What decay and integrability conditions on $\mathcal{F}^{-1}a$ ensure that the operator is trace-class or in $\mathcal{B}_p$?
• RQ5 How do the growth conditions on $m_1, m_2$ in the symbol class $\mathcal{S}^{m_1,m_2}$ affect the $L^1$-integrability of the inverse Fourier transform?

Key findings

• If $D_x^\alpha D_\xi^\beta a \in L^p(\mathbb{R}^n \times \mathbb{R}^n)$ for all $|\alpha|, |\beta| \leq [n/2] + 1$ and $1 \leq p < \infty$, then the associated pseudo-differential operator $A = a(x,D)$ belongs to the Schatten class $\mathcal{B}_p(L^2(\mathbb{R}^n))$.
• $\mathcal{F}^{-1}a$ is in $L^1(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$ whenever $m_1 < 0$ and $m_2 < 0$ in the symbol class $\mathcal{S}^{m_1,m_2}$, ensuring trace-class membership.
• For $m_1 + n_1 > 0$ and $m_2 + n_2 > 0$, the inverse Fourier transform $\mathcal{F}^{-1}a$ satisfies pointwise decay $|\mathcal{F}^{-1}a(x_1,x_2)| \leq C_N \langle x_1 \rangle^{-N} \langle x_2 \rangle^{-N} (1 + |x_1|^{-m_1 - n_1})(1 + |x_2|^{-m_2 - n_2})$.
• The method applies to both Weyl and Kohn-Nirenberg quantizations, as the $\ast$-algebraic structure on $\widehat{\mathcal{M}}(\mathfrak{S})$ is independent of the quantization scheme.
• The result generalizes the Calderón-Vaillancourt theorem to Schatten-class properties with minimal Hölder or $L^p$-regularity assumptions.
• The operator $a(P_{\mathbb{R}^{n_1}}, P_{\mathbb{R}^{n_2}})$ is bounded on $L^p(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$ for $1 \leq p \leq \infty$ if $m_1 < 0$ and $m_2 < 0$, under the symbol class $\mathcal{S}^{m_1,m_2}$.