Use of the Rellich Lemma in the proof of the Hodge Theorem
I am reading through the proof of the Hodge theorem that is given in Griffiths' and Harris' Principles of Algebraic Geometry, see pages 84-100.
The method of proof is to establish a weak solution of the equation $Delta varphi = psi$ using Hilbert space methods and elliptic regularity, first on the torus (the local picture), then on a general compact complex manifold.
Let us consider the local picture, i.e., obtaining the solution of $Delta varphi = psi$ on the torus. Indeed, assuming that $psi$ is perpendicular to the kernel of $Delta$ (since $Delta$ is self-adjoint), and assuming that $psi in L^2(mathbb{T}^n)$, a formal Fourier series solution of the Poisson equation is given by $$varphi = -sum_{xi neq 0} frac{1}{| xi |^2} varphi_{xi} e^{i langle xi, x rangle}.$$ Since $psi in L^2(mathbb{T}^n)$, then $varphi in L^2(mathbb{T}^n)$, and we define the Green's operator $G : H_s longrightarrow H_{s-2}$ by setting $$G(psi) = - sum_{xi neq 0} frac{1}{| xi |^2} psi_{xi} e^{i langle xi, x rangle};$$ this is a bounded linear operator.
By the Sobolev lemma, if $psi in mathscr{C}^{infty}(mathbb{T}^n)$, then $varphi in mathscr{C}^{infty}(mathbb{T}^n)$. At this point, have we not proved the Hodge decomposition theorem for smooth functions on the torus?
In the text, specifically on page 90, they continue: Finally, by the Rellich lemma $ G: L^2(mathbb{T}^n) longrightarrow L^2(mathbb{T}^n)$ is a compact, self-adjoint operator. The spectral decomposition for $G$ on $L^2(mathbb{T}^n)$ is just Fourier series.
Can someone point out to me why these extra two sentences are written (or in fact needed)? Please let me know if you would like any additional information regarding my question. Thanks in advance.
algebraic-geometry pde riemannian-geometry complex-geometry hodge-theory
asked Dec 7 at 22:17
AmorFati
391529
add a comment |
I am reading through the proof of the Hodge theorem that is given in Griffiths' and Harris' Principles of Algebraic Geometry, see pages 84-100.
The method of proof is to establish a weak solution of the equation $Delta varphi = psi$ using Hilbert space methods and elliptic regularity, first on the torus (the local picture), then on a general compact complex manifold.
Let us consider the local picture, i.e., obtaining the solution of $Delta varphi = psi$ on the torus. Indeed, assuming that $psi$ is perpendicular to the kernel of $Delta$ (since $Delta$ is self-adjoint), and assuming that $psi in L^2(mathbb{T}^n)$, a formal Fourier series solution of the Poisson equation is given by $$varphi = -sum_{xi neq 0} frac{1}{| xi |^2} varphi_{xi} e^{i langle xi, x rangle}.$$ Since $psi in L^2(mathbb{T}^n)$, then $varphi in L^2(mathbb{T}^n)$, and we define the Green's operator $G : H_s longrightarrow H_{s-2}$ by setting $$G(psi) = - sum_{xi neq 0} frac{1}{| xi |^2} psi_{xi} e^{i langle xi, x rangle};$$ this is a bounded linear operator.
By the Sobolev lemma, if $psi in mathscr{C}^{infty}(mathbb{T}^n)$, then $varphi in mathscr{C}^{infty}(mathbb{T}^n)$. At this point, have we not proved the Hodge decomposition theorem for smooth functions on the torus?
In the text, specifically on page 90, they continue: Finally, by the Rellich lemma $ G: L^2(mathbb{T}^n) longrightarrow L^2(mathbb{T}^n)$ is a compact, self-adjoint operator. The spectral decomposition for $G$ on $L^2(mathbb{T}^n)$ is just Fourier series.
Can someone point out to me why these extra two sentences are written (or in fact needed)? Please let me know if you would like any additional information regarding my question. Thanks in advance.
algebraic-geometry pde riemannian-geometry complex-geometry hodge-theory
asked Dec 7 at 22:17
AmorFati
391529
add a comment |
I am reading through the proof of the Hodge theorem that is given in Griffiths' and Harris' Principles of Algebraic Geometry, see pages 84-100.
The method of proof is to establish a weak solution of the equation $Delta varphi = psi$ using Hilbert space methods and elliptic regularity, first on the torus (the local picture), then on a general compact complex manifold.
Let us consider the local picture, i.e., obtaining the solution of $Delta varphi = psi$ on the torus. Indeed, assuming that $psi$ is perpendicular to the kernel of $Delta$ (since $Delta$ is self-adjoint), and assuming that $psi in L^2(mathbb{T}^n)$, a formal Fourier series solution of the Poisson equation is given by $$varphi = -sum_{xi neq 0} frac{1}{| xi |^2} varphi_{xi} e^{i langle xi, x rangle}.$$ Since $psi in L^2(mathbb{T}^n)$, then $varphi in L^2(mathbb{T}^n)$, and we define the Green's operator $G : H_s longrightarrow H_{s-2}$ by setting $$G(psi) = - sum_{xi neq 0} frac{1}{| xi |^2} psi_{xi} e^{i langle xi, x rangle};$$ this is a bounded linear operator.
By the Sobolev lemma, if $psi in mathscr{C}^{infty}(mathbb{T}^n)$, then $varphi in mathscr{C}^{infty}(mathbb{T}^n)$. At this point, have we not proved the Hodge decomposition theorem for smooth functions on the torus?
In the text, specifically on page 90, they continue: Finally, by the Rellich lemma $ G: L^2(mathbb{T}^n) longrightarrow L^2(mathbb{T}^n)$ is a compact, self-adjoint operator. The spectral decomposition for $G$ on $L^2(mathbb{T}^n)$ is just Fourier series.
Can someone point out to me why these extra two sentences are written (or in fact needed)? Please let me know if you would like any additional information regarding my question. Thanks in advance.
algebraic-geometry pde riemannian-geometry complex-geometry hodge-theory
asked Dec 7 at 22:17
AmorFati
391529
I am reading through the proof of the Hodge theorem that is given in Griffiths' and Harris' Principles of Algebraic Geometry, see pages 84-100.
The method of proof is to establish a weak solution of the equation $Delta varphi = psi$ using Hilbert space methods and elliptic regularity, first on the torus (the local picture), then on a general compact complex manifold.
Let us consider the local picture, i.e., obtaining the solution of $Delta varphi = psi$ on the torus. Indeed, assuming that $psi$ is perpendicular to the kernel of $Delta$ (since $Delta$ is self-adjoint), and assuming that $psi in L^2(mathbb{T}^n)$, a formal Fourier series solution of the Poisson equation is given by $$varphi = -sum_{xi neq 0} frac{1}{| xi |^2} varphi_{xi} e^{i langle xi, x rangle}.$$ Since $psi in L^2(mathbb{T}^n)$, then $varphi in L^2(mathbb{T}^n)$, and we define the Green's operator $G : H_s longrightarrow H_{s-2}$ by setting $$G(psi) = - sum_{xi neq 0} frac{1}{| xi |^2} psi_{xi} e^{i langle xi, x rangle};$$ this is a bounded linear operator.
By the Sobolev lemma, if $psi in mathscr{C}^{infty}(mathbb{T}^n)$, then $varphi in mathscr{C}^{infty}(mathbb{T}^n)$. At this point, have we not proved the Hodge decomposition theorem for smooth functions on the torus?
In the text, specifically on page 90, they continue: Finally, by the Rellich lemma $ G: L^2(mathbb{T}^n) longrightarrow L^2(mathbb{T}^n)$ is a compact, self-adjoint operator. The spectral decomposition for $G$ on $L^2(mathbb{T}^n)$ is just Fourier series.
Can someone point out to me why these extra two sentences are written (or in fact needed)? Please let me know if you would like any additional information regarding my question. Thanks in advance.
algebraic-geometry pde riemannian-geometry complex-geometry hodge-theory
algebraic-geometry pde riemannian-geometry complex-geometry hodge-theory
asked Dec 7 at 22:17
AmorFati
391529
asked Dec 7 at 22:17
AmorFati
391529
asked Dec 7 at 22:17
AmorFati
391529
asked Dec 7 at 22:17
AmorFati
391529
asked Dec 7 at 22:17
AmorFati
391529
391529
add a comment |
add a comment |
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