Hilbert scales and Sobolev spaces defined by associated Legendre functions

TL;DR

This paper explores Hilbert scales generated by associated Legendre functions, linking them to weighted Sobolev spaces and analyzing their spectral properties for various parameters.

Contribution

It provides new characterizations of these Hilbert spaces as weighted Sobolev spaces and establishes identities among spaces with different regularity levels.

Findings

Characterization of Hilbert scales via associated Legendre functions

Identification of these spaces as weighted Sobolev spaces

Derivation of identities among spaces with varying regularity

Abstract

In this paper we study the Hilbert scales defined by the associated Legendre functions for arbitrary integer values of the parameter. This problem is equivalent to study the left-definite spectral theory associated to the modified Legendre equation. We give several characterizations of the spaces as weighted Sobolev spaces and prove identities among the spaces corresponding to lower regularity index.

Equations

$$-((1-t^2)u^{\prime }{)}^{\prime }+\frac{m^2}{(1-t^2)}u$$

$$Y_n^m(\theta ,\phi ):=Q_n^{|m|}(\cos \theta )\exp (ım\phi ),n=0,1,\dots ,-n\le m\le n$$

$$\sum _{n=0}^{\infty }\sum _{m=-n}^n(2n+1{)}^{2s}|{\hat{g}}_{n,m}{|}^2<\infty ,$$

$$\sum_{m=-\infty}^{\infty}\Big{(}\sum_{n=|m|}^{\infty}(2n+1)^{2s}|\widehat{g}_{n,m}|^{2}\Big{)}.$$

$$\mathcal{D}(-1,1)=\{\phi \in {\mathcal{C}}^{\infty }(-1,1):\mathrm{supp}\phi \subset (-1,1)\}$$

$$H_{\mathrm{loc}}^k(-1,1)$$

$$W^{1,1}(-1,1):=\{u\in L^1(-1,1):u^{\prime }\in L^1(-1,1)\}$$

$${\int }_{-1}^{1}f={\int }_{-1}^{1}f(t)\mathrm{d}t$$

$${\int }_{-1}^1{g}_1=-{\int }_{-1}^1{g}_2.$$

$$\Vert u{\Vert }_s:={\left(\sum _{n=1}^{\infty }{\lambda }_n^{2s}|(u,{\psi }_n{)}_{H^0}{|}^2\right)}^{1/2},$$

$$H^s:=\{u\in H^0:\Vert u{\Vert }_s<\infty \}.$$

$$\Vert u{\Vert }_s\le (\underset{n}{\min }{\lambda }_n{)}^{-\epsilon }\Vert u{\Vert }_{s+\epsilon },\forall u\in H^{s+\epsilon },\forall \epsilon >0.$$

$$u=\sum_{n=1}^{\infty}(u,\psi_{n})_{H^{0}}\,\psi_{n}\qquad\mbox{(convergent in $H^{s}$)},$$

$$u\in X,\text{mbox}s.t.(u,v{)}_X=(f,v{)}_{H^0},\forall v\in X.$$

$$Gf=\sum _{n=1}^{\infty }{\lambda }_n^{-2}(f,{\varphi }_n{)}_{H^0}{\varphi }_n,$$

$$({\varphi }_n,v{)}_X={\lambda }_n^2({\varphi }_n,v{)}_{H^0}\forall v\in X.$$

$$H^s=\mathcal{R}(G^{s/2}),\forall s>0.$$

$$X=H^1\subset H^0\cong (H^0{)}^{\prime }\subset H^{-1}=X^{\prime }$$

$$P_{n}(t):=\frac{(-1)^{n}}{2^{n}n!}\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\big{(}(1-t^{2})^{n}\big{)},\qquad n\geq 0$$

$$P_n^m(t):=(1-t^2{)}^{m/2}{P}_n^{(m)}(t),0\le m\le n,$$

$$Q_n^m:=c_{n,m}{P}_n^m,c_{n,m}:={\left(\frac{2n+1}{2}\frac{(n-m)!}{(n+m)!}\right)}^{1/2},0\le m\le n.$$

$$\omega (t):=\sqrt{1-t^2}.$$

$${\mathcal{L}}_{m}u:=-({\omega }^2{u}^{\prime }{)}^{\prime }+m^2{\omega }^{-2}u$$

$${\mathcal{L}}_m{Q}_n^m=n(n+1)Q_n^m.$$

$$\Vert u{\Vert }_{m,s}^2=\sum _{n=m}^{\infty }(2n+1{)}^{2s}|(u,Q_n^m{)}_{L^2(-1,1)}{|}^2<\infty .$$

$$2\le \frac{2n+1}{\sqrt{n(n+1)}}\le \frac{3}{\sqrt{2}},n\ge 1,$$

$${\omega }^m\mathbb{P}=\{{\omega }^{m}p:p\in \mathbb{P}\}=\mathrm{span}\{Q_n^m:n\ge m\}=\mathrm{span}\{P_n^m:n\ge m\}.$$

$$({\omega }^{\beta }{)}^{\prime }=-\beta t{\omega }^{\beta -2},$$

$${\omega }^{\alpha }\in L^2(-1,1)⟺\alpha >-1.$$

$$H_0^k=\{u\in L^2(-1,1):{\omega }^{\ell }{u}^{(\ell )}\in L^2(-1,1),0\le \ell \le k\}$$

Full text

Key words: Legendre Functions, Sobolev Towers, Hilbert Spaces, Spherical Harmonics

MSC: 34B30, 46E35, 47B25, 33C55

1 Introduction and motivation

In this paper we deal with the Hilbert scales generated by the sets of Legendre functions. The concept of Hilbert scale is related (in many ways it is equivalent and leads to the same sets) to the left– and right–definite spectral theories for differential operators and to the construction of the so–called Sobolev towers. We will clarify this point in the following section. We first remark that the study of the Hilbert scales defined by Legendre polynomials (and actually by some other sets of orthogonal polynomials) has been subject of active research (see [3], [5], [10]). The aim is often the study of the spaces of the domain of definition of iterated powers (and also powers of the square root) of a given unbounded self–adjoint operator that stems from a differential equation, whose spectral set is a known sequence of orthogonal polynomials. Part of the interest of that study is being able to characterize the spaces as weighted Sobolev spaces.

In this paper we extend the study of the Hilbert scales for Legendre polynomials to the sequence of Hilbert scales defined by associated Legendre functions, which are the eigenfunctions of the modified Legendre operator in $(-1,1)$