Logarithmic negative Sobolev spaces for white noise

Let $({\psi }_{lk})$ be a countable complete orthonormal basis of the real Hilbert space $L^2([0,1])$ (Orthonormal families, complete orthonormal systems and Hilbert bases, Hilbert space, The space $L^p(\mu )$ as the quotient by null functions). For a wavelet basis, index levels by $l\ge 0$, let $K_l$ have $2^l$ indices, and put $a_l=2^l$ for $l\ge 1$ and $a_0=2$ for the finite coarsest level. This finite-level convention only prevents a zero logarithmic weight and does not change any convergence criterion below. For a trigonometric basis, use its integer frequency index $l$, bounded multiplicities, and $a_l=\max (2,\mathrm{\mid }l\mathrm{\mid })$.

Here an $S$-regular wavelet basis means a wavelet orthonormal basis whose members are $S$ times differentiable, with each derivative through order $S$ in $L^2([0,1])$, following the source convention. The weak BvM theorem below only uses the basis's completeness, orthonormality, and level counts.

For $\delta >0$, define $H(\delta )=H_2^{-1\mathrm{/}2,\delta }$ to be the space of coefficient sequences $c=(c_{lk})$ with norm

$\mathrm{\parallel }c{\mathrm{\parallel }}_{H(\delta )}^2:={\sum }_{l,k}{a}_l^{-1}(\log a_l{)}^{-2\delta }\mathrm{\mid }{c}_{lk}{\mathrm{\mid }}^2<\mathrm{\infty }\mathrm{.}$

Its inner product is the corresponding weighted coefficient inner product. Multiplying each coordinate by $a_l^{-1\mathrm{/}2}(\log a_l{)}^{-\delta }$ identifies it isometrically with ${\mathrm{\ell }}^2(I)$, where $I=\{(l,k):k\in K_l\}$ (or the corresponding trigonometric index set). This sequence space is complete: if $x^{(m)}$ is Cauchy, each coordinate converges to a scalar $x_i$; for every finite $F\subset I$, the Cauchy bound passes to the limit in ${\sum }_{i\in F}\mathrm{\mid }{x}_i^{(m)}-x_i^{(n)}{\mathrm{\mid }}^2$, so the supremum over finite $F$ shows $x^{(m)}\to x$ in ${\mathrm{\ell }}^2(I)$. It is separable because finite-support sequences with rational coordinates form a countable dense set. Thus $H(\delta )$ is a separable Hilbert space without an additional choice assumption.

A function $f\in L^2([0,1])$ is identified with its coefficient sequence $c_{lk}=⟨f,{\psi }_{lk}⟩$. For each finite set $F$ of indices, orthonormality makes ${\sum }_{(l,k)\in F}{c}_{lk}{\psi }_{lk}$ orthogonal to its residual in $L^2$, so the Pythagorean identity gives ${\sum }_F\mathrm{\mid }{c}_{lk}{\mathrm{\mid }}^2\le \mathrm{\parallel }f{\mathrm{\parallel }}_2^2$. Taking the supremum over finite $F$ and using the boundedness of the weights proves continuity of the coefficient map. If all coefficients of $f$ vanish, then $f$ is orthogonal to the linear span of the basis. Completeness makes this span dense, and continuity of $g\mapsto ⟨f,g⟩$ gives $⟨f,f⟩=0$, hence $f=0$ in $L^2$. Applying this to the difference of two functions proves injectivity. Thus the coefficient map is a continuous linear embedding $L^2([0,1])\hookrightarrow H(\delta )$.

The weight trace ${\sum }_{l,k}{a}_l^{-1}(\log a_l{)}^{-2\delta }$ is finite exactly when $\delta >1\mathrm{/}2$: for wavelets its tail is a constant multiple of ${\sum }_{l\ge 1}{l}^{-2\delta }$, while for a trigonometric basis it is comparable to ${\sum }_{l\ge 2}[l(\log l{)}^{2\delta }{]}^{-1}$. The product-prior theorem below uses the wavelet case.