Theorem: Singular primitive with explicit regularity

Status: Conditional; implication proved for displayed mathematical data.

Let $u_{\delta }\in {\mathcal{D}}^{\prime }(X;E)$ have scalar scale regularity. Suppose its logarithmic derivative has the decomposition

in scalar pairings almost everywhere. The polynomial coefficients are distributions and $H_{\delta }$ satisfies the Dini bounds of Theorem thm:Dini.

For $\lambda <0$, define

and define $Q_0(t)={{\int }_0}^t{P}_0(s)\mathrm{d}s$ when present. Absent terms are zero. Then

has the stated singular derivative and $u_{\delta }-S_{\delta }$ has a strong finite limit $T$, with the Dini tail rate.

The raw family converges if and only if every coefficient of $S_{\delta }$ vanishes. Otherwise one fixed compact test detects an unbounded raw pairing.

Proof. The finite formula telescopes to $\lambda Q_{\lambda }+Q_{\lambda }^{\prime }=P_{\lambda }$; also $Q_0^{\prime }=P_0$. Thus $D_{\delta }(u_{\delta }-S_{\delta })=H_{\delta }$. Scalar scale regularity permits integration, and Theorem thm:Dini gives the finite limit.

For a surviving singular coefficient, choose the smallest real exponent with a nonzero coefficient and the largest logarithmic degree at that exponent. A fixed compact test detects it. Its scalar factor dominates all remaining singular terms, the finite term, and the vanishing remainder. Its absolute value tends to infinity. If all singular coefficients vanish, the Dini estimate gives strong convergence.