Pome

Corollary 5.39.7380EB17A4350

Corollary 5.39.7.380EB17A4350 In the setting of theorem 5.39.6, let $V$ be as in the theorem. For $0<\beta \leq 1$, a modulus $m(r)\leq Cr^\beta $ gives $d_\eta ^ U(x,y)\leq C'|x-y|^{\beta /2}$ for sufficiently close $x,y\in V\setminus A$. If instead

\[ m(r)\leq C(-\log r)^{-a},\qquad a>2, \]

then for sufficiently close $x,y\in V\setminus A$,

\[ d_\eta ^ U(x,y)\leq C'(-\log |x-y|)^{1-a/2}. \]

These are only sufficient conditions; we do not claim that the Dini condition is necessary.

Proof. Integrating the square root of the power modulus gives a multiple of $r^{\beta /2}$. For the logarithmic modulus, the substitution $s=-\log t$ gives

\[ \int _0^ r\frac{(-\log t)^{-a/2}}t\, \mathrm{d}t =\frac{(-\log r)^{1-a/2}}{a/2-1}. \]

In either case the term $r$ in $D(r)$ is bounded by the asserted rate for small $r$. The theorem now gives both estimates; the fixed dilation of the argument of $D$ changes only the constants. $\square$

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