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I am starting to read Boundary Behavior of Holomorphic Funtions of Several Complex Varieties by E.M. Stein. I don't know the meaning of the following symbol:

The class $C^{1+\epsilon}$ would suffice...

Can you tell what does $C^{1+\epsilon}$ mean? This symbol appears in the first page and there is no definition of such symbol in this book. And $C^{2-\epsilon}$ appears when introducting the Green's function.

Any help will be appreciated.

hardmath
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Jason785
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    Without knowing the context, I'd say, $C^1$ functions, whose derivative is $\epsilon$-Hölder continuous. – martini Oct 13 '15 at 12:38
  • @martini Thank you. This symbol appears in the first page and there is no definition of such symbol in this book. And $C^{2-\epsilon}$ appears when introducting the Green's function. Holder continous should be $C^{1,\epsilon}$, I think – Jason785 Oct 13 '15 at 12:41
  • There is also no definition, of say $C^2$. Some authors use $C^{1+\epsilon}$ instead of $C^{1,\epsilon}$, and $C^{2-\epsilon}$ is just $C^{1,1-\epsilon}$ in that sense. – martini Oct 13 '15 at 12:47
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    While there are other definitions of fractional derivatives, it's hard to think of one that works well enough with "continutity" to be so introduced without a careful definition. – hardmath Oct 13 '15 at 12:51
  • From the context (the author starts to recall "known" results from potential theory), it is very unlikely, that something besides Hölder continuity is meant, I'd say. – martini Oct 13 '15 at 13:02
  • @martini Thank you. It is Holder continuous after I read over a 1960s book. – Jason785 Oct 13 '15 at 13:13
  • @hardmath Thank you for commenting. This symbol turns out to be an "old-fashioned" usage of Holder continity – Jason785 Oct 13 '15 at 13:16
  • @Jason785: May I suggest inviting martini to post an Answer? – hardmath Oct 13 '15 at 18:50
  • @hardmath I don't know how to invite martini to post an answer. Math StackExchange is totally new to me. Perhaps I think a comment is enought to inform freshmen like me to understand such symbols. Afterall this is not a key problem. – Jason785 Oct 14 '15 at 01:51

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$C^{1+\epsilon}$ means $C^1$ functions, whose derivative is $\epsilon$-Hölder continuous.

(quoting martini's comment so that the question doesn't appear unanswered)

Chris Culter
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