The restricted class of quasicircles sometimes called the
"Weil-Petersson-class" has been a subject of interest in the last decade. In
this paper we establish a Sokhotski-Plemelj jump formula for WP-class
quasicircles, for boundary data in a certain conformally invariant Besov space.
We show that this Besov space is precisely the set of traces on the boundary of
harmonic functions of finite Dirichlet energy on the WP-class quasidisk.
We apply this result to multiply connected domains, Sigma, which are the
complement of n+1 WP-class quasidisks. Namely, we give a bounded isomorphism
between the Dirichlet space D(Sigma) of Sigma and a direct sum of Dirichlet
spaces, D-, of the unit disk. Writing the quasidisks as images of the disk
under conformal maps (f_0,...,f_n), we also show that {(h \circ f_0,...,h \circ
f_n) : h \in D(Sigma)} is the graph of a certain bounded Grunsky operator on
D-.Comment: 24 pages. Introductory material revise
The Kepler Mission has discovered thousands of planets with radii $<4\
R_\oplus,pavingthewayforthefirststatisticalstudiesofthedynamics,formation,andevolutionofthesesubβNeptunesandsuperβEarths.Planetarymassesareanimportantphysicalpropertyforthesestudies,andyetthevastmajorityofKeplerplanetcandidatesdonothavetheirsmeasured.AkeyconcernisthereforehowtomapthemeasuredradiitomassestimatesinthisEarthβtoβNeptunesizerangewheretherearenoSolarSystemanalogs.Previousworkshavederiveddeterministic,oneβtoβonerelationshipsbetweenradiusandmass.However,iftheseplanetsspanarangeofcompositionsasexpected,thenanintrinsicscatteraboutthisrelationshipmustexistinthepopulation.Herewepresentthefirstprobabilisticmassβradiusrelationship(MβRrelation)evaluatedwithinaBayesianframework,whichbothquantifiesthisintrinsicdispersionandtheuncertaintiesontheMβRrelationparameters.Weanalyzehowtheresultsdependontheradiusrangeofthesample,andonhowthemassesweremeasured.AssumingthattheMβRrelationcanbedescribedasapowerlawwithadispersionthatisconstantandnormallydistributed,wefindthatM/M_\oplus=2.7(R/R_\oplus)^{1.3},ascatterinmassof1.9\ M_\oplus,andamassconstrainttophysicallyplausibledensities,isthe"bestβfit"probabilisticMβRrelationforthesampleofRVβmeasuredtransitingsubβNeptunes(R_{pl}<4\ R_\oplus$). More broadly, this work provides a
framework for further analyses of the M-R relation and its probable
dependencies on period and stellar properties.Comment: 14 pages, 5 figures, 2 tables. Accepted to the Astrophysical Journal
on April 28, 2016. Select posterior samples and code to use them to compute
the posterior predictive mass distribution are available at
https://github.com/dawolfgang/MRrelatio