The bivariate luminosity and mass functions of the local HRS galaxy sample. The stellar, dust, gas mass functions

We discuss the results of the relationships between the K-band and stellar mass, far-infrared luminosities, star formation rate, dust and gas masses of nearby galaxies computing the bivariate K-band Luminosity Function (BLF) and bivariate K-band Mass Function (BMF) of the Herschel Reference Survey (HRS), a volume-limited sample with full wavelength coverage. We derive the BLFs and BMFs from the K-band and stellar mass, far-infrared luminosities, star formation rate, dust and gas masses cumulative distributions using a copula method which is outlined in detail. The use of the bivariate computed taking into account the upper limits allows us to derive on a more solid statistical ground the relationship between the observed physical quantities. The analysis shows that the behaviour of the morphological (optically selected) subsamples is quite different. A statistically meaningful result can be obtained over the whole HRS sample only from the relationship between the K-band and the stellar mass, while for the remaining physical quantities (dust and gas masses, far-IR luminosity and star formation rate), the analysis is distinct for late-type (LT) and early-type galaxies (ETG). However, the number of ETGs is small to perform a robust statistical analysis, and in most of the case results are discussed only for the LTG subsample. The Luminosity and Mass Functions (LFs, MFs) of LTGs are generally dependent on the K-band and the various dependencies are discussed in detail. We are able to derive the corresponding LFs and MFs and compare them with those computed with other samples. Our statistical analysis allows us to characterise the HRS, that, although non homogeneously selected and partially biased towards low IR luminosities, may be considered as representative of the local LT galaxy population.