Hilbert-Kunz density function for graded domains

We prove the existence of HK density function for a graded pair (R,I), where R is an N-graded domain of finite type over a perfect field and I⊂R is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair (R,I), where, in addition R is standard graded. Other properties of the HK density functions also hold for the graded pairs: for example, it is a multiplicative function for Segre products, its maximum support is the F-threshold of an m-primary ideal provided ProjR is smooth, it has a closed formula when either I is generated by a system of parameters or R is of dimension two. As one of the consequences we show that if G is a finite group scheme acting linearly on a polynomial ring R of dimension d then the HK density function f_R^G,mG, of the pair (R^G,m_G), is a piecewise polynomial function of degree d−1. We also compute the HK density functions for (R^G,m_G), where G⊂SL₂(k) is a finite group acting linearly on the ring k[X,Y].