Doubling measure and regularity to K-quasiminimizers of double-phase energy

We consider an integral functional involving the double-phase operator in a metric measure space equipped with a doubling measure. On the basis of suitable hypotheses on the function governing the phase changes, we design a unifying approach to establish Sobolev-Poincar & eacute; inequalities. By using such inequalities together with a Caccioppoli type estimate, we obtain the local boundedness and the local H & ouml;lder continuity of K K -quasiminimizers of the double-phase energy. Here, we provide a direct approach for establishing the Sobolev-Poincar & eacute; inequalities and local H & ouml;lder continuity, that is, we do not use neither the method of separation of phases nor auxiliary frozen functional. Finally, we establish the local gradient higher integrability for K K -quasiminimizers of the integral functional. We prove our results via the De Giorgi's method, imposing that the involved measure is just doubling.