Sum-product estimates over arbitrary finite fields

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset {\mathbb{F}}_q$ we have $$\mathrm{\mid }(A-A{)}^2+(A-A{)}^2\mathrm{\mid }\gg \mathrm{\mid }A{\mathrm{\mid }}^{1+\frac{1}{21}}\mathrm{.}$$ This can be viewed as the Erd\H{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that $$\max \{\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }{A}^2+A^2\mathrm{\mid }\}\gg \mathrm{\mid }A{\mathrm{\mid }}^{1+\frac{1}{42}},\mathrm{\mid }A+A^2\mathrm{\mid }\gg \mathrm{\mid }A{\mathrm{\mid }}^{1+\frac{1}{84}}\mathrm{.}$$