Hölder's inequality for functions of multiple variables

Question

I have one question about the Hölder's inequality. We know that the functions in the continuous Holder inequality are f(x) and g(x). If the function were changed to f (x, y) and g (x, y) would the result still hold? $\int_\Omega f(x,y)g(x,y)dxdy\leq(\int_\Omega |f(x,y)|^pdxdy)^{\frac{1}{p}}(\int_\Omega |g(x,y)|^qdxdy)^{\frac{1}{q}}$

Answer 1

Sure does, and in fact the inequality generalizes to arbitrary measure spaces.