Let $f\in L^p(\mathbb R^2)$ for all $p\in[1,\bar p)$. Consider $$ g(y):=\int_{\mathbb R}f(x,y)\,dx \;.$$ Can I say that $g\in L^p(\mathbb R)$ for the same values of $p$ ?
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1Hint: Minkowski integral inequality. – Giuseppe Negro Oct 27 '21 at 17:09
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1@GiuseppeNegro if I understand correctly, this is triangular inequality for integrals. So I get $$|g|{L^p(\mathbb R)} \leq \int{\mathbb R}|f(x,\cdot)|_{L^p(\mathbb R)},d x ;.$$ But it is not guarantee that the r.h.s. integral is finite for $p>1$, am I wrong? – tituf Oct 27 '21 at 17:33