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It's well known (and proved by R. O'neil) that there is a version of Hölder's inequality for Lorentz spaces, namely

$$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{p_2, q_2}}$$

for all $0 < p, q, p_1, q_1, p_2, q_2 \leq \infty$ such that $1/p = 1/p_1 + 1/p_2$ and $1/q = 1/q_1 + 1/q_2$.

My question is whether anything is known about the best dependence on the exponents, and in particular best dependence on $p_2$ asymptotically for $p_2$ very large?

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