Best constant for Hölder inequality in Lorentz spaces

Question

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?

Answer 1

The Holder inequality for Lorentz Spaces in actually due to Hunt, proved in

Hunt, R.A. On L(p, q) spaces, Enseign. Math. (2) 12 (1966).

O'Neil extended the Young's inequality for convolutions to the setting of Lorentz Spaces in

O’Neil, R. Convolution operators and L(p, q) spaces. Duke Math. J. 30 (1963).