Reference for $f \in L^{p,\infty} \cap L^{q}$ then $f \in L^r$ for $p < r \leq q$

Question

Okay, so I think I've shown that if $f \in L^{p,\infty} \cap L^{q}$ with $p < q$ then $f \in L^r$ for $p < r \leq q$ where $L^{p, \infty}$ denotes the weak $L^p$ space. what I did was I wrote $$ \int_{\mathbb{R}^n} |f|^r dx = r \int_0^{\infty} t^{r-1} \lambda_f(t)dt = r \int_0^{1} t^{r-1}\lambda_f(t)dt + r \int_1^{\infty} t^{r-1} \lambda_f(t)dt $$ $$\leq r \int_0^{1} t^{r-1} \lambda_f(t)dt + r\int_1^{\infty} t^{q-1} \lambda_f(t)dt $$ and to bound the first term I use estimate on $\lambda_f(t)$ given by $L^{p,\infty}$ assumption while the second one is just a part of formula for $||f||_q$. My question is - does this result have a name? It seems like something that should come up relatively often and I don't want to bother the readers of what I'm writing with an obsolete proof. I looked for names like weak Holder inequality but all I got was a more general claim that isn't true in fact. (I thought it might be called that cause if we had strong $L^p$ we could show our claim using Holder)

Answer 1

It's the interpolation property of weak Lebesgue spaces. It can be refined to $$ \| u \|_{L^{r}} \leq C_{p,q,r} \| u \|_{L^{p,\infty}}^{\alpha} \| u \|_{L^{q,\infty}}^{1 - \alpha} $$ where $p<r<q$ and $$ \frac{1}{r} = \frac{\alpha}{p} + \frac{1 - \alpha}{q}$$ See this blog post or that blog post.

Or if you prefer books: Interpolation of Operators by Bennett and Sharpley, or Classical Fourier Analysis by Grafakos.