# Marcinkiewicz Interpolation for exponents $p_0, q_0, p_1, q_1 \in (0, \infty]$.

I would like to try and extend the Marcinkiewicz Theorem in Folland (Theorem 6.28) using a similar argument in the text. The theorem I am after is as follows:

Let $$p_0, p_1, q_0, q_1 \in (0, \infty]$$, with $$p_j \leq q_j$$ and $$q_0 \neq q_1$$. Let $$(X, \mathcal M, \mu), (Y, \mathcal N, \nu)$$ be measure spaces. For $$t \in (0,1)$$, put $$p^{-1} = \frac{1-t}{p_0} + \frac{t}{p_1}, q^{-1} = \frac{1-t}{q_0}+ \frac{t}{q_1}$$. Let $$T$$ be a quasilinear operator (with some quasilinearity constant $$K>0$$) which is weak type $$(p_j, q_j)$$. Then, $$T$$ is strong type $$(p,q)$$.

The only difference between this statement and Folland's is that I would like to allow the exponents to live in $$(0, \infty]$$ rather than in $$[1, \infty]$$ as in Folland's formulation (I've also relaxed sublinearity of $$T$$ to quasilinearity, but this doesn't seem to make much of a difference in the proof besides carrying an extra constant $$K$$ around.

The theorem above is proven in the case of sublinear $$T$$ here https://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/#lind on Terry Tao's blog (Theorem 27). The technique of proof there makes use of dyadic estimates.

My question is as follows: is it possible to use essentially the same proof strategy as in Folland and avoid a dyadic argument, or perhaps can the general case when $$p_j, q_j \in (0, \infty]$$ be deduced from the special case of $$p_j, q_j \in [1, \infty]$$?

For reference, images of Folland's argument are attached. The argument goes through up till (and including) the application of Minkowski's Integral inequality. The place where the argument seems to break down is the claim "If $$q_1>q_0$$, then $$q-q_0$$ and $$\sigma$$ are positive." Perhaps I'm not seeing something obvious, but any feedback is appreciated.