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There are two types of boundedness for operators that are commonly used as endpoints in interpolation theory:

  1. Weak-type $(1,1)$ boundedness;
  2. Hardy space $H^1$ boundedness.

Both of these serve as possible endpoint estimates in real interpolation methods. This raises the question:

Is there a strength or weakness relationship between (1) and (2)?

In other words, does $H^1$ boundedness imply weak-type $(1,1)$ boundedness, or vice versa?

Dang Dang
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xxxg
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    Weak type boundedness corresponds to boundedness in weak $L^1$ spaces, or Lorentz spaces $L^{1,\infty} = F^0_{(1,\infty),2}$, while $H^1 = F^0_{1,2} = F^0_{(1,1),2}$. These spaces are in general different and by the last line of the answer here https://math.stackexchange.com/questions/3654495/besov-or-triebel-lizorkin-spaces-versus-lorentz-spaces $H^1 \subset L^{1,\infty}$. However, inclusion of one space into another space does not tell you anything about the strength of operator boundedness on the space. – LL 3.14 Apr 20 '25 at 10:31

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