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The weak $L^p$ spaces, often denoted $L^{p,\infty}$, provide another class of commonly ocurring occurring function spaces for which the triangle inequality fails to hold. The most common of these is weak $L^1$ which is the set of all functions so that $\sup_{\alpha>0}\alpha\textrm{ }\mu(x:|f(x)|>\alpha)$ is finite. This quantity is called the weak-$L^1$ "norm" of $f$, though it is not a true norm because it does not satisfy the triangle inequality. The term "weak" is appropriate as every $L^1$ function is automatically in weak $L^1$ by Chebyshev's inequality; the function $1/x$ is in weak $L^1$, however.
These spaces arise when studying, for example, maximal operators, e.g. Hardy-Littlewood, and singular integral operators, e.g. the Hilbert transform. Both the Hardy-Littlewood maximal operator and the Hilbert transform are bounded from $L^p$ to $L^p$ for $p\in(1,\infty)$ but are not bounded $L^1$ to $L^1$. They are, however, bounded from $L^1$ to weak $L^1$. The weak Lebesgue spaces are very useful substitutes for the usual Lebesgue spaces because some interpolation theorems, e.g. the Marcinkiewicz interpolation theorem, allow one to interpolate between two weak estimates to produce strong estimates in between. So the fact that the Hardy-Littlewood maximal operator is bounded from $L^1$ to weak $L^1$ and $L^\infty$ to $L^\infty$ is enough to prove that it is bounded from $L^p$ to $L^p$ for $p\in(1,\infty)$.
The weak $L^p$ spaces, often denoted $L^{p,\infty}$, provide another class of commonly ocurring function spaces. The most common of these is weak $L^1$ which is the set of all functions so that $\sup_{\alpha>0}\alpha\textrm{ }\mu(x:|f(x)|>\alpha)$ is finite. This quantity is called the weak-$L^1$ "norm" of $f$, though it is not a true norm because it does not satisfy the triangle inequality. The term "weak" is appropriate as every $L^1$ function is automatically in weak $L^1$ by Chebyshev's inequality; the function $1/x$ is in weak $L^1$, however.
These spaces arise when studying, for example, maximal operators, e.g. Hardy-Littlewood, and singular integral operators, e.g. the Hilbert transform. Both the Hardy-Littlewood maximal operator and the Hilbert transform are bounded from $L^p$ to $L^p$ for $p\in(1,\infty)$ but are not bounded $L^1$ to $L^1$. They are, however, bounded from $L^1$ to weak $L^1$. The weak Lebesgue spaces are very useful substitutes for the usual Lebesgue spaces because some interpolation theorems, e.g. the Marcinkiewicz interpolation theorem, allow one to interpolate between two weak estimates to produce strong estimates in between. So the fact that the Hardy-Littlewood maximal operator is bounded from $L^1$ to weak $L^1$ and $L^\infty$ to $L^\infty$ is enough to prove that it is bounded from $L^p$ to $L^p$ for $p\in(1,\infty)$.