The Lebesgue spaces $L^p(\mathbb{R}^n)$ measure the size of a function by a single exponent $p$: the condition $f \in L^p$ asks that $\int |f|^p\,d\mathcal{L}^n < \infty$. This is a blunt instrument. Two [functions](/page/Function) can have the same $L^p$ norm yet have very different decay profiles — one might be large on a small set, the other small on a large set — and $L^p$ cannot distinguish them. The Lorentz spaces $L^{p,q}$ refine the Lebesgue scale by introducing a second exponent $q$ that controls the *rate* at which the [distribution](/page/Distribution) function $\lambda \mapsto \mathcal{L}^n(\{|f| > \lambda\})$ decays, providing a finer gradation of integrability. The most important special case is the **weak Lebesgue space** $L^{p,\infty}$, which consists of functions whose distribution function decays like $\lambda^{-p}$ but which may fail to be in $L^p$ by a logarithmic factor. This space arises naturally as the target of weak-type estimates: the [Hardy–Littlewood maximal inequality](/theorems/228) and the weak-type endpoint of the [Hardy–Littlewood–Sobolev inequality](/theorems/469) both produce bounds in $L^{p,\infty}$ rather than $L^p$. Understanding Lorentz spaces is essential for the [Marcinkiewicz interpolation theorem](/page/Marcinkiewicz%20Interpolation), which recovers strong-type bounds from weak-type estimates, and for the [weak Young inequality](/theorems/649), which extends [Young's convolution inequality](/theorems/463) to functions in weak Lebesgue spaces. [motivation] ### Why $L^p$ Is Not Enough Consider the function $f(x) = |x|^{-n/p}$ on $\mathbb{R}^n$ for $1 \le p < \infty$. A polar coordinate computation gives \begin{align*} \int_{B(0,R)} |f(x)|^p\,d\mathcal{L}^n(x) = n\omega_n \int_0^R r^{n-1}\cdot r^{-n}\,d\mathcal{L}^1(r) = n\omega_n \int_0^R \frac{1}{r}\,d\mathcal{L}^1(r) = +\infty \end{align*} for every $R > 0$, so $f \notin L^p(\mathbb{R}^n)$. Yet $f$ is "almost" in $L^p$: it fails by the mildest possible divergence (a logarithm). Its distribution function is \begin{align*} \mathcal{L}^n(\{|f| > \lambda\}) = \mathcal{L}^n(\{|x|^{-n/p} > \lambda\}) = \mathcal{L}^n(B(0, \lambda^{-p/n})) = \omega_n\,\lambda^{-p}, \end{align*} which decays like $\lambda^{-p}$ — exactly the rate one would expect from an $L^p$ function. The $L^p$ norm is infinite only because the borderline decay accumulates a logarithmic divergence in the layer cake integral. To capture this "almost $L^p$" behaviour, one needs a space that controls the distribution function directly. ### The Role of the Distribution Function For a measurable function $f: \mathbb{R}^n \to \mathbb{C}$, the **distribution function** $d_f: (0,\infty) \to [0,\infty]$ defined by $d_f(\lambda) := \mathcal{L}^n(\{|f| > \lambda\})$ encodes how the mass of $f$ is distributed across different amplitude levels. The [layer cake representation](/page/Integral) expresses the $L^p$ norm as a weighted integral of $d_f$: \begin{align*} \|f\|_{L^p}^p = p\int_0^\infty \lambda^{p-1}\,d_f(\lambda)\,d\mathcal{L}^1(\lambda). \end{align*} The idea behind Lorentz spaces is to replace the weight $\lambda^{p-1}$ with a more flexible family of weights, parametrised by a second exponent $q$, that can detect finer distinctions in the decay of $d_f$. [/motivation] ## Definition The definition of Lorentz spaces proceeds through two intermediate objects: the distribution function and the decreasing rearrangement. Throughout this page, $(E, \mathcal{E}, \mu)$ denotes a $\sigma$-finite measure space. In most applications $E = \mathbb{R}^n$ and $\mu = \mathcal{L}^n$, but the definitions are valid in full generality. The distribution function measures the size of the super-level [sets](/page/Set) of $|f|$. [definition:Distribution Function] Let $f: E \to \mathbb{C}$ be a measurable function. The **distribution function** of $f$ is the function \begin{align*} d_f: (0, \infty) &\to [0, \infty] \\ \lambda &\mapsto \mu(\{x \in E : |f(x)| > \lambda\}). \end{align*} [/definition] The distribution function is non-negative, non-increasing, and right-[continuous](/page/Continuity). Two functions with the same distribution function are said to be **equimeasurable**; equimeasurable functions have identical $L^p$ norms for every $p$. The decreasing rearrangement sorts the values of $|f|$ in decreasing order, producing a function on $(0,\infty)$ that depends only on the distribution function and hence only on the "shape" of $f$, not on the geometry of the domain. [definition:Decreasing Rearrangement] Let $f: E \to \mathbb{C}$ be a measurable function with $d_f(\lambda) < \infty$ for all $\lambda > 0$. The **decreasing rearrangement** of $f$ is the function \begin{align*} f^*: (0, \infty) &\to [0, \infty) \\ t &\mapsto \inf\{\lambda > 0 : d_f(\lambda) \le t\}. \end{align*} [/definition] The function $f^*$ is non-negative, non-increasing, and right-continuous. It is equimeasurable with $|f|$ in the sense that $\mathcal{L}^1(\{t > 0 : f^*(t) > \lambda\}) = d_f(\lambda)$ for all $\lambda > 0$. Consequently, the layer cake representation gives \begin{align*} \|f\|_{L^p}^p = p\int_0^\infty \lambda^{p-1} d_f(\lambda)\,d\mathcal{L}^1(\lambda) = \int_0^\infty (f^*(t))^p\,d\mathcal{L}^1(t) \end{align*} for $0 < p < \infty$, so the $L^p$ norm of $f$ equals the $L^p$ norm of $f^*$ on $(0,\infty)$ with Lebesgue measure. The Lorentz quasi-norm replaces the weight in this integral with a power of $t$ controlled by the second exponent. [definition:Lorentz Space] Let $0 < p < \infty$ and $0 < q \le \infty$. The **Lorentz space** $L^{p,q}(E, \mu)$ is the set of all measurable functions $f: E \to \mathbb{C}$ for which the **Lorentz quasi-norm** \begin{align*} \|f\|_{L^{p,q}} := \begin{cases} \displaystyle\left(\frac{q}{p}\int_0^\infty \left(t^{1/p}\,f^*(t)\right)^q \frac{d\mathcal{L}^1(t)}{t}\right)^{1/q} & \text{if } 0 < q < \infty, \\[6pt] \displaystyle\sup_{t > 0}\, t^{1/p}\,f^*(t) & \text{if } q = \infty, \end{cases} \end{align*} is finite. [/definition] The normalising factor $q/p$ is a convention that ensures $\|f\|_{L^{p,p}} = \|f\|_{L^p}$ (see below). Some authors omit it; the choice does not affect which functions belong to $L^{p,q}$. The quantity $\|\cdot\|_{L^{p,q}}$ is a **quasi-norm** (it satisfies the triangle inequality up to a multiplicative constant) but not a true norm in general. For $1 < p < \infty$ and $1 \le q \le \infty$, an equivalent true norm can be obtained by replacing $f^*$ with its maximal average $f^{**}(t) := t^{-1}\int_0^t f^*(s)\,d\mathcal{L}^1(s)$, but we will not pursue this here. ## Relationship to Lebesgue Spaces The Lorentz spaces refine the Lebesgue scale in a precise sense: $L^{p,q}$ becomes a smaller space as $q$ decreases. [example:Lorentz Equals Lebesgue At The Diagonal] When $q = p$, the Lorentz quasi-norm reduces to the $L^p$ norm. For $0 < p < \infty$: \begin{align*} \|f\|_{L^{p,p}}^p = \frac{p}{p}\int_0^\infty \left(t^{1/p} f^*(t)\right)^p \frac{d\mathcal{L}^1(t)}{t} = \int_0^\infty (f^*(t))^p\,d\mathcal{L}^1(t) = \|f\|_{L^p}^p, \end{align*} where the last equality is the equimeasurability identity. Therefore $L^{p,p}(E, \mu) = L^p(E, \mu)$ with equal (quasi-)norms. [/example] For different values of $q$, the Lorentz spaces are ordered by strict inclusion. If $0 < q_1 < q_2 \le \infty$, then $L^{p,q_1} \subset L^{p,q_2}$ with $\|f\|_{L^{p,q_2}} \le C\,\|f\|_{L^{p,q_1}}$ for a constant $C$ depending on $p$, $q_1$, $q_2$. In particular: \begin{align*} L^{p,1} \subset L^{p,q_1} \subset L^{p,p} = L^p \subset L^{p,q_2} \subset L^{p,\infty} \end{align*} for any $1 \le q_1 < p < q_2 < \infty$. Thus $L^{p,1}$ is the smallest and $L^{p,\infty}$ is the largest of the Lorentz spaces with first index $p$. ## The Weak Lebesgue Space The most important Lorentz space beyond $L^p$ itself is the endpoint $q = \infty$, called the **weak $L^p$ space** or **weak Lebesgue space**, and sometimes written $L^p_w(\mathbb{R}^n)$. From the definition, $f \in L^{p,\infty}$ means $\sup_{t > 0} t^{1/p} f^*(t) < \infty$. An equivalent and often more convenient characterisation is in terms of the distribution function directly. [definition:Weak Lebesgue Space] Let $0 < p < \infty$. A measurable function $f: E \to \mathbb{C}$ belongs to the **weak Lebesgue space** $L^{p,\infty}(E, \mu)$ if \begin{align*} \|f\|_{L^{p,\infty}} := \sup_{\lambda > 0}\, \lambda\,\bigl(d_f(\lambda)\bigr)^{1/p} < \infty. \end{align*} Equivalently, $f \in L^{p,\infty}$ if and only if there exists a constant $C > 0$ such that \begin{align*} d_f(\lambda) \le C\,\lambda^{-p} \qquad \text{for all } \lambda > 0, \end{align*} and the infimum of such $C^{1/p}$ equals $\|f\|_{L^{p,\infty}}$. [/definition] The two expressions for $\|f\|_{L^{p,\infty}}$ — the one from the Lorentz space definition using $f^*$ and the one using $d_f$ — agree because $t^{1/p}f^*(t) = \sup\{\lambda\,d_f(\lambda)^{1/p} : d_f(\lambda) \ge t\}$ by the definition of $f^*$ as a generalised inverse of $d_f$. The condition $d_f(\lambda) \le C\,\lambda^{-p}$ is precisely the conclusion of **Chebyshev's inequality** applied to $|f|^p$: if $f \in L^p$, then $d_f(\lambda) \le \lambda^{-p}\|f\|_{L^p}^p$, giving $\|f\|_{L^{p,\infty}} \le \|f\|_{L^p}$. Therefore $L^p \subseteq L^{p,\infty}$ — the weak space is strictly larger than the strong space. [example: Power Law Singularity in Weak $L^p$] The function $f(x) = |x|^{-n/p}$ on $\mathbb{R}^n$ computed in the motivation satisfies $d_f(\lambda) = \omega_n\,\lambda^{-p}$, so $\|f\|_{L^{p,\infty}} = \omega_n^{1/p} < \infty$. Yet $f \notin L^p(\mathbb{R}^n)$ (the $L^p$ integral diverges logarithmically). This is the prototypical example: $L^{p,\infty}$ contains functions that are "borderline" for $L^p$, failing integrability by at most a logarithmic factor. [/example] [example: Riesz Kernel in Weak $L^p$] The Riesz kernel $I_\alpha(x) = |x|^{-(n-\alpha)}$ on $\mathbb{R}^n$ for $0 < \alpha < n$ satisfies $d_{I_\alpha}(\lambda) = \omega_n\,\lambda^{-n/(n-\alpha)}$, so $I_\alpha \in L^{n/(n-\alpha),\infty}(\mathbb{R}^n)$. Since $n/(n-\alpha) > 1$, the kernel lies in a weak Lebesgue space but not in the corresponding strong $L^{n/(n-\alpha)}$ space. This observation is the starting point for the [Hardy–Littlewood–Sobolev inequality](/theorems/469): the [convolution](/page/Convolution) $I_\alpha * f$ is controlled not by the (non-existent) $L^{n/(n-\alpha)}$ norm of $I_\alpha$, but by its weak norm, combined with a Young-type inequality adapted to weak spaces. [/example] ## Weak-Type Estimates Lorentz spaces provide the natural language for **weak-type estimates**, which arise when an operator is bounded on $L^p$ for $p$ in an open interval but fails at an endpoint. An operator $T$ is said to be of **strong type $(p,q)$** if $\|Tf\|_{L^q} \le C\|f\|_{L^p}$, and of **weak type $(p,q)$** if $\|Tf\|_{L^{q,\infty}} \le C\|f\|_{L^p}$, i.e., \begin{align*} d_{Tf}(\lambda) \le \left(\frac{C\|f\|_{L^p}}{\lambda}\right)^q \qquad \text{for all } \lambda > 0. \end{align*} Every strong-type $(p,q)$ bound implies the corresponding weak-type bound (since $L^q \subseteq L^{q,\infty}$), but not conversely. The power of weak-type estimates is that they can hold at endpoints where strong-type estimates fail, and the Marcinkiewicz interpolation theorem recovers strong-type bounds at interior points from weak-type bounds at the endpoints. ## Convolution in Lorentz Spaces [Young's convolution inequality](/theorems/463) states that $\|f * g\|_{L^r} \le \|f\|_{L^p}\|g\|_{L^q}$ when $1/p + 1/q = 1 + 1/r$ and $f \in L^p$, $g \in L^q$. This requires both $f$ and $g$ to belong to strong Lebesgue spaces. When one factor (such as a singular kernel) lies only in a weak space, the strong Young inequality does not apply. The **weak Young inequality** extends the convolution estimate to this setting: if $g \in L^{q,\infty}$ (rather than $L^q$), then under the same exponent relation, the convolution $f * g$ still belongs to $L^r$ (or $L^{r,\infty}$ at endpoints), but with the $L^q$ norm of $g$ replaced by its $L^{q,\infty}$ quasi-norm. This is a key tool in harmonic analysis — it is, for instance, the mechanism behind the [Hardy–Littlewood–Sobolev inequality](/theorems/469), where the Riesz kernel $I_\alpha \in L^{n/(n-\alpha),\infty}$ plays the role of $g$. ## References - Grafakos, L. *Classical Fourier Analysis*. 3rd ed. Springer, 2014. Chapter 1. - Stein, E. M. and Weiss, G. *Introduction to Fourier Analysis on Euclidean Spaces*. Princeton University Press, 1971. Chapter V. - Bergh, J. and Löfström, J. *Interpolation Spaces: An Introduction*. Springer, 1976. - Hunt, R. A. On $L(p,q)$ spaces. *L'Enseignement Mathématique* 12 (1966), 249–276.