Riesz and Bessel Potentials and the Hardy-Littlewood-Sobolev Inequality

Intuition Beginner

Imagine a sharp signal — a spike of charge, a burst of heat, a jagged density. Differentiating sharpens it further; integrating smooths it out. The Riesz potential is a knob that lets you smooth by a fractional amount: you can integrate "one and a half times" instead of only once or twice. The number you turn the knob to is a smoothing order, written as a positive amount $\alpha$. Turn it a little and you round off the sharpest corners; turn it more and the signal spreads into a gentle hill. This single adjustable operation interpolates continuously between leaving a function alone and integrating it many whole times.

Why bother with fractional smoothing? Because the right amount of smoothing is often not a whole number. A function might be just barely too rough to be continuous, and the honest description of how rough it is uses a fraction. The Riesz potential gives you the vocabulary for "this function is exactly one-half of a derivative away from being nice," which whole-number derivatives cannot express.

The headline fact is a trade you get for free: smoothing by amount $\alpha$ does not just make a function look nicer, it makes it live in a better space of functions. If you measure a function's size by how its values pile up — its averaged $p$-th power — then after smoothing by $\alpha$ you can measure with a more demanding exponent $q$ and still get a finite answer, with the new exponent tied to the old one by a clean bookkeeping rule. That guaranteed upgrade is the Hardy-Littlewood-Sobolev inequality, and it is the engine behind why solutions of equations are smoother than the data that produced them.

The one-sentence takeaway: the Riesz potential smooths a function by an adjustable fractional amount, and the price you pay in spread is repaid by a guaranteed improvement in how the function's size is measured — that guaranteed improvement is the Hardy-Littlewood-Sobolev inequality.

Visual Beginner

Picture a tall narrow spike sitting at the origin of a line. Apply a small amount of smoothing: the spike becomes a rounded bump, lower and wider. Apply more smoothing: the bump flattens into a broad gentle hill that decays slowly out to the sides. The amount of smoothing is the dial $\alpha$; larger $\alpha$ means more spreading. The width of the spread is governed by a kernel that decays like a power of the distance — close to the source it is large, far away it tapers off at a rate set by the dimension and by $\alpha$.

The companion picture is the bookkeeping rule. Draw a dial labelled with the input measuring-exponent $p$ on the left and the output measuring-exponent $q$ on the right; turning the smoothing knob to amount $\alpha$ slides the output exponent up to a value determined by the equation that subtracts a fixed fraction from the reciprocal of $p$. The Bessel potential is the same picture with the long power-law tails clipped to decay rapidly, which is the version that matches the everyday Sobolev spaces.

Worked example Beginner

We work out the smoothing-exponent bookkeeping in a concrete case so the rule is no longer abstract.

Step 1. Work in ordinary three-dimensional space, so the dimension is $n=3$. Choose to smooth by the amount $\alpha =1$. We start with a function whose size is measured with the input exponent $p=3/2$, meaning we look at the average of its values raised to the power $3/2$.

Step 2. The bookkeeping rule says the reciprocal of the output exponent equals the reciprocal of the input exponent minus the smoothing amount divided by the dimension. In numbers: one over $q$ equals one over $(3/2)$ minus $1$ divided by $3$.

Step 3. Compute the right side. One over $(3/2)$ is $2/3$. The smoothing amount over the dimension is $1/3$. Subtract: $2/3-1/3=1/3$. So one over $q$ equals $1/3$.

Step 4. Invert to find the output exponent: $q=3$. The smoothing by one unit raised the measuring exponent from $3/2$ all the way to $3$.

What this tells us: starting from a function controlled only in the weak $3/2$ averaged sense, one unit of fractional smoothing produces a function controlled in the much more demanding $q=3$ sense — a genuine upgrade in regularity bought purely by the smoothing operation. This is the exact arithmetic behind the Sobolev embedding: a function with one derivative in the $3/2$ sense automatically sits in the space of functions with finite cube-average, with no further hypotheses.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $\hat{f}(\xi )={\int }_{{\mathbb{R}}^n}f(x)e^{-2\pi ix\cdot \xi }dx$ is the Fourier transform with the unitary normalisation, $|E|$ is Lebesgue measure, and $L^p=L^p({\mathbb{R}}^n)$. The Hardy-Littlewood maximal operator $M$ and its weak-type $(1,1)$ and strong-type $(p,p)$ bounds are taken from 02.19.01; the Sobolev space $W^{\alpha ,p}$ from 24.01.01; the Newtonian-potential prototype from 02.13.02.

Definition (Riesz potential). For $0<\alpha <n$ the Riesz potential of order $\alpha$ is the convolution operator $$I_{\alpha }f(x)=\frac{1}{{\gamma }_n(\alpha )}{\int }_{{\mathbb{R}}^n}\frac{f(y)}{|x-y{|}^{n-\alpha }}dy,{\gamma }_n(\alpha )=\frac{{\pi }^{n/2}{2}^{\alpha }\Gamma (\alpha /2)}{\Gamma ((n-\alpha )/2)}.$$ The kernel $K_{\alpha }(x)={\gamma }_n(\alpha {)}^{-1}|x{|}^{-(n-\alpha )}$ is locally integrable precisely because $n-\alpha <n$, and the normalisation ${\gamma }_n(\alpha )$ is fixed so that on the Fourier side $$\widehat{I_{\alpha }f}(\xi )=(2\pi |\xi |{)}^{-\alpha }\hat{f}(\xi ),$$ i.e. $I_{\alpha }=(-\Delta {)}^{-\alpha /2}$ as a Fourier multiplier. The Newtonian potential of 02.13.02 is the case $\alpha =2$, $n\ge 3$.

Definition (Bessel potential). For $\alpha >0$ the Bessel potential of order $\alpha$ is the operator $J_{\alpha }=(I-\Delta {)}^{-\alpha /2}$ defined by the Fourier multiplier $$\widehat{J_{\alpha }f}(\xi )=(1+4{\pi }^2|\xi {|}^2{)}^{-\alpha /2}\hat{f}(\xi ).$$ Its convolution kernel $G_{\alpha }$, the Bessel kernel, is a positive integrable function: $G_{\alpha }\in L^1({\mathbb{R}}^n)$ with $\int G_{\alpha }=1$, smooth away from the origin, with the same $|x{|}^{-(n-\alpha )}$ singularity as the Riesz kernel near $0$ but with exponential rather than power decay at infinity. Concretely $G_{\alpha }={\mathcal{F}}^{-1}[(1+4{\pi }^2|\xi {|}^2{)}^{-\alpha /2}]$, and the multiplier expands as $(1+4{\pi }^2|\xi {|}^2{)}^{-\alpha /2}=(2\pi |\xi |{)}^{-\alpha }(1+O(|\xi {|}^{-2}))$ at high frequency, so $J_{\alpha }$ and $I_{\alpha }$ differ by an operator bounded on every $L^p$.

Definition (Bessel-potential space). For $1\le p<\infty$ and $\alpha >0$ the Bessel-potential space is $$L_{\alpha }^p=J_{\alpha }(L^p)=\{J_{\alpha }g:g\in L^p\},\Vert f{\Vert }_{L_{\alpha }^p}=\Vert J_{-\alpha }f{\Vert }_{L^p}=\Vert (I-\Delta {)}^{\alpha /2}f{\Vert }_{L^p}.$$ Because $J_{\alpha }$ is injective (its multiplier never vanishes), every $f\in L_{\alpha }^p$ has a unique $g=J_{-\alpha }f\in L^p$, and the norm is well defined.

Definition (weak-type and the scaling line). A sublinear operator $T$ is of weak type $(p,q)$ if $|\{|Tf|>\lambda \}|\le (C\Vert f{\Vert }_p/\lambda {)}^q$ for all $\lambda >0$. The Sobolev scaling line for $I_{\alpha }$ is the relation $$\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n},$$ forced by dimensional analysis: under the dilation $f_t(x)=f(x/t)$ one has $I_{\alpha }{f}_t=t^{\alpha }(I_{\alpha }f{)}_t$, and matching the $L^p\to L^q$ norms of both sides leaves this single admissible relation.

Counterexamples to common slips Intermediate+

• The Hardy-Littlewood-Sobolev inequality fails at $p=1$. The scaling line at $p=1$ gives $1/q=1-\alpha /n$, but $I_{\alpha }$ is not bounded $L^1\to L^{n/(n-\alpha )}$; only the weak-type $(1,n/(n-\alpha ))$ estimate holds. Testing on an approximate identity concentrating at the origin shows the strong bound must fail, exactly as the Newtonian potential fails strong type at $p=1$.

• The inequality fails at the upper endpoint $p=n/\alpha$. There $1/q=0$, i.e. $q=\infty$, and $I_{\alpha }$ does not map $L^{n/\alpha }$ into $L^{\infty }$: the Riesz potential of an $L^{n/\alpha }$ function can be unbounded (a logarithmic blow-up). The correct endpoint statement is $I_{\alpha }:L^{n/\alpha }\to \mathrm{BMO}$, the space of functions of bounded mean oscillation.

• Riesz potentials do not preserve $L^p$. For $\alpha >0$ the operator $I_{\alpha }$ is never bounded from $L^p$ to itself; it always changes the exponent. The Bessel potential $J_{\alpha }$ does map $L^p$ to $L^p$ (its kernel is integrable), which is the structural reason $J_{\alpha }$, not $I_{\alpha }$, is the right tool for defining the function spaces $L_{\alpha }^p$.

• The semigroup law has a range restriction. The composition law $I_{\alpha }{I}_{\beta }=I_{\alpha +\beta }$ holds only while $\alpha +\beta <n$, because $I_{\alpha +\beta }$ is undefined once the kernel exponent $n-(\alpha +\beta )$ becomes non-positive. The Bessel potentials carry no such restriction: $J_{\alpha }{J}_{\beta }=J_{\alpha +\beta }$ for all $\alpha ,\beta >0$, since their multipliers multiply cleanly.

Key theorem with proof Intermediate+

Theorem (Hardy-Littlewood-Sobolev inequality; Hardy-Littlewood 1928 Math. Z. 27, Sobolev 1938 Mat. Sb. 4). Let $0<\alpha <n$ and let $1<p<q<\infty$ satisfy the scaling relation $1/q=1/p-\alpha /n$. Then there is a constant $C=C(n,\alpha ,p)$ with $$\Vert I_{\alpha }f{\Vert }_{L^q({\mathbb{R}}^n)}\le C\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}(f\in L^p).$$

Proof. Normalise $\Vert f{\Vert }_{L^p}=1$; by homogeneity it suffices to show $\Vert I_{\alpha }f{\Vert }_{L^q}\le C$. The argument splits the kernel at a radius $R>0$ to be optimised pointwise, bounds the near part by the maximal function and the far part by Hölder, and then runs Marcinkiewicz interpolation. Write, for fixed $x$ and any $R>0$, $${\gamma }_n(\alpha )I_{\alpha }f(x)=\underset{=A_R(x)}{\underbrace{{\int }_{|y-x|\le R}\frac{f(y)}{|x-y{|}^{n-\alpha }}dy}}+\underset{=B_R(x)}{\underbrace{{\int }_{|y-x|>R}\frac{f(y)}{|x-y{|}^{n-\alpha }}dy}}.$$

Step 1 (near part is controlled by the maximal function). Decompose the near ball into dyadic annuli $\{2^{-k-1}R<|y-x|\le 2^{-k}R\}$, $k\ge 0$. On the $k$-th annulus $|x-y{|}^{-(n-\alpha )}\le (2^{-k-1}R{)}^{-(n-\alpha )}$, so $$|A_R(x)|\le \sum _{k\ge 0}(2^{-k-1}R{)}^{-(n-\alpha )}{\int }_{|y-x|\le 2^{-k}R}|f(y)|dy\le \sum _{k\ge 0}(2^{-k-1}R{)}^{-(n-\alpha )}{\omega }_n(2^{-k}R{)}^{n}Mf(x),$$ where ${\omega }_n=|B(0,1)|$ and $Mf$ is the centred maximal function of 02.19.01. The $k$-th term is ${\omega }_n{2}^{n-\alpha }{2}^{-k\alpha }{R}^{\alpha }Mf(x)$, and summing the geometric series in $2^{-k\alpha }$ gives $$|A_R(x)|\le C_1{R}^{\alpha }Mf(x),C_1={\omega }_n{2}^{n-\alpha }(1-2^{-\alpha }{)}^{-1}.$$

Step 2 (far part is controlled by Hölder). Let $p^{\prime }$ be the conjugate exponent, $1/p+1/p^{\prime }=1$. By Hölder, $$|B_R(x)|\le \Vert f{\Vert }_{L^p}({\int }_{|y-x|>R}|x-y{|}^{-(n-\alpha )p^{\prime }}dy{)}^{1/p^{\prime }}.$$ The radial integral converges at infinity exactly when $(n-\alpha )p^{\prime }>n$, i.e. when $\alpha <n/p=n(1-1/p^{\prime })$, equivalently $1/q=1/p-\alpha /n>0$ — precisely the hypothesis $q<\infty$. Computing in polar coordinates, $${\int }_{|y-x|>R}|x-y{|}^{-(n-\alpha )p^{\prime }}dy=|S^{n-1}|{\int }_R^{\infty }{r}^{-(n-\alpha )p^{\prime }+n-1}dr=\frac{|S^{n-1}|}{(n-\alpha )p^{\prime }-n}{R}^{n-(n-\alpha )p^{\prime }}.$$ Raising to the power $1/p^{\prime }$ and using $\Vert f{\Vert }_{L^p}=1$, $$|B_R(x)|\le C_2{R}^{n/p^{\prime }-(n-\alpha )}=C_2{R}^{\alpha -n/p},$$ where $C_2=(|S^{n-1}|/((n-\alpha )p^{\prime }-n){)}^{1/p^{\prime }}$ and $n/p^{\prime }-(n-\alpha )=n-n/p-n+\alpha =\alpha -n/p$.

Step 3 (optimise the splitting radius). Combining Steps 1 and 2, $${\gamma }_n(\alpha )|I_{\alpha }f(x)|\le C_1{R}^{\alpha }Mf(x)+C_2{R}^{\alpha -n/p}.$$ The two terms balance when $R^{\alpha }Mf(x)\approx R^{\alpha -n/p}$, i.e. at $R=R(x)=(Mf(x){)}^{-p/n}$ (any $x$ with $Mf(x)>0$; where $Mf(x)=0$ the function vanishes near $x$ and $I_{\alpha }f(x)=0$). Substituting this $R$, both terms become a constant multiple of $(Mf(x){)}^{1-\alpha p/n}$, so $$|I_{\alpha }f(x)|\le C_3(Mf(x){)}^{1-\alpha p/n}=C_3(Mf(x){)}^{p/q},$$ using $1-\alpha p/n=p(1/p-\alpha /n)=p/q$.

Step 4 (Marcinkiewicz / direct integration). Raise to the $q$-th power and integrate: $${\int }_{{\mathbb{R}}^n}|I_{\alpha }f{|}^{q}dx\le C_3^q{\int }_{{\mathbb{R}}^n}(Mf{)}^{p}dx=C_3^q\Vert Mf{\Vert }_{L^p}^p\le C_3^q{A}_{n,p}^p\Vert f{\Vert }_{L^p}^p=C_3^q{A}_{n,p}^p,$$ where $A_{n,p}$ is the strong-type $(p,p)$ bound for $M$ from 02.19.01 (valid since $p>1$), and $\Vert f{\Vert }_{L^p}=1$. Taking $q$-th roots gives $\Vert I_{\alpha }f{\Vert }_{L^q}\le C_3{A}_{n,p}^{p/q}=:C$, the claim. (The pointwise bound $|I_{\alpha }f|\le C_3(Mf{)}^{p/q}$ together with the maximal theorem is the Hedberg form of the argument; it is the Marcinkiewicz interpolation between the weak endpoints made explicit.) $\square$

Bridge. The Hardy-Littlewood-Sobolev inequality builds toward the full Sobolev embedding theorem and appears again in the Calderón-Zygmund regularity theory of 02.13.02, where the second derivatives of the Newtonian potential are controlled in $L^p$ by exactly the same kind of singular-kernel estimate. The central insight is that fractional integration is tamed by the maximal function: the pointwise bound $|I_{\alpha }f|\le C(Mf{)}^{p/q}$ converts a global convolution estimate into a local averaging estimate, and this is exactly the foundational reason the proof needs no Fourier analysis at all. The bridge is that the maximal theorem of 02.19.01 and the Sobolev embedding of 24.01.01 are two faces of one inequality: putting these together, the weak-type $(1,1)$ bound for $M$ supplies, via the splitting-radius optimisation, the strong-type $(p,q)$ bound for $I_{\alpha }$, which generalises the Newtonian-potential mapping $I_2:L^p\to L^q$ to every fractional order.

Exercises Intermediate+

Advanced results Master

Theorem 1 (strong-type HLS and its proof structure; Hardy-Littlewood 1928, Sobolev 1938). For $0<\alpha <n$ and $1<p<q<\infty$ on the scaling line $1/q=1/p-\alpha /n$, the Riesz potential $I_{\alpha }:L^p\to L^q$ is bounded. The proof reduces to the Hedberg pointwise inequality $|I_{\alpha }f|\le C(Mf{)}^{p/q}\Vert f{\Vert }_{L^p}^{1-p/q}$ and the maximal theorem of 02.19.01; no Fourier analysis or singular-integral machinery is required, which is the distinctive economy of the maximal-function route ^{[Stein 1970]}.

Theorem 2 (sharp constant and extremals; Lieb 1983). The optimal constant in the HLS inequality in its bilinear form $$|\iint \frac{f(x)g(y)}{|x-y{|}^{\lambda }}dxdy|\le C_{n,\lambda ,p}\Vert f{\Vert }_{L^p}\Vert g{\Vert }_{L^{p^{\prime }}},\lambda =n-\alpha ,$$ is attained, and in the conformally invariant diagonal case $p=p^{\prime }=2n/(2n-\lambda )$ the extremals are exactly the translates, dilates, and scalar multiples of $(1+|x{|}^2{)}^{-(2n-\lambda )/2}$. Lieb computed the constant by symmetric-decreasing rearrangement, identifying the extremal profiles as the stereographic images of constants on the sphere ^{[Lieb 1983]}. The diagonal sharp HLS is dual to the sharp Sobolev inequality of Talenti and Aubin and underlies the Yamabe problem in conformal geometry.

Theorem 3 (Bessel-potential spaces equal Sobolev spaces; Calderón 1961, Aronszajn-Smith 1961). For $1<p<\infty$ and every non-negative integer $k$, $$L_k^p=W^{k,p}({\mathbb{R}}^n)\text{with equivalent norms},$$ and for non-integer $\alpha >0$ the Bessel-potential space $L_{\alpha }^p$ is the natural fractional-order interpolant. The norm equivalence $\Vert f{\Vert }_{W^{k,p}}\approx \Vert (I-\Delta {)}^{k/2}f{\Vert }_{L^p}=\Vert J_{-k}f{\Vert }_{L^p}$ is the content of the Calderón-Zygmund theory applied to the Riesz-transform symbols ${\xi }^{\beta }/(1+|\xi {|}^2{)}^{k/2}$, which are $L^p$-bounded Fourier multipliers for $1<p<\infty$ ^{[Calderon 1961] [Aronszajn-Smith 1961]}. At $p=2$ the identification is exact by Plancherel: $\Vert f{\Vert }_{W^{k,2}}^2=\int (1+4{\pi }^2|\xi {|}^2{)}^k|\hat{f}(\xi ){|}^{2}d\xi$.

Theorem 4 (the borderline endpoint $I_{\alpha }:L^{n/\alpha }\to \mathrm{BMO}$). At the upper endpoint $p=n/\alpha$, the scaling line gives $q=\infty$ but $I_{\alpha }$ fails to map into $L^{\infty }$. The correct statement is that $I_{\alpha }$ maps $L^{n/\alpha }$ continuously into the space $\mathrm{BMO}$ of functions of bounded mean oscillation: $$\Vert I_{\alpha }f{\Vert }_{\mathrm{BMO}}\le C\Vert f{\Vert }_{L^{n/\alpha }},\Vert u{\Vert }_{\mathrm{BMO}}=\underset{B}{\sup }\frac{1}{|B|}{\int }_B|u-u_B|.$$ This is the harmonic-analysis shadow of the Trudinger-Moser borderline of Sobolev embedding: at the critical exponent the embedding into $L^{\infty }$ fails, and the honest target is $\mathrm{BMO}$ (or, for the exponential-integrability refinement, the Orlicz space $e^L$). The full John-Nirenberg theory of $\mathrm{BMO}$ — that bounded mean oscillation forces exponential-tail distributional decay — is developed in 02.20.01 ^{[Stein 1970]}.

Theorem 5 (Riesz transforms and the factoring of $I_{\alpha }$). The Riesz transforms ${\mathcal{R}}_j$, the singular-integral operators with Fourier multipliers $-i{\xi }_j/|\xi |$, factor the relation between fractional integration and differentiation. One has ${\partial }_j={\mathcal{R}}_j(-\Delta {)}^{1/2}$ on the Fourier side, hence ${\partial }_j{I}_{\alpha }={\mathcal{R}}_j{I}_{\alpha -1}$ for $1<\alpha <n$, so a derivative of a Riesz potential is a Riesz transform of a Riesz potential of one lower order. Because each ${\mathcal{R}}_j$ is bounded on every $L^p$ ($1<p<\infty$) by the Calderón-Zygmund theorem, the Sobolev mapping properties of $I_{\alpha }$ transfer to all of its derivatives, which is the mechanism behind the regularity gain "$\alpha$ orders of smoothness" being uniform across derivatives ^{[Stein 1970]}.

Theorem 6 (the semigroup, fractional Laplacian, and stable processes). The Riesz potentials form the negative-power semigroup of the Laplacian: $I_{\alpha }=(-\Delta {)}^{-\alpha /2}$ with $I_{\alpha }{I}_{\beta }=I_{\alpha +\beta }$ for $\alpha +\beta <n$. Their inverses are the fractional Laplacians $(-\Delta {)}^s=I_{-2s}$, the generators of $2s$-stable Lévy processes for $0<s<1$, with the pointwise singular-integral representation $(-\Delta {)}^{s}u(x)=c_{n,s}p.v.\int (u(x)-u(y))|x-y{|}^{-n-2s}dy$. This identifies the analytic theory of Riesz potentials with the probabilistic theory of jump processes and with the modern theory of nonlocal elliptic equations ^{[Riesz 1949]}.

Synthesis. The Riesz potential is the foundational reason that the gain of regularity in elliptic and Sobolev theory is fractional and scale-invariant rather than tied to whole derivatives, and this is exactly the structural fact that unifies the maximal-function, singular-integral, and Sobolev-embedding strands of the chapter. The central insight is the Hedberg dictionary: fractional integration is dominated pointwise by a power of the maximal function, so the entire $L^p\to L^q$ theory of $I_{\alpha }$ is dual to the weak-type theory of $M$, and putting these together the Sobolev embedding of 24.01.01 is revealed as the $\alpha =1$ case of one inequality whose engine is the covering lemma of 02.19.01.

The pattern generalises in three directions that recur throughout harmonic analysis: vertically, from the Riesz potential to the Bessel potential, where clipping the power-law tail to exponential decay produces the $L^p$-bounded operator $J_{\alpha }$ that identifies $L_{\alpha }^p$ with $W^{\alpha ,p}$ and is dual to the Calderón-Zygmund multiplier theory; horizontally, from the strong-type interior of the scaling line to its two endpoints, where the lower endpoint $p=1$ degrades to weak type and the upper endpoint $p=n/\alpha$ degrades to $\mathrm{BMO}$, the bridge to the John-Nirenberg theory of 02.20.01; and structurally, from negative powers $(-\Delta {)}^{-\alpha /2}$ to positive powers $(-\Delta {)}^s$, the fractional Laplacians that generalise the local operator $-\Delta$ of 02.13.02 to the nonlocal generators of stable processes. The central insight binding all three is that $I_{\alpha }$ inverts a fractional power of the Laplacian by convolution against a homogeneous kernel, exactly as the Newtonian potential inverts $-\Delta$, and the Sobolev-scaling line is the unique exponent relation this homogeneity permits.

Full proof set Master

Proposition 1 (local integrability and well-definedness of $I_{\alpha }$ on $L^p$). For $0<\alpha <n$ and $1\le p<n/\alpha$, the integral defining $I_{\alpha }f(x)$ converges absolutely for almost every $x$ and every $f\in L^p$.

Proof. Split the kernel at radius $1$: $K_{\alpha }=K_{\alpha }{\chi }_{B(0,1)}+K_{\alpha }{\chi }_{B(0,1{)}^c}$. The near piece $K_{\alpha }{\chi }_{B(0,1)}$ lies in $L^1$ since ${\int }_{|z|<1}|z{|}^{-(n-\alpha )}dz=|S^{n-1}|{\int }_0^1{r}^{\alpha -1}dr=|S^{n-1}|/\alpha <\infty$, so its convolution with $f\in L^p$ is in $L^p$ by Young, finite a.e. The far piece $K_{\alpha }{\chi }_{B(0,1{)}^c}$ lies in $L^{p^{\prime }}$ exactly when $(n-\alpha )p^{\prime }>n$, i.e. $p<n/\alpha$, since then ${\int }_{|z|>1}|z{|}^{-(n-\alpha )p^{\prime }}dz<\infty$; its convolution with $f\in L^p$ is therefore bounded by $\Vert f{\Vert }_{L^p}\Vert K_{\alpha }{\chi }_{B(0,1{)}^c}{\Vert }_{L^{p^{\prime }}}$ pointwise by Hölder, finite everywhere. The sum converges a.e. $\square$

Proposition 2 (Fourier-multiplier identity for $I_{\alpha }$). On the Schwartz class, $\widehat{I_{\alpha }f}(\xi )=(2\pi |\xi |{)}^{-\alpha }\hat{f}(\xi )$, and this fixes the constant ${\gamma }_n(\alpha )$.

Proof. The homogeneous distribution $|x{|}^{-(n-\alpha )}$ has Fourier transform a constant multiple of $|\xi {|}^{-\alpha }$, by the Riesz formula for the Fourier transform of homogeneous functions: for $0<\beta <n$, $\mathcal{F}[|x{|}^{-\beta }]={\pi }^{\beta -n/2}\frac{\Gamma ((n-\beta )/2)}{\Gamma (\beta /2)}|\xi {|}^{\beta -n}$. Set $\beta =n-\alpha$, so $\beta -n=-\alpha$ and the prefactor is ${\pi }^{(n-\alpha )-n/2}\Gamma (\alpha /2)/\Gamma ((n-\alpha )/2)$. Dividing the kernel by ${\gamma }_n(\alpha )={\pi }^{n/2}{2}^{\alpha }\Gamma (\alpha /2)/\Gamma ((n-\alpha )/2)$ converts this prefactor into $(2\pi {)}^{-\alpha }$, giving $\widehat{K_{\alpha }}(\xi )=(2\pi |\xi |{)}^{-\alpha }$. Since convolution becomes multiplication under $\mathcal{F}$, $\widehat{I_{\alpha }f}=\widehat{K_{\alpha }}\hat{f}=(2\pi |\xi |{)}^{-\alpha }\hat{f}$. $\square$

Proposition 3 (Hedberg pointwise inequality). For $1\le p<n/\alpha$ and $f\in L^p$, $|I_{\alpha }f(x)|\le C(Mf(x){)}^{1-\alpha p/n}\Vert f{\Vert }_{L^p}^{\alpha p/n}$.

Proof. This is Exercise 4: split at radius $R$, bound the near part by $C_1{R}^{\alpha }Mf(x)$ via dyadic annuli against $M$, bound the far part by $C_2{R}^{\alpha -n/p}\Vert f{\Vert }_{L^p}$ via Hölder, and optimise $R=(\Vert f{\Vert }_{L^p}/Mf(x){)}^{p/n}$ to balance the two. The balanced value is $C(Mf(x){)}^{1-\alpha p/n}\Vert f{\Vert }_{L^p}^{\alpha p/n}$. $\square$

Proposition 4 (strong-type HLS from Hedberg and the maximal theorem). For $1<p<n/\alpha$ and $q$ on the scaling line, $\Vert I_{\alpha }f{\Vert }_{L^q}\le C\Vert f{\Vert }_{L^p}$.

Proof. Raise the Hedberg bound to the $q$-th power: $|I_{\alpha }f{|}^q\le C^q(Mf{)}^{(1-\alpha p/n)q}\Vert f{\Vert }_{L^p}^{(\alpha p/n)q}$. The exponent $(1-\alpha p/n)q=p$ since $1-\alpha p/n=p(1/p-\alpha /n)=p/q$. Integrate: $$\Vert I_{\alpha }f{\Vert }_{L^q}^q\le C^q\Vert f{\Vert }_{L^p}^{(\alpha p/n)q}\int (Mf{)}^p=C^q\Vert f{\Vert }_{L^p}^{(\alpha p/n)q}\Vert Mf{\Vert }_{L^p}^p\le C^q{A}_{n,p}^p\Vert f{\Vert }_{L^p}^{(\alpha p/n)q+p}.$$ The total power of $\Vert f{\Vert }_{L^p}$ is $(\alpha p/n)q+p=q(\alpha p/n+p/q)=q(\alpha p/n+1-\alpha p/n)=q$. Taking $q$-th roots gives $\Vert I_{\alpha }f{\Vert }_{L^q}\le C{A}_{n,p}^{p/q}\Vert f{\Vert }_{L^p}$. The hypothesis $p>1$ is used precisely to invoke the strong-type maximal theorem $\Vert Mf{\Vert }_{L^p}\le A_{n,p}\Vert f{\Vert }_{L^p}$ of 02.19.01. $\square$

Proposition 5 (Bessel kernel is a positive $L^1$ probability density). The Bessel kernel $G_{\alpha }$ satisfies $G_{\alpha }\ge 0$, $G_{\alpha }\in L^1$, $\int G_{\alpha }=1$.

Proof. Use the subordination formula obtained by writing the multiplier as a Laplace transform: for $a>0$, $$a^{-\alpha /2}=\frac{1}{\Gamma (\alpha /2)}{\int }_0^{\infty }{t}^{\alpha /2-1}{e}^{-ta}dt.$$ Apply with $a=1+4{\pi }^2|\xi {|}^2$ and take inverse Fourier transforms, using ${\mathcal{F}}^{-1}[e^{-4{\pi }^{2}t|\xi {|}^2}](x)=(4\pi t{)}^{-n/2}{e}^{-|x{|}^2/(4t)}$ (the heat kernel at time $t$): $$G_{\alpha }(x)=\frac{1}{\Gamma (\alpha /2)}{\int }_0^{\infty }{t}^{\alpha /2-1}{e}^{-t}(4\pi t{)}^{-n/2}{e}^{-|x{|}^2/(4t)}dt.$$ Every factor in the integrand is positive, so $G_{\alpha }>0$. Integrating in $x$ and using ${\int }_{{\mathbb{R}}^n}(4\pi t{)}^{-n/2}{e}^{-|x{|}^2/(4t)}dx=1$ gives $\int G_{\alpha }=\frac{1}{\Gamma (\alpha /2)}{\int }_0^{\infty }{t}^{\alpha /2-1}{e}^{-t}dt=1$. Hence $G_{\alpha }$ is an $L^1$ probability density, and $J_{\alpha }=G_{\alpha }\ast$ is a contraction on every $L^p$. $\square$

Proposition 6 ($L^2$ identification $L_{\alpha }^2=H^{\alpha }$). For every $\alpha >0$, $L_{\alpha }^2=H^{\alpha }({\mathbb{R}}^n)$ with equal norms, where $H^{\alpha }=\{f:(1+4{\pi }^2|\xi {|}^2{)}^{\alpha /2}\hat{f}\in L^2\}$.

Proof. By definition $f\in L_{\alpha }^2$ means $J_{-\alpha }f\in L^2$, i.e. $(1+4{\pi }^2|\xi {|}^2{)}^{\alpha /2}\hat{f}\in L^2$ by Proposition 2's multiplier description applied to $J_{-\alpha }$ and Plancherel. That is exactly the defining condition of $H^{\alpha }$, and the norms coincide: $$\Vert f{\Vert }_{L_{\alpha }^2}^2=\Vert J_{-\alpha }f{\Vert }_{L^2}^2=\int (1+4{\pi }^2|\xi {|}^2{)}^{\alpha }|\hat{f}(\xi ){|}^{2}d\xi =\Vert f{\Vert }_{H^{\alpha }}^2.$$ For integer $\alpha =k$ this equals $\Vert f{\Vert }_{W^{k,2}}^2$ after expanding the binomial $(1+4{\pi }^2|\xi {|}^2{)}^k$ and matching each $|{\xi }^{\beta }\hat{f}{|}^2$ term to $\Vert {\partial }^{\beta }f{\Vert }_{L^2}^2$ via Plancherel, recovering 24.01.01's definition at $p=2$. $\square$

Connections Master

• Hardy-Littlewood maximal function and Vitali covering 02.19.01. The direct engine of the proof. The Hedberg pointwise bound $|I_{\alpha }f|\le C(Mf{)}^{p/q}$ reduces the entire $L^p\to L^q$ mapping theory of the Riesz potential to the strong-type $(p,p)$ maximal inequality, and the weak-type $(1,1)$ bound supplies the $p=1$ endpoint. The covering-lemma economy of that unit is exactly why HLS needs no Fourier analysis, and the dyadic-annulus splitting in Step 1 of the Key Theorem is the same scale-decomposition used there.

• $L^p$ spaces, Hölder, Minkowski, Marcinkiewicz interpolation 02.07.06. The function-space substrate. Hölder's inequality bounds the far part of the kernel split; the conjugate-exponent arithmetic determines the convergence threshold $p<n/\alpha$; and Marcinkiewicz interpolation between the weak endpoints is the abstract form of the Hedberg balancing argument. Young's inequality from this unit gives the $L^p$-boundedness of the Bessel potential.

• Sobolev spaces $H^s$ and $W^{k,p}$ 24.01.01. The destination. The Bessel-potential space $L_{\alpha }^p=J_{\alpha }(L^p)$ is identified with $W^{\alpha ,p}$, giving an intrinsic, Fourier-analytic definition of fractional Sobolev regularity; and the Sobolev embedding $W^{1,p}\hookrightarrow L^{p^{\ast }}$ is re-derived here as the $\alpha =1$ case of HLS applied to the gradient, complementing that unit's direct Gagliardo-Nirenberg-Sobolev proof.

• Poisson equation, fundamental solution, Newtonian potential 02.13.02. The prototype and the $\alpha =2$ special case. The Newtonian potential $\Phi \ast f$ is $I_{2}f$, so HLS upgrades the Newtonian-potential mapping from a special exponent to the whole scaling line, and the fractional Laplacians $(-\Delta {)}^s=I_{-2s}$ generalise that unit's local operator $-\Delta$ to nonlocal generators of stable processes.

• John-Nirenberg inequality and BMO [forward: 02.20.01]. The endpoint successor. The upper-endpoint statement $I_{\alpha }:L^{n/\alpha }\to \mathrm{BMO}$ derived here as a closing preview is completed there, where the John-Nirenberg theorem shows bounded mean oscillation forces exponential distributional decay, the borderline refinement of the Sobolev embedding into $L^{\infty }$.

Historical & philosophical context Master

The fractional integral $I_{\alpha }$ traces to the Riemann-Liouville fractional calculus of the nineteenth century, but its modern multidimensional theory begins with Marcel Riesz, who in his 1949 Acta Mathematica memoir L'intégrale de Riemann-Liouville et le problème de Cauchy ^{[Riesz 1949]} introduced the $n$-dimensional Riesz potentials with the normalisation ${\gamma }_n(\alpha )$ that makes them a convolution semigroup and identifies them as negative powers of the Laplacian. The boundedness theorem that bears the Hardy-Littlewood-Sobolev name was assembled in two stages: Godfrey Harold Hardy and John Edensor Littlewood proved the one-dimensional fractional-integration inequality in their 1928 Mathematische Zeitschrift paper Some properties of fractional integrals. I ^{[Hardy-Littlewood 1928]}, and Sergei Lvovich Sobolev extended it to ${\mathbb{R}}^n$ in his 1938 Matematicheskii Sbornik paper On a theorem of functional analysis ^{[Sobolev 1938]}, in the course of establishing the embedding theorems that now carry his name.

The identification of the Bessel-potential spaces $L_{\alpha }^p$ with the Sobolev spaces $W^{k,p}$ is due to Alberto Calderón, whose 1961 Proceedings of Symposia in Pure Mathematics article Lebesgue spaces of differentiable functions and distributions ^{[Calderon 1961]} used the Calderón-Zygmund multiplier theory to prove the norm equivalence for all $1<p<\infty$, with the systematic theory of Bessel potentials developed in parallel by Nachman Aronszajn and Kennard Smith in their 1961 Annales de l'Institut Fourier paper ^{[Aronszajn-Smith 1961]}. The maximal-function proof of HLS presented here, which dispenses with Fourier analysis entirely, is the route taken by Elias Stein in his 1970 monograph ^{[Stein 1970]}, where the Riesz and Bessel potentials open the chapter that culminates in the Sobolev embedding theorems. The sharp constants and the conformal extremals were found by Elliott Lieb in his 1983 Annals of Mathematics paper ^{[Lieb 1983]}, connecting the inequality to the Yamabe problem and to symmetric-decreasing rearrangement.

Bibliography Master

@article{Riesz1949,
  author  = {Riesz, Marcel},
  title   = {L'int\'egrale de Riemann-Liouville et le probl\`eme de Cauchy},
  journal = {Acta Mathematica},
  volume  = {81},
  year    = {1949},
  pages   = {1--223}
}

@article{HardyLittlewood1928,
  author  = {Hardy, G. H. and Littlewood, J. E.},
  title   = {Some properties of fractional integrals. I},
  journal = {Mathematische Zeitschrift},
  volume  = {27},
  year    = {1928},
  pages   = {565--606}
}

@article{Sobolev1938,
  author  = {Sobolev, S. L.},
  title   = {On a theorem of functional analysis},
  journal = {Matematicheskii Sbornik (N.S.)},
  volume  = {4},
  year    = {1938},
  pages   = {471--497}
}

@book{Stein1970,
  author    = {Stein, Elias M.},
  title     = {Singular Integrals and Differentiability Properties of Functions},
  publisher = {Princeton University Press},
  year      = {1970}
}

@article{Calderon1961,
  author  = {Calder\'on, Alberto P.},
  title   = {Lebesgue spaces of differentiable functions and distributions},
  journal = {Proceedings of Symposia in Pure Mathematics},
  volume  = {IV},
  year    = {1961},
  pages   = {33--49}
}

@article{AronszajnSmith1961,
  author  = {Aronszajn, Nachman and Smith, Kennard T.},
  title   = {Theory of Bessel potentials. I},
  journal = {Annales de l'Institut Fourier},
  volume  = {11},
  year    = {1961},
  pages   = {385--475}
}

@article{Lieb1983,
  author  = {Lieb, Elliott H.},
  title   = {Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities},
  journal = {Annals of Mathematics},
  volume  = {118},
  year    = {1983},
  pages   = {349--374}
}

@book{LiebLoss2001,
  author    = {Lieb, Elliott H. and Loss, Michael},
  title     = {Analysis},
  edition   = {2},
  publisher = {American Mathematical Society},
  year      = {2001}
}