Sobolev Inequalities: the Gagliardo-Nirenberg-Sobolev and Morrey Inequalities

Intuition Beginner

A Sobolev inequality is a precise statement of an idea you already trust: if you know that a function does not change too fast, then the function itself cannot be too big or too wild. The rate of change is captured by the derivative, the gradient; the inequality says that controlling the size of the gradient automatically controls the size of the function. You give up information about the slope, and in exchange you get information about the height.

Why would anyone want this trade? Because in most of mathematical physics the natural quantity you can bound is an energy, and energy is built from the gradient. The energy of a stretched membrane, the energy stored in an electric field, the kinetic energy of a flow: each is an integral of the square of a derivative. A Sobolev inequality is the bridge that converts a bound on this energy into a bound on the thing you actually care about, the displacement, the potential, the velocity itself.

There are two faces of the same coin, and which one you get depends on a competition between two numbers: how many derivatives you control, and how many dimensions you live in. When dimension wins, the function need not even be bounded, but its overall size in an averaged sense improves: you started controlling the function in one averaging scale and you end up controlling it in a stronger one. This is the Gagliardo-Nirenberg-Sobolev regime. When the derivative wins, you get something much stronger and more tangible: the function is genuinely continuous, with no jumps, and in fact it cannot change value too quickly between any two nearby points. This is the Morrey regime.

The dividing line is set by comparing the number of derivatives you control against the dimension. Below the line you buy improved averaged size; above the line you buy honest continuity. Right on the line is a delicate borderline case where the function is almost bounded, missing it only by a whisker, and the correct statement involves an exponential rather than a power.

A useful everyday picture: imagine pouring a fixed amount of paint that must be spread so that its slope is never steep. In a cramped low-dimensional space, the paint has nowhere to hide and piles up into a visible, continuous coat. In a roomy high-dimensional space, the same slope budget lets the paint thin out and spread, so it need not form a continuous coat, but its total spread-out concentration still improves. Dimension is room to spread; derivative control is the leash on the slope. The Sobolev inequalities measure exactly how these two forces balance.

Visual Beginner

The single most useful picture is the competition between two numbers, drawn as a number line.

The number line is the whole story. Put your finger at the dimension. Slide left, where the integrability of the gradient is smaller than the dimension: you are in the regime where you buy averaged size, and the averaging scale you end up with, the critical exponent, is determined by a formula that gets larger and larger as you approach the dimension from below. Slide right, where the integrability of the gradient exceeds the dimension: you buy continuity, and a quantitative smoothness number that measures how gently the function is allowed to vary. The exact crossing point, where the two numbers are equal, is the borderline case, where the function is almost but not quite bounded.

The second picture is the paint cartoons. A fixed slope budget in a small room forces a visible continuous coat; the same slope budget in a large room lets the paint thin and spread without ever forming a coat. The Sobolev inequalities turn this qualitative intuition into an exact accounting.

Worked example Beginner

We test the central formula on a concrete shape and watch the bookkeeping work out. Take three-dimensional space, so the dimension is three. Suppose we control the gradient in the averaging scale with exponent two, that is, we control the integral of the squared gradient, the most common energy in physics. We want to know the improved averaging scale for the function itself.

Step 1. Identify the numbers. The dimension is three. The exponent controlling the gradient is two. We need the dimension to beat the exponent for this regime, and indeed three is bigger than two, so we are on the Gagliardo-Nirenberg-Sobolev side.

Step 2. Apply the critical-exponent formula. The improved averaging scale is the dimension times the exponent, divided by the dimension minus the exponent. Substitute: the numerator is three times two, which is six. The denominator is three minus two, which is one. So the improved averaging scale is six divided by one, namely six.

Step 3. Read the conclusion. Controlling the integral of the squared gradient of a function in three-dimensional space controls the integral of the sixth power of the function itself. We started controlling the function's slope at the level of squares and we ended up controlling the function at the level of sixth powers, a genuinely stronger grip.

Step 4. Sanity-check the direction. The improved number, six, is larger than the starting number, two. A larger averaging exponent is a stronger form of control, because it weighs tall narrow spikes more heavily and so forbids them more strictly. The trade gave us something better than we put in, which is the whole point.

Step 5. Watch the formula misbehave as a warning. Keep the dimension at three but imagine pushing the gradient exponent up toward three. The denominator, the dimension minus the exponent, shrinks toward zero, and the improved averaging scale blows up. This is the inequality announcing the borderline: when the gradient exponent equals the dimension, the clean power-law conclusion breaks and a subtler exponential statement takes over.

What this tells us: the formula is a bookkeeping device that converts control of the slope into stronger control of the function, with the exact strength dictated by a competition between the gradient exponent and the dimension. The blow-up as the two approach each other is not a flaw; it is the inequality pointing at its own boundary.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $n\ge 1$ is the dimension, $U\subseteq {\mathbb{R}}^n$ is open, and $1\le p<\infty$. The Sobolev space $W^{1,p}(U)$ and its norm are taken as already defined 24.01.01: $u\in W^{1,p}(U)$ means $u\in L^p(U)$ together with weak partial derivatives $D_{i}u\in L^p(U)$ for $i=1,\dots ,n$, with $\Vert u{\Vert }_{W^{1,p}(U)}=(\Vert u{\Vert }_{L^p(U)}^p+\Vert Du{\Vert }_{L^p(U)}^p{)}^{1/p}$, where $Du=(D_{1}u,\dots ,D_{n}u)$ and $|Du|=({\sum }_i(D_{i}u{)}^2{)}^{1/2}$. We write $W_0^{1,p}(U)$ for the closure of $C_c^{\infty }(U)$ in $W^{1,p}(U)$. The $L^p$ machinery, Hölder's inequality, and Minkowski's inequality are taken as available 02.07.06.

Definition (critical Sobolev exponent). For $1\le p<n$, the *critical Sobolev exponent** is $$p^{\ast }=\frac{np}{n-p},\text{equivalently}\frac{1}{p^{\ast }}=\frac{1}{p}-\frac{1}{n}.$$ Note $p^{\ast }>p$ and $p^ \to \infty$as$p \uparrow n$.

The exponent $p^{\ast }$ is forced by scaling. If $u\in C_c^{\infty }({\mathbb{R}}^n)$ and $u_{\lambda }(x)=u(\lambda x)$ for $\lambda >0$, then a change of variables gives $\Vert u_{\lambda }{\Vert }_{L^q}={\lambda }^{-n/q}\Vert u{\Vert }_{L^q}$ and $\Vert D{u}_{\lambda }{\Vert }_{L^p}={\lambda }^{1-n/p}\Vert Du{\Vert }_{L^p}$. For an inequality of the form $\Vert u{\Vert }_{L^q}\le C\Vert Du{\Vert }_{L^p}$ to hold for all $u$ with a constant $C$ independent of $\lambda$, the two powers of $\lambda$ must match: $-n/q=1-n/p$, which rearranges to $q=p^{\ast }$. Any other exponent fails by sending $\lambda \to 0$ or $\lambda \to \infty$.

Definition (Hölder space). For $0<\gamma \le 1$ and $U$ open, the Hölder space $C^{0,\gamma }(\bar{U})$ consists of bounded continuous $u:\bar{U}\to \mathbb{R}$ for which the seminorm $$[u{]}_{C^{0,\gamma }(\bar{U})}=\underset{\begin{array}{c}x,y\in U\\ x\ne y\end{array}}{\sup }\frac{|u(x)-u(y)|}{|x-y{|}^{\gamma }}$$ is finite, normed by $\Vert u{\Vert }_{C^{0,\gamma }(\bar{U})}={\sup }_U|u|+[u{]}_{C^{0,\gamma }(\bar{U})}$. A function in $C^{0,\gamma }$ is Hölder continuous with exponent $\gamma$; $\gamma =1$ is Lipschitz continuity.

The two regimes. The Gagliardo-Nirenberg-Sobolev (GNS) inequality covers $1\le p<n$ and asserts a continuous embedding $W^{1,p}({\mathbb{R}}^n)\hookrightarrow L^{p^{\ast }}({\mathbb{R}}^n)$. The Morrey inequality covers $n<p<\infty$ (here $p\le \infty$ is admissible, with the seminorm read off below) and asserts $W^{1,p}({\mathbb{R}}^n)\hookrightarrow C^{0,\gamma }({\mathbb{R}}^n)$ with $\gamma =1-n/p$, after redefinition of $u$ on a set of measure zero. The borderline $p=n$ falls under neither power-law conclusion: $u$ need not be bounded (the standard example $u(x)=\log \log (1+1/|x|)$ near the origin in dimension $n$ lies in $W^{1,n}$ of a ball yet is unbounded), and the correct endpoint statement is exponential integrability in the sense of Trudinger-Moser, or membership in BMO ^{[Trudinger 1967] [Brezis-Wainger 1980]}.

Counterexamples to common slips Intermediate+

• The GNS exponent is not a free parameter. The embedding $W^{1,p}\hookrightarrow L^q$ for $p<n$ holds with the single critical $q=p^{\ast }$ on all of ${\mathbb{R}}^n$; lower exponents $q<p^{\ast }$ fail on ${\mathbb{R}}^n$ because constants and slowly-decaying functions have finite gradient norm but infinite $L^q$ norm. On a bounded domain the full ladder $W^{1,p}\hookrightarrow L^q$ for $p\le q\le p^{\ast }$ holds, because the finite measure lets Hölder interpolate down from $p^{\ast }$, but the gain stops at $p^{\ast }$.
• Morrey requires $p$ strictly above $n$. At $p=n$ the candidate Hölder exponent $1-n/p$ degenerates to zero, and continuity genuinely fails: $\log \log (1+1/|x|)$ has gradient in $L^n$ near the origin but is unbounded, so no Hölder estimate can hold.
• The embeddings are for the right space. The GNS inequality $\Vert u{\Vert }_{L^{p^{\ast }}}\le C\Vert Du{\Vert }_{L^p}$ with the gradient norm alone on the right needs either $u\in W_0^{1,p}$ or $u$ decaying at infinity; for general $u\in W^{1,p}(U)$ on a bounded domain one needs the full norm $\Vert u{\Vert }_{W^{1,p}}$ on the right and a Lipschitz (or extension-domain) boundary, because the bare gradient cannot see additive constants.
• Weak derivatives, not classical. All statements use the weak gradient. The Morrey conclusion is that the equivalence class of $u$ contains a (unique) Hölder-continuous representative; the original $L^p$ function may have been defined arbitrarily on a null set.

Key theorem with proof Intermediate+

Theorem (Gagliardo-Nirenberg-Sobolev inequality). Let $1\le p<n$. There is a constant $C=C(n,p)$ such that for every $u\in C_c^1({\mathbb{R}}^n)$, $$\Vert u{\Vert }_{L^{p^{\ast }}({\mathbb{R}}^n)}\le C\Vert Du{\Vert }_{L^p({\mathbb{R}}^n)},p^{\ast }=\frac{np}{n-p}$$ ^{[Gagliardo 1958] [Nirenberg 1959] [Evans 2010 §5.6.1]}.

Proof. The heart is the case $p=1$, where $p^{\ast }=n/(n-1)$; the general case follows by applying the $p=1$ result to a power of $|u|$ and using Hölder's inequality.

Step 1 (fundamental theorem of calculus on each axis). Fix $u\in C_c^1({\mathbb{R}}^n)$. For each $i\in \{1,\dots ,n\}$ and each $x\in {\mathbb{R}}^n$, integrate the $i$-th partial derivative along the $i$-th coordinate line from $-\infty$ up to $x_i$. Because $u$ has compact support, $u(x)={\int }_{-\infty }^{x_i}{D}_{i}u(x_1,\dots ,t,\dots ,x_n)dt$, and therefore $$|u(x)|\le {\int }_{-\infty }^{\infty }|D_{i}u(x_1,\dots ,t,\dots ,x_n)|dt=:g_i({\hat{x}}_i),$$ where ${\hat{x}}_i=(x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_n)$ records all coordinates except the $i$-th, and $g_i$ depends only on ${\hat{x}}_i$. This holds for every $i$ simultaneously.

Step 2 (multiply the $n$ estimates). Raise to the power $1/(n-1)$ and multiply over $i$: $$|u(x){|}^{n/(n-1)}=\prod _{i=1}^n|u(x){|}^{1/(n-1)}\le \prod _{i=1}^n{g}_i({\hat{x}}_i{)}^{1/(n-1)}.$$ The left side is the target exponent: $n/(n-1)=1^{\ast }$. The right side is a product of $n$ functions, the $i$-th of which is independent of the variable $x_i$.

Step 3 (Loomis-Whitney / iterated generalized Hölder). Integrate the inequality of Step 2 over ${\mathbb{R}}^n$, one variable at a time, pulling out the factor that does not depend on the current integration variable and applying the generalized Hölder inequality with $n-1$ exponents each equal to $n-1$ to the remaining factors. Integrate first in $x_1$. The factor $g_1({\hat{x}}_1{)}^{1/(n-1)}$ is constant in $x_1$ and pulls out; the remaining $n-1$ factors $g_i({\hat{x}}_i{)}^{1/(n-1)}$ ($i\ge 2$) each depend on $x_1$, and generalized Hölder with exponents $n-1$ gives $${\int }_{\mathbb{R}}\prod _{i=1}^n{g}_i^{1/(n-1)}d{x}_1\le g_1^{1/(n-1)}\prod _{i=2}^n({\int }_{\mathbb{R}}{g}_{i}d{x}_1{)}^{1/(n-1)}.$$ Repeat the procedure successively in $x_2,\dots ,x_n$. After all $n$ integrations, each $g_i$ has been integrated over all of its $n-1$ variables exactly once inside a $1/(n-1)$ power, yielding $${\int }_{{\mathbb{R}}^n}|u{|}^{n/(n-1)}dx\le \prod _{i=1}^n({\int }_{{\mathbb{R}}^n}|D_{i}u|dx{)}^{1/(n-1)}\le ({\int }_{{\mathbb{R}}^n}|Du|dx{)}^{n/(n-1)},$$ the last step using $|D_{i}u|\le |Du|$ and the arithmetic-geometric mean inequality on the $n$ factors. This is precisely the $p=1$ case: $$\Vert u{\Vert }_{L^{n/(n-1)}}\le {\int }_{{\mathbb{R}}^n}|Du|dx=\Vert Du{\Vert }_{L^1}.$$ (The combinatorial estimate just executed is the Loomis-Whitney inequality: the $L^1$ size of a function is bounded by the geometric mean of the $L^1$ sizes of its $n$ coordinate marginals.)

Step 4 (bootstrap from $p=1$ to general $p$). For $1<p<n$ apply the $p=1$ inequality to $v=|u{|}^{\gamma }$ with the exponent $\gamma =p(n-1)/(n-p)>1$ chosen below. Then $|Dv|=\gamma |u{|}^{\gamma -1}|Du|$ almost everywhere, so $$(\int |u{|}^{\gamma n/(n-1)}{)}^{(n-1)/n}\le \gamma \int |u{|}^{\gamma -1}|Du|\le \gamma (\int |u{|}^{(\gamma -1)p^{\prime }}{)}^{1/p^{\prime }}(\int |Du{|}^p{)}^{1/p},$$ where $p^{\prime }=p/(p-1)$ is the Hölder conjugate. The exponent $\gamma$ is fixed by the requirement that the two powers of $|u|$ match: $\gamma n/(n-1)=(\gamma -1)p^{\prime }$. Solving gives $\gamma =p(n-1)/(n-p)$, and then $\gamma n/(n-1)=p^{\ast }$. Dividing both sides by the common $|u|$-integral factor (finite because $u\in C_c^1$) leaves $$(\int |u{|}^{p^{\ast }}{)}^{(n-1)/n-1/p^{\prime }}\le \gamma (\int |Du{|}^p{)}^{1/p}.$$ The exponent on the left simplifies to $1/p^{\ast }$, giving $\Vert u{\Vert }_{L^{p^{\ast }}}\le C(n,p)\Vert Du{\Vert }_{L^p}$ with $C(n,p)=\gamma =p(n-1)/(n-p)$. $\square$

Bridge. The proof is the foundational reason the critical exponent is what it is: $p^{\ast }$ is not chosen for convenience but forced, first by the scaling identity that opened the formal-definition section and again here by the algebra that makes the two $|u|$-powers in Step 4 coincide. This is exactly the same move — integrate a derivative back to the function, then balance exponents — that the Morrey estimate below performs with a single radial integration instead of $n$ axis integrations, so the two inequalities are dual faces of one fundamental-theorem-of-calculus argument. Putting these together with a density argument extends both from $C_c^1$ to all of $W^{1,p}$, which builds toward the general embedding ladder for $W^{k,p}$ and the compactness refinement (Rellich-Kondrachov). The GNS inequality appears again in 02.16.02 as the existence engine for weak solutions of elliptic equations via the direct method, and the sharp-constant version generalises to the isoperimetric inequality, recovered as the $p=1$ case applied to the indicator of a smooth set.

Exercises Intermediate+

Advanced results Master

The first-order inequalities organize a much larger structure: the full Sobolev embedding ladder, the sharp constants and their extremals, the compactness refinement, the borderline endpoints, and the fractional and trace generalizations. Each is a refinement of the two fundamental-theorem-of-calculus arguments above.

Theorem 1 (general Sobolev embedding; Sobolev 1938). Let $U\subseteq {\mathbb{R}}^n$ be open and bounded with $C^1$ boundary, $1\le p<\infty$, and $k\in \mathbb{N}$. If $kp<n$, then $W^{k,p}(U)\hookrightarrow L^q(U)$ for $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$, the embedding being continuous. If $kp>n$, then $W^{k,p}(U)\hookrightarrow C^{m,\gamma }(\bar{U})$, where $m=k-\lfloor n/p\rfloor -1$ and $\gamma =\lfloor n/p\rfloor +1-n/p$ if $n/p\notin \mathbb{Z}$, and $\gamma$ any number in $(0,1)$ if $n/p\in \mathbb{Z}$ ^{[Sobolev 1938] [Evans 2010 §5.6.3]}. The two clauses are the iterated GNS and iterated Morrey arguments of Exercise 8, with the reciprocal-exponent accounting $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$ supplying the unified bookkeeping: integrability and differentiability trade at the fixed rate $1/n$ per derivative.

Theorem 2 (Rellich-Kondrachov compactness; Rellich 1930, Kondrachov 1945). Let $U$ be bounded with $C^1$ boundary and $1\le p<n$. Then for every $q$ with $1\le q<p^{\ast }$ (strictly subcritical), the embedding $W^{1,p}(U)\hookrightarrow \hookrightarrow L^q(U)$ is compact: bounded sequences in $W^{1,p}$ have subsequences converging strongly in $L^q$. Compactness fails at the critical exponent $q=p^{\ast }$ itself, where concentration (a bubble $u_{\epsilon }(x)={\epsilon }^{-(n-p)/p}\varphi (x/\epsilon )$) and translation to infinity both produce bounded non-convergent sequences. The loss of compactness at $p^{\ast }$ is the central analytic difficulty in critical elliptic problems and is quantified by the concentration-compactness principle.

Theorem 3 (sharp constant and extremals; Talenti 1976, Aubin 1976). For $1<p<n$, the best constant in $\Vert u{\Vert }_{L^{p^{\ast }}({\mathbb{R}}^n)}\le C\Vert Du{\Vert }_{L^p({\mathbb{R}}^n)}$ is $$C_{n,p}={\pi }^{-1/2}{n}^{-1/p}\left(\frac{p-1}{n-p}{)}^{1-1/p}\right(\frac{\Gamma (1+n/2)\Gamma (n)}{\Gamma (n/p)\Gamma (1+n-n/p)}{)}^{1/n},$$ and the extremal functions (where equality holds) are exactly the **Aubin-Talenti bubbles** $$u(x)=(a+b|x-x_0{|}^{p/(p-1)}{)}^{-(n-p)/p},a,b>0,x_0\in {\mathbb{R}}^n$$ ^{[Talenti 1976] [Aubin 1976]}. The extremals are unique modulo the symmetry group (translations, dilations, scalar multiples), and for $p=2$ they are the standard bubbles solving the critical Lane-Emden equation $-\Delta u=u^{(n+2)/(n-2)}$ that appear in the Yamabe problem.

Theorem 4 (Trudinger-Moser borderline; Trudinger 1967, Moser 1971). At the borderline $p=n$ the embedding $W^{1,n}\hookrightarrow L^{\infty }$ fails, but exponential integrability holds: there are constants ${\alpha }_n,C_n>0$ such that for $u\in W_0^{1,n}(U)$, $U$ bounded, with $\Vert Du{\Vert }_{L^n}\le 1$, $${\int }_U\exp ({\alpha }_n|u{|}^{n/(n-1)})dx\le C_n|U|,$$ and the constant ${\alpha }_n=n{\omega }_{n-1}^{1/(n-1)}$ (with ${\omega }_{n-1}$ the surface area of the unit sphere) is sharp: for any larger $\alpha$ the supremum is infinite ^{[Trudinger 1967] [Moser 1971]}. The Trudinger-Moser inequality is the correct endpoint of the Sobolev ladder, replacing the failed $L^{\infty }$ bound by membership in the exponential Orlicz class; an alternative endpoint reading places $W^{1,n}$ in BMO and the John-Nirenberg space ^{[Brezis-Wainger 1980]}.

Theorem 5 (fractional Sobolev and trace; Gagliardo 1957, Aronszajn-Slobodeckij). For $s\in (0,1)$ and $sp<n$, the fractional Sobolev space $W^{s,p}({\mathbb{R}}^n)$, normed by the Gagliardo seminorm $[u{]}_{s,p}^p=\iint \frac{|u(x)-u(y){|}^p}{|x-y{|}^{n+sp}}dxdy$, embeds continuously into $L^q$ with $\frac{1}{q}=\frac{1}{p}-\frac{s}{n}$, and the trace operator $u\mapsto u{|}_{\partial U}$ maps $W^{1,p}(U)$ onto $W^{1-1/p,p}(\partial U)$ for $p>1$. The trace theorem is the precise statement that boundary values of $W^{1,p}$ functions lose exactly $1/p$ of a derivative, the foundational fact for boundary-value problems in the weak formulation. The fractional scale interpolates the integer Sobolev spaces and is the natural setting for the nonlocal operators $(-\Delta {)}^s$ and for boundary integral equations.

Synthesis. The Sobolev inequalities are the foundational reason the calculus of variations and the weak theory of PDE work at all, and the entire structure is generated by a single principle made precise in two ways: integrate a derivative to recover the function, then balance exponents by scaling. The GNS inequality is exactly this with $n$ axis integrations and the Loomis-Whitney product estimate; the Morrey inequality is exactly this with one radial integration and a Riesz potential; and the general embedding ladder of Theorem 1 is the iteration of these two, with the reciprocal-exponent rule $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$ as the bridge that makes integrability and differentiability a single tradeable currency. Putting these together, the central insight is that the critical exponent $p^{\ast }$ is not a parameter but a scaling invariant, and this is exactly why compactness fails there (Theorem 2): the dilation symmetry that fixes $p^{\ast }$ also produces the non-compact bubbling sequences, and the same Aubin-Talenti bubbles that saturate the sharp constant (Theorem 3) are the bubbles that obstruct compactness and that solve the critical Euler-Lagrange equations.

The borderline $p=n$ generalises the power-law conclusion to the exponential Trudinger-Moser endpoint, and the whole edifice generalises further: to fractional orders and trace spaces (Theorem 5), which appears again in 02.16.04 as the device that gives meaning to boundary data; to Riemannian manifolds, where the sharp constant controls the Yamabe invariant; and to the abstract theory of interpolation spaces, where the Sobolev embedding is a single instance of the real and complex interpolation functors. The arc from Sobolev's 1938 averaging lemma to the modern concentration-compactness method is one continuous refinement of the same fundamental-theorem-of-calculus argument, generalised until it became the load-bearing inequality of twentieth-century analysis.

Full proof set Master

Proposition 1 (Loomis-Whitney inequality). Let $f_1,\dots ,f_n$ be non-negative measurable functions on ${\mathbb{R}}^{n-1}$, and for $x\in {\mathbb{R}}^n$ write ${\hat{x}}_i$ for $x$ with the $i$-th coordinate deleted. Then $${\int }_{{\mathbb{R}}^n}\prod _{i=1}^n{f}_i({\hat{x}}_i{)}^{1/(n-1)}dx\le \prod _{i=1}^n({\int }_{{\mathbb{R}}^{n-1}}{f}_i{)}^{1/(n-1)}.$$

Proof. Induct on $n$ using the generalized Hölder inequality with $n-1$ equal exponents $n-1$. For $n=2$ the claim is ${\int }_{{\mathbb{R}}^2}{f}_1(x_2)f_2(x_1)dx=(\int f_1)(\int f_2)$, an equality by Tonelli. For the inductive step, integrate first in $x_n$. The factor $f_n({\hat{x}}_n{)}^{1/(n-1)}$ does not depend on $x_n$ and pulls out; the remaining $n-1$ factors $f_i({\hat{x}}_i{)}^{1/(n-1)}$, $i<n$, each depend on $x_n$, and generalized Hölder with $n-1$ exponents equal to $n-1$ gives $${\int }_{\mathbb{R}}\prod _{i=1}^n{f}_i^{1/(n-1)}d{x}_n\le f_n^{1/(n-1)}\prod _{i=1}^{n-1}({\int }_{\mathbb{R}}{f}_{i}d{x}_n{)}^{1/(n-1)}.$$ Now integrate the result over the remaining variables $(x_1,\dots ,x_{n-1})$ and apply the inductive hypothesis in dimension $n-1$ to the $n-1$ functions $g_i={\int }_{\mathbb{R}}{f}_{i}d{x}_n$ (each a function of $n-2$ of the variables, after the $x_n$-integration), absorbing the leftover $f_n$ factor by Tonelli. The exponents collect to give the stated product bound. $\square$

Proposition 2 (GNS by density on $W^{1,p}({\mathbb{R}}^n)$). The inequality $\Vert u{\Vert }_{L^{p^{\ast }}}\le C(n,p)\Vert Du{\Vert }_{L^p}$ extends from $C_c^1({\mathbb{R}}^n)$ to all $u\in W^{1,p}({\mathbb{R}}^n)$, $1\le p<n$.

Proof. Let $u\in W^{1,p}({\mathbb{R}}^n)$. Smooth functions with compact support are dense in $W^{1,p}({\mathbb{R}}^n)$ 24.01.01, so choose $u_m\in C_c^{\infty }$ with $u_m\to u$ in $W^{1,p}$; in particular $D{u}_m\to Du$ in $L^p$. The Key Theorem gives $\Vert u_m-u_{\ell }{\Vert }_{L^{p^{\ast }}}\le C\Vert D(u_m-u_{\ell }){\Vert }_{L^p}\to 0$, so $(u_m)$ is Cauchy in $L^{p^{\ast }}$, with limit $v\in L^{p^{\ast }}$. Passing to a subsequence, $u_m\to u$ almost everywhere (from $L^p$ convergence) and $u_m\to v$ almost everywhere (from $L^{p^{\ast }}$ convergence), so $v=u$ a.e. Then $\Vert u{\Vert }_{L^{p^{\ast }}}=\lim \Vert u_m{\Vert }_{L^{p^{\ast }}}\le \lim C\Vert D{u}_m{\Vert }_{L^p}=C\Vert Du{\Vert }_{L^p}$. $\square$

Proposition 3 (Morrey inequality). Let $n<p\le \infty$. There is $C=C(n,p)$ so that every $u\in C^1({\mathbb{R}}^n)$ has, for all $x,y\in {\mathbb{R}}^n$, $$|u(x)-u(y)|\le C\Vert Du{\Vert }_{L^p({\mathbb{R}}^n)}|x-y{|}^{1-n/p},$$ and consequently $u$ (after redefinition on a null set, in the $W^{1,p}$ statement) lies in $C^{0,1-n/p}$.

Proof. Fix $x,y$, set $r=|x-y|$, and let $W=B(x,r)\cap B(y,r)$, a set of measure $\ge c(n)r^n$. For $z\in W$, the oscillation bound of Exercise 7 applied at $x$ and at $y$ gives ${\text{fint}}_{B(x,r)}|u-u(x)|\le C\Vert Du{\Vert }_{L^p}{r}^{1-n/p}$ and likewise at $y$. Average $u(x)-u(y)=(u(x)-u(z))+(u(z)-u(y))$ over $z\in W$: $$|u(x)-u(y)|\le {\text{fint}}_W|u(x)-u(z)|dz+{\text{fint}}_W|u(z)-u(y)|dz.$$ Since $W\subseteq B(x,r)$ and $|W|\ge c(n)|B(x,r)|$, each average over $W$ is bounded by a constant times the corresponding average over $B(x,r)$, hence by $C\Vert Du{\Vert }_{L^p}{r}^{1-n/p}$. Adding, $|u(x)-u(y)|\le C(n,p)\Vert Du{\Vert }_{L^p}|x-y{|}^{1-n/p}$. Taking $\sup |u|\le |u(x_0)|+C\Vert Du{\Vert }_{L^p}$ on any fixed ball bounds the sup norm; for $u\in W^{1,p}$ one applies the estimate to a smooth approximating sequence, which is then uniformly Hölder, hence uniformly convergent to a continuous representative. $\square$

Proposition 4 (failure at $p=n$). In dimension $n\ge 2$, $W^{1,n}(B_1)/\hookrightarrow L^{\infty }(B_1)$.

Proof. Take $u(x)=\log \log (1+1/|x|)$ near the origin (smoothly cut off near $|x|=1$). In polar coordinates, $|Du(x)|\sim \frac{1}{|x|(1+|x|)\log (1+1/|x|)}\sim \frac{1}{|x|\log (1/|x|)}$ as $|x|\to 0$. Then ${\int }_{B_{1/2}}|Du{|}^{n}dx\sim {\omega }_{n-1}{\int }_0^{1/2}\frac{r^{n-1}}{r^n(\log (1/r){)}^n}dr={\omega }_{n-1}{\int }_0^{1/2}\frac{dr}{r(\log (1/r){)}^n}$, which converges for $n\ge 2$ by the substitution $s=\log (1/r)$, ${\int }^{\infty }{s}^{-n}ds<\infty$. So $u\in W^{1,n}(B_1)$. Yet $u(x)\to \infty$ as $|x|\to 0$, so $u\notin L^{\infty }$. Hence the embedding into $L^{\infty }$ fails exactly at the borderline $p=n$, where the Morrey exponent $1-n/p$ degenerates to zero. $\square$

Connections Master

• The Sobolev space scaffolding — the definition of $W^{k,p}$, the weak derivative, the density of smooth functions, and the extension and approximation theorems — is supplied by 24.01.01, which surveys the embedding theorems proved here in full. This unit owns the deep embedding theorems; 24.01.01 owns the space. The split follows the spec discipline that the foundational unit defines the object and the downstream unit builds the heavy theory.

• The $L^p$ apparatus on which every estimate rests — Hölder's inequality, the generalized Hölder inequality with several exponents, Minkowski's inequality, and completeness — is developed in 02.07.06. The Loomis-Whitney product estimate is iterated generalized Hölder, and the bootstrap from $p=1$ to general $p$ in the Key Theorem is a single application of Hölder with conjugate exponents $p$ and $p^{\prime }$.

• The fundamental-theorem-of-calculus and integration-along-segments steps, together with the chain rule $|D|u{|}^{\gamma }|=\gamma |u{|}^{\gamma -1}|Du|$ used in the bootstrap, are the multivariable differentiation results of 02.05.04; the Morrey oscillation bound is a multivariable mean-value estimate along the segment joining two points.

• The embeddings are the existence engine for weak solutions of elliptic boundary-value problems in 02.16.02: the direct method of the calculus of variations minimizes an energy over $W^{1,p}$, and Rellich-Kondrachov compactness (Theorem 2) is exactly what upgrades a minimizing sequence's weak limit to a genuine minimizer. The trace theorem (Theorem 5) gives meaning to the boundary data, developed in 02.16.04.

• The critical-exponent obstruction reappears in geometric analysis: the $p=2$ sharp constant (Theorem 3) is the Yamabe constant of the round sphere, and the Aubin-Talenti bubbles are the concentration profiles in the Yamabe problem and in critical semilinear equations $-\Delta u=u^{(n+2)/(n-2)}$, linking to the variational and geometric PDE material in 02.16.05.

Historical & philosophical context Master

Sergei Sobolev introduced the spaces now bearing his name and proved the foundational embedding lemma in his 1938 Matematicheskii Sbornik paper ^{[Sobolev 1938]}, motivated by the Cauchy problem for hyperbolic equations and the need for a function-space framework in which weak solutions could be sought and bounded. Sobolev's averaging method (mollification by a smooth kernel) and his integral lemma gave the first general embedding $W^{k,p}\hookrightarrow L^q$, establishing the reciprocal-exponent trade $1/q=1/p-k/n$ that organizes the entire theory.

The sharp first-order inequalities were brought to their modern form by Emilio Gagliardo in his 1958 Ricerche di Matematica paper ^{[Gagliardo 1958]} and independently by Louis Nirenberg in his 1959 Annali della Scuola Normale Superiore di Pisa paper on elliptic equations ^{[Nirenberg 1959]}; the combinatorial product estimate at the heart of the $p=1$ case had appeared a decade earlier in the 1949 note of Lynn Loomis and Hassler Whitney ^{[Loomis-Whitney 1949]}, who proved it as a discrete-geometric inequality bounding a set's measure by its coordinate projections. Charles Morrey's 1940 Duke Mathematical Journal paper ^{[Morrey 1940]} established the complementary regime, the embedding into Hölder spaces for $p>n$, in his study of the differentiability of solutions of variational problems.

The sharp constants and extremal functions were found independently in 1976 by Giorgio Talenti ^{[Talenti 1976]}, using symmetric decreasing rearrangement and the radial Euler-Lagrange equation, and by Thierry Aubin ^{[Aubin 1976]} in the context of the Yamabe problem in Riemannian geometry; the resulting Aubin-Talenti bubbles became the canonical concentration profiles of critical-exponent problems. The borderline $p=n$ was settled by Neil Trudinger's 1967 Journal of Mathematics and Mechanics paper ^{[Trudinger 1967]} establishing exponential integrability, with the sharp constant determined by Jürgen Moser's 1971 Indiana University Mathematics Journal paper ^{[Moser 1971]}. The complementary BMO and John-Nirenberg reading of the endpoint was clarified by Haïm Brezis and Stephen Wainger in 1980 ^{[Brezis-Wainger 1980]}.

Bibliography Master

@article{Sobolev1938,
  author  = {Sobolev, Sergei L.},
  title   = {On a theorem of functional analysis},
  journal = {Matematicheskii Sbornik},
  volume  = {4(46)},
  year    = {1938},
  pages   = {471--497},
  note    = {English transl. AMS Translations (2) 34 (1963), 39--68}
}

@article{Gagliardo1958,
  author  = {Gagliardo, Emilio},
  title   = {Propriet\`a di alcune classi di funzioni in pi\`u variabili},
  journal = {Ricerche di Matematica},
  volume  = {7},
  year    = {1958},
  pages   = {102--137}
}

@article{Nirenberg1959,
  author  = {Nirenberg, Louis},
  title   = {On elliptic partial differential equations},
  journal = {Annali della Scuola Normale Superiore di Pisa, Serie 3},
  volume  = {13},
  year    = {1959},
  pages   = {115--162}
}

@article{Morrey1940,
  author  = {Morrey, Charles B.},
  title   = {Functions of several variables and absolute continuity, II},
  journal = {Duke Mathematical Journal},
  volume  = {6},
  year    = {1940},
  pages   = {187--215}
}

@article{LoomisWhitney1949,
  author  = {Loomis, Lynn H. and Whitney, Hassler},
  title   = {An inequality related to the isoperimetric inequality},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {55},
  year    = {1949},
  pages   = {961--962}
}

@article{Talenti1976,
  author  = {Talenti, Giorgio},
  title   = {Best constant in {S}obolev inequality},
  journal = {Annali di Matematica Pura ed Applicata},
  volume  = {110},
  year    = {1976},
  pages   = {353--372}
}

@article{Aubin1976,
  author  = {Aubin, Thierry},
  title   = {Probl\`emes isop\'erim\'etriques et espaces de {S}obolev},
  journal = {Journal of Differential Geometry},
  volume  = {11},
  year    = {1976},
  pages   = {573--598}
}

@article{Trudinger1967,
  author  = {Trudinger, Neil S.},
  title   = {On imbeddings into {O}rlicz spaces and some applications},
  journal = {Journal of Mathematics and Mechanics},
  volume  = {17},
  year    = {1967},
  pages   = {473--483}
}

@article{Moser1971,
  author  = {Moser, J\"urgen},
  title   = {A sharp form of an inequality by {N}. {T}rudinger},
  journal = {Indiana University Mathematics Journal},
  volume  = {20},
  year    = {1971},
  pages   = {1077--1092}
}

@article{BrezisWainger1980,
  author  = {Brezis, Ha\"im and Wainger, Stephen},
  title   = {A note on limiting cases of {S}obolev embeddings and convolution inequalities},
  journal = {Communications in Partial Differential Equations},
  volume  = {5},
  year    = {1980},
  pages   = {773--789}
}

@book{Evans2010,
  author    = {Evans, Lawrence C.},
  title     = {Partial Differential Equations},
  edition   = {2},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {19},
  year      = {2010}
}

@book{GilbargTrudinger1983,
  author    = {Gilbarg, David and Trudinger, Neil S.},
  title     = {Elliptic Partial Differential Equations of Second Order},
  edition   = {2},
  publisher = {Springer},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {224},
  year      = {1983}
}

@book{AdamsFournier2003,
  author    = {Adams, Robert A. and Fournier, John J. F.},
  title     = {Sobolev Spaces},
  edition   = {2},
  publisher = {Academic Press},
  year      = {2003}
}

@book{Mazya2011,
  author    = {Maz'ya, Vladimir},
  title     = {Sobolev Spaces},
  edition   = {2},
  publisher = {Springer},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {342},
  year      = {2011}
}

@book{LiebLoss2001,
  author    = {Lieb, Elliott H. and Loss, Michael},
  title     = {Analysis},
  edition   = {2},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {14},
  year      = {2001}
}