Inequalities are the working estimates of analysis. In \(L^p\) spaces they control products, averages, convolutions, interpolation, and tail probabilities.
Jensen inequality
Convexity moves through integration.
Theorem. If \(\mu\) is a probability measure, \(\phi\) is convex, and \(f,\phi(f)\in L^1\), then \(\phi(\int f\,\mathrm{d}\mu)\le\int\phi(f)\,\mathrm{d}\mu\).
Proof. For any point \(x_0\) in the interior of the domain of \(\phi\), convexity gives a supporting affine function \(\ell\) with \(\ell\le\phi\) and \(\ell(x_0)=\phi(x_0)\). Apply this with \(x_0=\int f\). Then \(\phi(x_0)=\ell(\int f)=\int \ell(f)\le\int\phi(f)\).
H?lder and generalized H?lder
Products are integrable when reciprocal exponents add to one.
Theorem. If \(p_i\in[1,\infty]\) and \(\sum_i1/p_i=1\), then \(\|\prod_i f_i\|_1\le\prod_i\|f_i\|_{p_i}\).
Proof. The two-factor case is H?lder. The finite-factor case follows by induction: combine the first \(n-1\) factors into one function with exponent \(r\) satisfying \(1/r=\sum_{i\lt n}1/p_i\), then apply the two-factor inequality to that product and \(f_n\).
Interpolation
Intermediate norms are controlled by endpoint norms.
Theorem. If \(1\le p\lt r\lt q\le\infty\) and \(1/r=\theta/p+(1-\theta)/q\), then \(\|f\|_r\le\|f\|_p^\theta\|f\|_q^{1-\theta}\).
Proof. For \(q\lt \infty\), write \(|f|^r=|f|^{\theta r}|f|^{(1-\theta)r}\) and apply H?lder with exponents \(p/(\theta r)\) and \(q/((1-\theta)r)\). The \(q=\infty\) case follows directly from \(\int |f|^r\le\|f\|_\infty^{r-p}\int|f|^p\).
Young convolution inequality
Convolution smooths while obeying norm bounds.
Theorem (Young convolution inequality). On \(\mathbb R^d\), if \(1\le p,q,r\le\infty\) and \[ 1+\frac1r=\frac1p+\frac1q, \] then \[ \|f*g\|_r\le\|f\|_p\|g\|_q. \]
Proof outline. The endpoint \[ \|f*g\|_\infty\le\|f\|_p\|g\|_{p'} \] is H?lder, while \[ \|f*g\|_p\le\|f\|_p\|g\|_1 \] follows from Minkowski's integral inequality. Applying the Riesz-Thorin interpolation theorem to the convolution operator \(T_gf=f*g\), and then symmetrizing in \(f,g\), yields every exponent triple on the plane \(1+1/r=1/p+1/q\). A direct proof is obtained by splitting the integrand into powers and applying generalized H?lder twice; interpolation packages exactly those exponent calculations.
Theorem (Minkowski integral inequality). For \(1\le p<\infty\), \[ \left( \int_X\left|\int_YF(x,y)\,\mathrm{d}\nu(y)\right|^p\mathrm{d}\mu(x) \right)^{1/p} \le \int_Y\left(\int_X|F(x,y)|^p\,\mathrm{d}\mu(x)\right)^{1/p}\mathrm{d}\nu(y), \] whenever the right side is finite.
Proof. For simple \(F\) in the \(y\)-variable, this is repeated Minkowski inequality in \(L^p(X)\). Approximate a general integrable \(F\) by such simple functions and use monotone convergence for nonnegative majorants, or Bochner integration in the Banach space \(L^p(X)\). The inequality is precisely the norm bound \[ \left\|\int_YF(\cdot,y)\,\mathrm{d}\nu(y)\right\|_{L^p(X)} \le\int_Y\|F(\cdot,y)\|_{L^p(X)}\,\mathrm{d}\nu(y). \]
Markov and Chebyshev inequalities
Integral bounds imply tail bounds.
Proposition. If \(f\ge0\), then \(\mu(f>\lambda)\le\lambda^{-1}\int f\,\mathrm{d}\mu\). More generally, \(\mu(|X|>\lambda)\le\lambda^{-p}\|X\|_p^p\).
Proof. On the set \(\{f>\lambda\}\), one has \(f\ge\lambda\mathbf 1_{\{f>\lambda\}}\). Integrating gives Markov's inequality. Apply it to \(f=|X|^p\) for the \(L^p\) form.
Corollary (Chebyshev and moment tails). For a random variable \(X\) with finite variance, \[ \mathbb P(|X-\mathbb EX|\ge t)\le\frac{\operatorname{Var}(X)}{t^2}. \] More generally, any finite \(p\)-th moment gives \[ \mathbb P(|X|\ge t)\le t^{-p}\mathbb E|X|^p. \]