A Synthesis Examination in \(L^p\)
The previous tutorial showed how to approximate \(L^p\) functions on Euclidean space by continuous functions of compact support. This examination draws on that result and on the norm, Hölder, Minkowski, and convergence theorems developed earlier in the course. The emphasis is on choosing an estimate that fits the conclusion: a norm bound, an integral bound, or convergence in a different space.
Unless a problem specifies otherwise, functions are real-valued measurable functions modulo equality almost everywhere. For \(1\leq r<\infty\), \(L^r(X,\mu)\) has norm \(\|f\|_r=(\int_X|f|^r\,d\mu)^{1/r}\). When conjugate exponents are used, \(1<r<\infty\) and \(r'=r/(r-1)\), so \(1/r+1/r'=1\). We will cite earlier results such as Hölder’s Inequality and Minkowski’s Inequality by name rather than reproduce their proofs.
Worked Problems: Estimates and Norm Structure
Worked Example: Estimating an Integral from an \(L^2\) Norm
Let \(f(x)=x^{-1/4}\) on \((0,1)\), with Lebesgue measure. First check that \(f\in L^2(0,1)\):
Since the interval has measure one, the constant function \(1\) has \(L^2\) norm \(1\). Hölder’s Inequality, applied to \(f\) and \(1\), gives
Direct integration confirms the estimate and gives its exact value:
The final inequality holds because both sides are positive and \((4/3)^2=16/9<2=(\sqrt{2})^2\). This calculation illustrates a reusable proof move: on a finite-measure set, pair a function with the indicator of that set and apply Hölder. Integrability of \(f\) alone does not provide an \(L^2\) estimate; here the calculation of \(\|f\|_2\) verifies the needed hypothesis.
Worked Example: The Norm of a Sum with Disjoint Supports
On \(\mathbb{R}\), let \(f=3\mathbf{1}_{[0,1]}\) and \(g=4\mathbf{1}_{[2,3]}\), and take \(1\leq p<\infty\). The supports are disjoint except possibly at endpoints, which have measure zero. Thus at almost every point, at most one of \(f\) and \(g\) is nonzero. It follows that
Both intervals have measure one, so integrating and taking the \(p\)th root gives
For example, at \(p=2\), the norm is \(\sqrt{3^2+4^2}=5\), whereas \(\|f\|_2+\|g\|_2=3+4=7\). There is no contradiction with Minkowski’s Inequality: it asserts an upper bound of \(7\), not equality. Disjointness lets us calculate the norm exactly and can make the triangle inequality strict.
Products of Convergent Sequences
An \(L^p\) estimate often controls a product by pairing two factors with conjugate exponents. The next result uses this idea to transfer convergence in two different spaces into convergence of the product in \(L^1\). The norms of the convergent sequences need not be estimated term by term: the reverse triangle inequality shows that a norm-convergent sequence has bounded norms.
Proof. Since \(g_n\to g\) in \(L^q\), the reverse triangle inequality gives \(\bigl|\|g_n\|_q-\|g\|_q\bigr|\leq\|g_n-g\|_q\to0\). Therefore the numbers \(\|g_n\|_q\) are bounded; choose \(C<\infty\) such that \(\|g_n\|_q\leq C\) for all \(n\). Pointwise almost everywhere, the algebraic identity \(f_ng_n-fg=(f_n-f)g_n+f(g_n-g)\) holds. Hölder’s Inequality shows that both terms on the right are integrable and yields
The first term on the far right tends to zero by \(L^p\) convergence, and the second tends to zero by \(L^q\) convergence. Hence \(\|f_ng_n-fg\|_1\to0\), as required. \(\square\)
Worked Example: Products of Powers on the Unit Interval
On \((0,1)\), set \(f_n(x)=x^n\) and \(g_n(x)=1+x^n\). For any fixed \(1<p<\infty\),
Thus \(f_n\to0\) in \(L^p\). With \(q=p/(p-1)\), \(g_n\to1\) in \(L^q\), because
The product is \(f_ng_n=x^n+x^{2n}\), and its \(L^1\) error from the limiting product \(0\cdot1=0\) is exactly
This direct computation agrees with the product convergence theorem. In applications, the theorem is more useful than calculating the product integral: its proof needs only the two separate norm convergences and Hölder’s Inequality.
Convergence of \(p\)th Powers
The product theorem also yields a stronger conclusion than convergence of norms. For \(p>1\), \(L^p\) convergence implies that the functions \(|f_n|^p\) converge to \(|f|^p\) in \(L^1\). This controls the entire difference of the power functions in integral, rather than only the difference between their integrals.
Proof. When \(p=1\), the reverse triangle inequality for real numbers gives \(||f_n|-|f||\leq|f_n-f|\) almost everywhere. Integrating proves the claim in this case.
Now suppose \(p>1\). For nonnegative \(a,b\), the mean value theorem applied to \(t\mapsto t^p\), followed by \(||f_n|-|f||\leq|f_n-f|\), gives the pointwise estimate
Indeed, the derivative \(pt^{p-1}\) on the interval between \(a\) and \(b\) is at most \(p\max(a,b)^{p-1}\), and \(\max(a,b)\leq a+b\). Set \(q=p/(p-1)\). Hölder’s Inequality and Minkowski’s Inequality now give
The reverse triangle inequality implies \(\|f_n\|_p\to\|f\|_p\), so the factor \((\|f_n\|_p+\|f\|_p)^{p-1}\) is bounded. The remaining factor \(\|f_n-f\|_p\) tends to zero. This proves the stated \(L^1\) convergence. \(\square\)
A useful consequence follows by applying the absolute-value bound for integrals to the difference: \(\int_X|f_n|^p\,d\mu\to\int_X|f|^p\,d\mu\). This is consistent with the earlier theorem on convergence of norms and \(p\)th moments, but the result just proved also controls the \(L^1\) distance between the power functions themselves.
Worked Problems: Testing the Hypotheses
Worked Example: Pointwise Convergence Need Not Give \(L^p\) Convergence
On the positive integers with counting measure, let \(e_n(k)=1\) if \(k=n\), and \(e_n(k)=0\) otherwise. For each fixed \(k\), \(e_n(k)=0\) whenever \(n>k\); hence \(e_n(k)\to0\) pointwise. But for every \(1\leq p<\infty\),
Thus \(e_n\) does not converge to zero in \(L^p\). It also does not converge to zero in \(L^\infty\), since its essential supremum is one. This example pinpoints a hypothesis that cannot be omitted from a dominated convergence argument: pointwise convergence by itself does not control the size of the \(L^p\) error over the whole space.
Proof Choices and Common Pitfalls
These problems illustrate several checks that make an \(L^p\) proof more reliable. First, identify which exponents are needed before applying Hölder. To place a product in \(L^1\), the two exponents must be conjugate. Second, when a proof uses a bound on \(\|g_n\|_q\), explain why that bound is uniform; convergence in \(L^q\) and the reverse triangle inequality provide it. Third, an algebraic identity for a difference is useful only when each resulting term has an integrable norm, which is exactly what Hölder verifies in the product theorem.
The disjoint-support calculation has a different basis: at almost every point, one summand vanishes, so the \(p\)th power of the sum is the sum of the \(p\)th powers. Without that pointwise condition, the exact formula need not hold. Finally, pointwise convergence and norm convergence are distinct conclusions. The sequence of coordinate indicators has pointwise limit zero while keeping norm one; a proof of norm convergence needs additional information, such as an applicable domination hypothesis.
Check Your Understanding
For each question, identify the estimate or hypothesis that makes the proposed conclusion valid.
- In the product convergence theorem, why must \(\|g_n\|_q\) be bounded, and which earlier norm inequality establishes this?
- When \(f\) and \(g\) have disjoint supports almost everywhere, why does \(|f+g|^p=|f|^p+|g|^p\) hold almost everywhere?
- For \(p>1\), which pair of conjugate exponents is used to estimate the \(L^1\) difference of the \(p\)th powers?
- What feature of the sequence \(e_n\) prevents its pointwise convergence to zero from being convergence in \(L^p\)?
- What additional pointwise or integrability information, beyond \(L^p\) convergence alone, is typically needed to invoke dominated convergence?