Question 1.1. Suppose that \(f _ { j } \in L ^ { 2 } ( \mathbb { R } ^ { d } ) , j = 1 , 2 , . . . ,\) and \(f \in L ^ { 2 } ( { \mathbb { R } } ^ { d } )\) satisfy
\[ \operatorname* { l i m } _ { j \to \infty } \int _ { \mathbb { R } ^ { d } } f _ { j } g = \int _ { \mathbb { R } ^ { d } } f g \]
for all \(g \in L ^ { 2 } ( { \mathbb { R } } ^ { d } )\) . That is, \(f _ { j }\) converges to f weakly in \(L ^ { 2 } .\) . Suppose that the sequence satisfies the uniform bound
\[ \operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } ( 1 + | x | ) ^ { d } | f _ { j } ( x ) | \leq M < \infty .\tag{A} \]
Determine whether condition (A), together with weak convergence, implies \[ \|f_j\|_2\to\|f\|_2 \qquad\text{and hence}\qquad \|f_j-f\|_2\to0. \] If not, give a counterexample.
Solution
1. Construct a sequence satisfying the weighted pointwise bound.
Let \[ Q=[0,1]^d, \qquad f_j(x)=\mathbf1_Q(x)\sin(2\pi jx_1), \qquad f=0. \] Then \(f_j\in L^2(\mathbb R^d)\) and, on \(Q\), \[ (1+|x|)^d|f_j(x)|\le (1+\sqrt d)^d. \] Outside \(Q\), \(f_j=0\). Hence condition (A) holds with \[ M=(1+\sqrt d)^d. \]
2. Prove weak convergence to zero.
Fix \(g\in L^2(\mathbb R^d)\). Since \(Q\) has finite measure, \[ g\mathbf1_Q\in L^1(Q) \] by Cauchy–Schwarz. Fubini gives an \(L^1(0,1)\) function \[ h(t):=\int_{[0,1]^{d-1}}g(t,x')\,dx'. \] Therefore \[ \int_{\mathbb R^d}f_j(x)g(x)\,dx =\int_0^1 h(t)\sin(2\pi jt)\,dt. \] By the Riemann–Lebesgue lemma, the right-hand side tends to \(0\). Thus \[ f_j\rightharpoonup0 \quad\text{in }L^2(\mathbb R^d). \]
3. Show that strong convergence fails.
For every positive integer \(j\), \[ \begin{aligned} \|f_j\|_2^2 &=\int_Q\sin^2(2\pi jx_1)\,dx\\ &=\int_0^1\sin^2(2\pi jt)\,dt\\ &=\frac12. \end{aligned} \] Hence \[ \|f_j\|_2=2^{-1/2} \] for every \(j\), while \(\|f\|_2=0\). Consequently neither \[ \|f_j\|_2\to\|f\|_2 \] nor \[ \|f_j-f\|_2\to0 \] holds.
Thus condition (A) together with weak convergence does not imply strong convergence: the sequence \(f_j\) satisfies the weighted pointwise bound (A) and oscillates at frequency \(j\) on \(Q\).