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Fall 2015 - Problem 1

Lp spaces

Let $\set{g_n}_n$ be a sequence of measurable functions on $\R^d$ , such that $\abs{g_n\p{x}} \leq 1$ for all $x$ , and assume that $g_n \to 0$ almost everywhere. Let $f \in L^1\p{\R^d}$ . Show that the sequence

\p{f * g_n}\p{x} = \int f\p{x - y}g_n\p{y} \,\diff{y} \to 0

uniformly on each compact subset of $\R^d$ , as $n \to \infty$ .

Solution.

Let $K$ be a compact set and $\epsilon > 0$ . Since $f \in L^1\p{\R^d}$ , there exists $h \in C_0\p{\R^d}$ such that $\norm{f - h}_{L^1} < \epsilon$ by density and regularity of the Lebesgue measure. Then if $E$ denotes the support of $h$ and $M$ the maximum of $H$ , we have by Hölder's inequality for any $x \in K$

\begin{aligned} \abs{\int f\p{x - y}g_n\p{y} \,\diff{y}} &\leq \abs{\int h\p{x - y}g_n\p{y} \,\diff{y}} + \int \abs{f\p{x - y} - h\p{x - y}}\abs{g_n\p{y}} \,\diff{y} \\ &\leq \abs{\int_{x + E} h\p{x - y} g_n\p{y} \,\diff{y}} + \int \abs{f\p{x - y} - h\p{x - y}} \,\diff{y} \\ &\leq M \int_{x + E} \abs{g_n\p{y}} \,\diff{y} + \norm{f - h}_{L^1}. \end{aligned}

Observe that because $K$ and $E$ are compact, there exists $R > 0$ such that $K + E \subseteq B\p{0, R}$ , e.g., $R > \sup_K \abs{x} + \sup_E \abs{x}$ . Hence, by dominated convergence ( $1 \in L^1\p{B\p{0,R}}$ ), there exists $N \in \N$ such that

\int_{B\p{0,R}} \abs{g_n\p{y}} \,\diff{y} < \epsilon

for all $n \geq N$ . Thus,

\abs{\p{f * g_n}\p{x}} \leq M\epsilon + \epsilon,

independent of $x$ , which completes the proof.