Let \(\varphi:\mathbb R\to\mathbb R\) be continuous with compact support.
If \(1\le p\le q\le\infty\), prove that there is a constant \(A\) such that \[ \|f*\varphi\|_q\le A\|f\|_p \qquad(f\in L^p). \]
Show by example that no such general estimate can hold when \(p>q\).
Solution
1. Endpoint bounds.
For \(1\le p<\infty\), Minkowski’s integral inequality gives \[ \begin{aligned} \|f*\varphi\|_p &=\left\|\int_{\mathbb R}\varphi(y)f(\cdot-y)\,dy\right\|_p\\ &\le\int_{\mathbb R}|\varphi(y)|\,\|f(\cdot-y)\|_p\,dy\\ &=\|\varphi\|_1\|f\|_p. \end{aligned} \] Also, by Hölder's inequality, if \(p'\) is conjugate to \(p\), \[ |(f*\varphi)(x)|\le\|f\|_p\,\|\varphi(x-\cdot)\|_{p'}=\|f\|_p\|\varphi\|_{p'}, \] so \[ \|f*\varphi\|_\infty\le\|\varphi\|_{p'}\|f\|_p. \] For \(p=\infty\), necessarily \(q=\infty\), and directly \[ \|f*\varphi\|_\infty\le\|\varphi\|_1\|f\|_\infty. \]
2. Interpolate the output norm for \(p\le q\le\infty\).
1. This proves part (a).
3. Failure when \(p>q\).
Choose a nonzero compactly supported continuous kernel \(\varphi\). Let \[ \psi(x)=\overline{\varphi(-x)}, \] so \(\psi\in C_c\) and \[ (\psi*\varphi)(0)=\int_{\mathbb R}|\varphi(y)|^2\,dy>0. \] Thus \(h:=\psi*\varphi\) is not identically zero.
Choose \(L>0\) so large that the translates \(\psi(\cdot-kL)\) have pairwise disjoint supports, and likewise the translates \(h(\cdot-kL)\) have pairwise disjoint supports. Define \[ f_N=\sum_{k=1}^N\psi(\cdot-kL). \] Then \[ \|f_N\|_p=N^{1/p}\|\psi\|_p, \] while \[ f_N*\varphi=\sum_{k=1}^N h(\cdot-kL) \] and therefore \[ \|f_N*\varphi\|_q=N^{1/q}\|h\|_q. \] If an estimate \(\|f*\varphi\|_q\le A\|f\|_p\) held with \(p>q\), then \[ N^{1/q-1/p}\le A\frac{\|\psi\|_p}{\|h\|_q} \] for every \(N\), impossible because \(1/q-1/p>0\). Hence no such general estimate can hold for \(p>q\).
4. Q.E.D.
Steps <1>1–<1>2 prove part (a), and step 3 proves part (b).