Behavior in \(L^\infty\) of convolution transforms with dilated kernels (Q6167847)

In this paper, one considers the convolution of a bounded function with the dilations of a kernel in $L^1(\mathbb{R})$. For $K(x)$ in $L^1(\mathbb{R})$, $t > 0$, and $K_t(x) = 1/t K(x/t)$ take the convolution $K_t\ast f(x)$ of $K_t$ with a function $f$ in $L^p$, $1 \leq p < \infty$. The behavior of this convolution when $t$ tends to $0$, with respect to the $L^p$ norm, is well known and one has for $f$ in $L^p$, \[ \lim K_t\ast f(x) = K^0f(x),\text{ in }L^p(\mathbb{R}),\] when $t$ tends to 0, \(1 \leq p < \infty\). With additional conditions on $K(x)$ one can conclude that the above límit is also valid pointwise almost everywhere, when $1 \leq p \leq \infty$. The behavior of the above convolution when $t$ tends to infinity has been studied for $1 \leq p < \infty$; when $p > 1$ it is known that \[ \lim K_t\ast f(x) = 0,\text{ in }L^p,\text{ when $t$ tends to } \infty . \] The case $p = \infty$ and $t$ tending to infinity is considered here. The main result is the following one: If $f(x)$ is in $L^\infty$ then, as $t$ tends to $\infty$, $K_t\ast f(x)$ either \begin{itemize} \item[(i)] converges uniformly for $x$ on compact subsets of $\mathbb{R}$ to a constant $c$ independent of $x$, or \item[(ii)] fails to converge for every $x$ in $\mathbb{R}$. \end{itemize} As examples of functions $f$ such that the limit of $K_t\ast f$ converges as $t$ tends to $\infty$ for every $K$ in $L^1$, one has the case that $\lim f(x) = 0$ when $x$ tends to $\infty$, or the case that the Fourier transform of $f$ is integrable in a neighborhood of the origin. In both cases, one can conclude that \[ \lim K_t\ast f(x) = 0,\text{ uniformly, when $t$ tends to }\infty. \] The above results, for functions of one real variable, can be extended with minor modifications to functions of several variables. With appropriate $K$ and $f$, the convolution $K_t\ast f$ represents the solution of some differential equations in the half-space $\mathbb{R}_+^{n+1}$, for instance of the heat equation. The results given here can be used to describe the behavior of bounded solutions to the initial value problem for these equations.