Ha Huy Bang, Vu Nhat Huy
abstract:
Let $\mathbb{T}=[-\pi,\pi]$, $1\leq p\leq \infty$ and ${Q}(x)$ be a polynomial.
In this paper, we introduce the notion called $Q$-primitives of a function in
$\mathcal{S}'(\mathbb{R})$ and apply it to examine the existence and uniqueness
of solutions in $L^p(\mathbb{T})$ of the non-homogeneous equation ${Q}(D)f=\psi
\in L^p(\mathbb{T})$. The explicit solutions of the equation are given. In
particular, we show that the condition ${Q}(x) \ne 0$
$\forall\,x\in\supp\widehat{\psi}$ is the criterion for the existence of a
$Q$-primitive in $L^p(\mathbb{T})$ of $f$. Note that every $Q$-primitive in
$L^p(\mathbb{T})$ of $f$ is a solution of the equation ${Q}(D)f=\psi$. Moreover,
an inequality for higher order $Q$-primitives is also given.
Mathematics Subject Classification: 26D10, 42A38, 46E30, 34A05
Key words and phrases: $L^p$-spaces, explicit solutions, periodic functions, Fourier transform