**Ha Huy Bang, Vu Nhat Huy**

##
*Q*-Primitives and Explicit Solutions of Non-Homogeneous Equations with
Constant Coefficients in L^{p}(T)

**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