Ha Huy Bang, Vu Nhat Huy

Q-Primitives and Explicit Solutions of Non-Homogeneous Equations with Constant Coefficients in Lp(T)

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