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arXiv 1607.04797math.CAmath.FA
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Sturm-Liouville方程的容许空间

Admissible Spaces for the Sturm-Liouville Equation

N. A. Chernyavskaya, L. A. Shuster

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摘要：

我们考虑方程\begin{equation} -y''(x)+q(x)y(x)=f(x),\quad x\in \mathbb R \end{equation}其中$ f \in L_p^{loc}(\mathbb R),$ $p \in [1,\infty) $和$ 0 < q \in L_1^{loc}(\mathbb R).$通过这个方程的解，我们意味着任何函数$ y,$绝对连续连同它的导数和满足方程几乎处处在$ \mathbb R.$让正和连续的函数$ μ(x) $和$ θ(x) $为$ x \in \mathbb R $给出。引入空间$ L_p(\mathbb R,μ) = \{f \in L_p^{loc}(\mathbb R): ||f||_{L_p(\mathbb R,μ)}^p =\int_{-\infty}^\infty|μ(x)f(x)|^p dx < \infty\}, $ $ L_p(\mathbb R,θ) = \{f\in L_p^{loc}(\mathbb R):||f||_{L_p(\mathbb R,θ)}^p = \int_{-\infty}^\infty|θ(x)f(x)|^p dx < \infty\}. $本文给出了函数$μ,θ$和$q$的条件：1）对于每个函数$f \in L_p(\mathbb R,θ) $，方程$y \in L_p(\mathbb R,μ)$存在唯一解; 2）存在一个绝对常数$ c(p) \in (0,\infty) $，使得无论选择何种函数$ f \in L_p(\mathbb R,θ) $，方程的解满足不等式$$ \|y\|_{L_p(\mathbb R,μ)} < c(p)\|f\|_{L_p(\mathbb R,θ)}. $$

英文摘要：

We consider the equation \begin{equation} -y''(x)+q(x)y(x)=f(x),\quad x\in \mathbb R \end{equation} where $ f \in L_p^{loc}(\mathbb R),$ $p \in [1,\infty) $ and $ 0 < q \in L_1^{loc}(\mathbb R).$ By a solution of this equation we mean any function $ y,$ absolutely continuous together with its derivative and satisfying the equation almost everywhere in $ \mathbb R.$ Let positive and continuous functions $ μ(x) $ and $ θ(x) $ for $ x \in \mathbb R $ be given. Let us introduce the spaces $ L_p(\mathbb R,μ) = \{f \in L_p^{loc}(\mathbb R): ||f||_{L_p(\mathbb R,μ)}^p =\int_{-\infty}^\infty|μ(x)f(x)|^p dx < \infty\}, $ $ L_p(\mathbb R,θ) = \{f\in L_p^{loc}(\mathbb R):||f||_{L_p(\mathbb R,θ)}^p = \int_{-\infty}^\infty|θ(x)f(x)|^p dx < \infty\}. $ In the present paper, we obtain requirements to the functions $μ,θ$ and $q$ under which 1) for every function $f \in L_p(\mathbb R,θ) $ there exists a unique solution of the equation $y \in L_p(\mathbb R,μ)$ ; 2) there is an absolute constant $ c(p) \in (0,\infty) $ such that regardless of he choice of a function $ f \in L_p(\mathbb R,θ) $ the solution of the equation satisfies the inequality $$ \|y\|_{L_p(\mathbb R,μ)} < c(p)\|f\|_{L_p(\mathbb R,θ)}. $$

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