Four Jacobi settings are considered in the context of Hardy’s inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy’s inequality is proved for the type parameters ${\alpha ,\beta \in (-1,\infty )}^d$, whereas in the function systems for ${\alpha ,\beta \in [-1∕2,\infty )}^d$. The ranges of these parameters are the widest in which the corresponding orthonormal bases are composed of bounded functions. Moreover, the sharp $L^1$-analogues of Hardy’s inequality are obtained with the same restrictions on the parameters $\alpha$ and $\beta$.

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Sharp Hardy对于Jacobi和对称Jacobi三角展开式的不等式

在Hardy不等式的上下文中考虑了四个Jacobi设置：三角多项式和函数，以及相应的对称系统。在多项式情况下，针对类型参数证明了尖锐的Hardy不等式${\alpha ，\beta \in （-\mathrm{1个}，\infty ）}^d$，而在功能系统中 ${\alpha ，\beta \in [-\mathrm{1个}∕2，\infty ）}^d$。这些参数的范围是最宽的，其中相应的正交基数由有界函数组成。而且，犀利${\mathrm{大号}}^{\mathrm{1个}}$-在参数相同的限制下获得哈代不等式的模拟 $\alpha$ 和 $\beta$。