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Fractional order Hardy-type inequality in fractional \(h\)-discrete calculus. (English) Zbl 1421.39004

Let \(1<p<+\infty\), \(0<\alpha<1\), and \(f(x):=D_hF(x)\). The author deduces a Hardy-type inequality of the following form: \[ \left[\int_0^{\infty}\int_0^{\infty}\frac{\big|F(x)-F(y)\big|^p}{\left[\big(|x-y|+3h\big)_h^{\left(\frac{1}{p}+\alpha\right)}\right]^p}\ d_hx\ d_hy\right]^{\frac{1}{p}}\le C\left[\int_0^{\infty}\frac{\big|f(x)\big|^p}{\left[(x+h)_h^{(\alpha-1)}\right]^p}\ d_hx\right]^{\frac{1}{p}}. \] The constant in the above inequality is defined by \[ C=\frac{2^{\frac{1}{p}}\alpha^{-1}}{(p-p\alpha)^{\frac{1}{p}}}. \]
The author makes use of the \(h\)-difference, which is a generalization of the classical forward difference. In particular, letting \(\delta_h(t)=t+h\) for some \(h>0\), one defines \[ D_hf(t)=\frac{f\big(\delta_h(t)\big)-f(t)}{h}, \] for each \(t\in\{a,a+h,a+2h,\dots\}\). Then one may define a falling factorial sort of function within this \(h\)-difference calculus. This is the function \(t_h^{(\alpha)}\), for \(t\), \(\alpha\in\mathbb{R}\), defined by \[ t_h^{(\alpha)}=h^{\alpha}\frac{\Gamma\left(\frac{t}{h}+1\right)}{\Gamma\left(\frac{t}{h}+1-\alpha\right)}. \]
The paper is clearly written and relatively easy to follow.

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
26A33 Fractional derivatives and integrals
39A12 Discrete version of topics in analysis
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
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