Abstract
We introduce and study fractional Sobolev spaces of functions taking their values in a Banach space. Our approach is based on Riemann-Liouville derivative. In this regard, paper is a continuation of the paper [D. Idczak, S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. of Function Spaces and Appl. 2013 (2013), Art. ID 128043, 15 pp.], where real-valued functions are investigated. Moreover, we study the relation between the fractional derivative of an abstract function of one variable and partial fractional derivative of a real-valued function of two variables.
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Notes
It is known that a function \(\mathbf {f}(t)=c+\int _{a}^{t}\mathbf {g}(s)ds\) with \(\mathbf {g}\in L^{p}([a,b],X)\) possesses the classical derivative \(\mathbf {f}^{\prime }\) a.e. on [a, b] and \(\mathbf {f}^{\prime }=\mathbf {g}\) a.e. on [a, b]; if \(\mathbf {g}\ \)is continuous, then \(\mathbf {f}^{\prime }\) exists everywhere on [a, b] and \(\mathbf {f}^{\prime }=\mathbf {g}\) everywhere on [a, b].
We identify the function \(I_{a+}^{n-\alpha }f\) with its representative belonging to \(IR_{a+}^{n,1}.\)
We have the following property:
$$\begin{aligned} D_{a+}^{\beta }I_{a+}^{\beta }f=\frac{d}{dt}I_{a+}^{1-\beta }I_{a+}^{\beta }f=\frac{d}{dt}I_{a+}^{1}f=f \end{aligned}$$for \(f\in L^{1}([a,b],X),\beta \in (0,1).\)
If \(n<\alpha <n+1\), then, for \(i=2,...,n+1\), \(i-1<\alpha -(n+1)+i<i\) and, consequently,
$$\begin{aligned} D_{b-}^{\alpha -((n\!+\!1)-i)}\varphi :=\!(-1)^{i}(\frac{d}{dt})^{i}(I_{b-} ^{n\!+\!1\!-\!\alpha }\varphi )\!=\! (-1)^{i\!-\!1}(\frac{d}{dt})^{i\!-\!1}I_{b-}^{n\!+\!1\!-\!\alpha } D_{b-}^{1}\varphi =\!...\!=I_{b-}^{n\!+\!1\!-\!\alpha }D_{b-}^{i}\varphi \end{aligned}$$The inclusion \(H\subset V^{*}\) is meant in the following way: to each \(h\in H\) there corresponds \(\overline{h}\in V^{*}\) given by
$$\begin{aligned} \left\langle \overline{h},v\right\rangle _{V^{*}\times V}=(h,v)_{H} \end{aligned}$$(4.3)for \(v\in V\). In the other words, we identify H and \(H^{*}\) and use the fact that the restriction to V of an element from \(H^{*}\) belongs to \(V^{*}\). From the reflexivity of V it follows that H is dense in \(V^{*}\).
In [5], the case of \(\Omega =(c,d)\subset \mathbb {R}\) was considered.
To obtain the second equality it is sufficient to observe that functions
$$\begin{aligned}&[a,t]\times \overline{\Omega }\ni (\tau ,s)\longmapsto \frac{u(\tau ,s)}{(t-\tau )^{1-(n-\alpha )}}\in \mathbb {R},\\&[a,t]\times \overline{\Omega }\ni (\tau ,s)\longmapsto \widetilde{u} _{t}(\tau ,s)\in \mathbb {R} \end{aligned}$$are measurable. So, the set of points of their equality is measurable. From (5.13) it follows that all \(\tau \)-sections of this set are of full measure in \(\overline{\Omega }\). It means that this set has the full measure in \([a,t]\times \overline{\Omega }\). In consequence, almost all s-sections have the full measure in [a, t].
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Idczak, D. Riemann-Liouville derivatives of abstract functions and Sobolev spaces. Fract Calc Appl Anal 25, 1260–1293 (2022). https://doi.org/10.1007/s13540-022-00058-8
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DOI: https://doi.org/10.1007/s13540-022-00058-8