Function spaces related to continuous negative definite functions: \(\psi\)-Bessel potential spaces (Q2760590)

This long paper is an excellent study of Bessel potential spaces associated with a real-valued continuous negative definite function. The historical background is well presented, and a wide bibliography is given. The starting point of the present paper is the introduction by \textit{A. Beurling} and \textit{J. Deny} of Dirichlet spaces [Acta Math. 99, 203-224 (1958; Zbl 0089.08106), Proc. Nat. Acad. Sci. USA 45, 208-215 (1959; Zbl 0089.08201)]. These Dirichlet spaces are built upon \(L^{2}\). Here, the authors develop the corresponding \(L^p\)-theory. NEWLINENEWLINENEWLINELet \(\psi: \mathbb R^n\rightarrow \mathbb R\) be a continuous negative definite function, which means that \(\psi(0)=0\) and that, for all \(t>0\), the function \(\xi\mapsto e^{-t\psi(\xi)}\) is positive definite, in the sense that, for all \(k\in \mathbb N\) and all \(\xi_{1},\ldots,\xi_{k}\in \mathbb R^n\), the matrix \((e^{-t\psi(\xi^j-\xi^l)})_{1\leq j,l\leq k}\) is positive hermitian. Such a function looks like temperate weight functions considered by \textit{L. Hörmander} in [``Linear partial differential operators'', Springer-Verlag (1963; Zbl 0108.09301)]. One associates to \(\psi\) a convolution semigroup \((\mu_{t})_{t\geq 0}\) of sub-probability measures on \(\mathbb R^n\) by NEWLINE\[NEWLINE \widehat{\mu_{t}}(\xi)=(2\pi)^{-n/2}e^{-t\psi(\xi)}.NEWLINE\]NEWLINE Let \((T_{t}^{(p)})_{t\geq 0}\) be the submarkovian semigroup on \(L^p(\mathbb R^n)\) (\(1\leq p<+\infty\)) associated with \((\mu_{t})_{t\geq 0}\) and \((A^{(p)})_{t\geq 0}\) its infinitesimal generator with domain \(D(A^{(p)})\). NEWLINENEWLINENEWLINEFor all \(s>0\), the space \(H_{p}^{\psi,s}\) is defined by NEWLINE\[NEWLINE H_{p}^{\psi,s}=(I-A^{(p)})^{-s/2}(L^p). NEWLINE\]NEWLINE It coincides with the domain of \((I-A^{(p)})^{s/2}\) in \(L^p\). If \(u\in H_{p}^{\psi,s}\), define its norm by NEWLINE\[NEWLINE \left\|u\right\|_{H^p_{\psi,s}} = \left\|(I-A^{(p)})^{s/2}u\right\|_{p}. NEWLINE\]NEWLINE Notice that, when \(\psi(\xi)=|\xi|^{s/2}\), \(H_{p}^{\psi,s}\) coincides with the usual Bessel potential space \(H_{p}^s\) [introduced by \textit{N. Aronszajn} and \textit{K. T. Smith} in Ann. Inst. Fourier 11, 385-475 (1961; Zbl 0102.32401)]. NEWLINENEWLINENEWLINEBessel potential spaces \(H_{p}^{\psi,s}\) for \(s<0\) are also introduced, and it is shown that the dual of \(H_{p}^{\psi,s}\) is \(H_{p'}^{\psi,-s}\). NEWLINENEWLINENEWLINEEmbedding results are also obtained, generalizing well-known statements for classical Bessel potential spaces. In particular, it is shown that \(H_{p}^{\psi,s} \hookrightarrow C_{\infty}\) if, and only if, \((1+\psi)^{-s/2}\) is a Fourier multiplier of type \((p,\infty)\). NEWLINENEWLINENEWLINEComplex interpolation for \(H_{p}^{\psi,s}\) is investigated. As in the usual case, it is proved that NEWLINE\[NEWLINE[H_{p_{0}}^{\psi,s_{0}},H_{p_{1}}^{\psi,s_{1}}]_{\theta}=H_{p}^{\psi,s}NEWLINE\]NEWLINE with NEWLINE\[NEWLINE0<\theta<1,\quad 1<p_{0},p_{1}<+\infty,\quad (1-\theta)/p_{0} + \theta/p_{1}=1/pNEWLINE\]NEWLINE and \(s=(1-\theta)s_{0}+\theta s_{1}\). NEWLINENEWLINENEWLINEFinally, a notion of capacity associated with \(H_{p}^{\psi,s}\) is introduced. For each function \(u\in H_{p}^{\psi,s}\), there exists a function \(\widetilde{u} \in H_{p}^{\psi,s}\) which is quasi-continuous (which means that, for all \(\varepsilon>0\), there exists \(G\subset \mathbb R^n\) with capacity \(<\varepsilon\) and \(\widetilde{u}\) is continuous outside \(G\)) and coincides with \(u\) almost everywhere. Actually, this fact follows from a result by \textit{M. Fukushima} [Lect. Notes Math. 1563, 21-53 (1993; Zbl 0810.47035)].