Functional calculus on non-homogeneous operators on nilpotent groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator $$\mathcal {L}$$ L on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator $$\mathcal {L}_0$$ L 0 on the ‘local’ contraction $$G_0$$ G 0 of G, as well as of the corresponding Rockland operator $$\mathcal {L}_\infty $$ L ∞ on the ‘global’ contraction $$G_\infty $$ G ∞ of G. We provide asymptotic estimates of the Riesz potentials associated with $$\mathcal {L}$$ L at 0 and at $$\infty $$ ∞ , as well as of the kernels associated with functions of $$\mathcal {L}$$ L satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for $$\mathcal {L}$$ L which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with $$\mathcal {L}$$ L from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

Smoothness of functions versus smoothness of approximation processes

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.

The Holomorphic Hörmander Functional Calculus

TThe topic of this thesis is functional calculus in connection with abstract multiplier theorems. In 1960, Hörmander showed how the uniform boundedness of certain integral means of a function m in L ∞ (R^d) and its weak derivatives imply that m yields a bounded Lp -Fourier multiplier. Nowadays, this is known as the Hörmander multiplier theorem, sometimes Hörmander--Mikhlin multiplier theorem. A noteworthy detail is that a radial function m(|x|) satisfies Hörmander's condition if and only if m (|x|²) does. Hence, Hörmander's theorem is also a result on the functional calculus of the negative Laplacian -Δ. Hörmander's result has inspired a lot of research, and authors have also proven similar results for other operators such as certain Schrödinger operators, Sublaplacians on Lie groups, and later certain differential operators on spaces of homogeneous type. For us, the work of Kriegler and Weis is of particular interest. Starting with the PhD thesis of Kriegler in 2009, they showed how abstract multiplier theorems can be proven in a more general context. Namely, considering a certain class of 0-sectorial and 0-strip type operators on a general Banach space, one can construct an abstract Hörmander functional calculus based on the classical holomorphic calculus. Then, by using probalistic techniques from Banach space geometry involving so-called R-boundedness one can derive multiplier results in this generalized setting. In 2001, García-Cuerva, Mauceri, Meda, Sjögren, and Torrea proved an abstract multiplier theorem for generators of symmetric contraction semigroups, where a bounded Hörmander calculus is inferred from growth conditions on the imaginary powers of the generator. As the considered operators need not be 0-sectorial, this result is not covered by the methods of Kriegler and Weis. However, the result is based on Meda's earlier work, where he derived a bounded Hörmander if the given imaginary powers only grow polynomially fast. In this case, the operator is 0-sectorial, and Kriegler and Weis were able to recover the result while improving the order of the calculus. In this thesis, we introduce a generalized class of Hörmander functions defined on strips and sectors. Based on this and the classical holomorphic calculus, we construct a holomorphic Hörmander calculus for a class of operators which may also have strip type or angle of sectoriality greater than zero. The main result is a generalization of the multiplier theorem of García-Cuerva et al. to Banach spaces of finite cotype and Banach spaces with Pisier's property (α), where we retain and even improve the order given by Kriegler and Weis for the 0-sectorial case.

Well-posedness of fractional degenerate differential equations in Banach spaces

We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.