arXiv subjects
Publications and source records attributed to Dorothee Frey.
We establish global-in-time Strichartz estimates without loss of derivatives for wave equations with time-dependent Lipschitz coefficients, which satisfy an additional structural assumption. Our approach is based on a parametrix construction through the Phillips functional calculus. We furthermore obtain the well-posedness of such wave equations with Lipschitz coefficients in $H^1$.
We consider magnetic Schr\"odinger equations with sublinear magnetic potentials and subquadratic electric potentials on $\mathbb{R}^{d}$, as well as generalizations thereof. We obtain new results on the global well-posedness of the Cauchy problem with initial data in magnetic modulation spaces $M^{p}_{A}(\mathbb{R}^{d})$. Our results are achieved by approximating the solution in phase space using the magnetic Hamiltonian flow. This method includes the potentials as part of the generalized Schr\"odinger operator instead of treating them as perturbations, and thereby allows us to deal with unbounded potentials. For $A \equiv 0$, the space $M^{p}_{A}(\mathbb{R}^{d})$ reduces to the usual modulation space $M^{p}(\mathbb{R}^{d})$, for which relevant known results for the usual Schr\"odinger equation can be recovered.
Sharp Strichartz estimates are proved for Schrödinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how dispersive properties are inherited from the constant coefficient case. Global Strichartz estimates follow provided that the derivatives of the coefficients are integrable. The estimates extend to structured coefficients of bounded variations. As applications we derive Strichartz estimates with additional derivative loss for wave equations with Hölder-continuous coefficients and solve nonlinear Schrödinger equations. Finally, we record spectral multiplier estimates, which follow from the Strichartz estimates by well-known means.
Peral/Miyachi's celebrated theorem on fixed time $L^{p}$ estimates with loss of derivatives for the wave equation states that the operator $(I-Δ)^{- \fracα{2}}\exp(i \sqrt{-Δ})$ is bounded on $L^{p}(\mathbb{R}^{d})$ if and only if $α\geq s_{p}:=(d-1)|\frac{1}{p}-\frac{1}{2}|$. We extend this result to operators of the form $\mathcal{L} = -\sum \limits _{j=1} ^{d} a_{j+d}\partial_{j}a_{j}\partial_{j}$, such that, for $j=1,...,d$, the functions $a_{j}$ and $a_{j+d}$ only depend on $x_{j}$, are bounded above and below, but are merely Lipschitz continuous. This is below the $C^{1,1}$ regularity that is known to be necessary in general for Strichartz estimates in dimension $d \geq 2$. Our proof is based on an approach to the boundedness of Fourier integral operators recently developed by Hassell, Rozendaal, and the second author. We construct a scale of adapted Hardy spaces on which $\exp(i\sqrt{ \mathcal{L}} )$ is bounded by lifting $L^{p}$ functions to the tent space $T^{p,2}(\mathbb{R}^{d})$, using a wave packet transform adapted to the Lipschitz metric induced by the coefficients $a_j$. The result then follows from Sobolev embedding properties of these spaces.
We consider operators $T$ satisfying a sparse domination property \[ |\langle Tf,g\rangle|\leq c\sum_{Q\in\mathscr{S}}\langle f\rangle_{p_0,Q}\langle g\rangle_{q_0',Q}|Q| \] with averaging exponents $1\leq p_0<q_0\leq\infty$. We prove weighted strong type boundedness for $p_0<p<q_0$ and use new techniques to prove weighted weak type $(p_0,p_0)$ boundedness with quantitative mixed $A_1$-$A_\infty$ estimates, generalizing results of Lerner, Ombrosi, and P\'erez and Hyt\"onen and P\'erez. Even in the case $p_0=1$ we improve upon their results as we do not make use of a H\"ormander condition of the operator $T$. Moreover, we also establish a dual weak type $(q_0',q_0')$ estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.
We consider abstract Sobolev spaces of Bessel-type associated with an operator. In this work, we pursue the study of algebra properties of such functional spaces through the corresponding semigroup. As a follow-up of [4], we show that under the extra property of a "carré du champ identity" , this algebra property holds in a wider range than previously shown.
We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are Riesz transforms / multipliers or paraproducts associated with a second order elliptic operator. It also applies to such operators whose unweighted continuity is restricted to Lebesgue spaces with certain ranges of exponents $(p_0,q_0)$ where $1\le p_0<2<q_0\le \infty$. The norm estimates obtained are powers $α$ of the characteristic used by Auscher and Martell. The critical exponent in this case is $\mathfrak{p}=1+\frac{p_0}{q'_0}$. We prove $α=\frac1{p-p_0}$ when $p_0<p\le \mathfrak{p}$ and $α= \frac{q_0-1}{q_0-p}$ when $\mathfrak{p}\le p<q_0$. In particular, we are able to obtain the sharp $A_2$ estimates for non-integral singular operators which do not fit into the class of Calderón-Zygmund operators. These results are new even in the Euclidean space and are the first ones for operators whose kernel does not satisfy any regularity estimate.
On a doubling metric measure space $(M,d,μ)$ endowed with a "carré du champ", let $\mathcal{L}$ be the associated Markov generator and $\dot L^{p}_α(M,\mathcal{L},μ)$ the corresponding homogeneous Sobolev space of order $0<α<1$ in $L^p$, $1 0}$ for the spaces $\dot L^{p}_α(M,\mathcal{L},μ) \cap L^\infty(M,μ)$ to be algebras for the pointwise product. Two approaches are developed, one using paraproducts (relying on extrapolation to prove their boundedness) and a second one through geometrical square functionals (relying on sharp estimates involving oscillations). A chain rule and a paralinearisation result are also given. In comparison with previous results ([29,11]), the main improvements consist in the fact that we neither require any Poincaré inequalities nor $L^p$-boundedness of Riesz transforms, but only $L^p$-boundedness of the gradient of the semigroup. As a consequence, in the range $p\in(1,2]$, the Sobolev algebra property is shown under Gaussian upper estimates of the heat kernel only.
We develop a strategy making extensive use of tent spaces to study parabolic equa-tions with quadratic nonlinearities as for the Navier-Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in \R^n with small initial data in BMO^{-1}(\R^n). We then study another model where neither pointwise kernel bounds nor self-adjointness are available.
We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $ω$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \in [2,p)$. This improves results of \cite{ACDH} and \cite{AC}, where the stronger assumption of a Poincaré inequality and the assumption $e^{-tL}(1)=1$ were made. The Poincaré inequality assumption is also weakened in the setting of a sectorial operator $L$. In the last section, we study elliptic perturbations of Riesz transforms.
On a doubling metric measure space endowed with a "carré du champ", we consider $L^p$ estimates $(G_p)$ of the gradient of the heat semigroup and scale-invariant $L^p$ Poincaré inequalities $(P_p)$. We show that the combination of $(G_p)$ and $(P_p)$ for $p\ge 2$ always implies two-sided Gaussian heat kernel bounds. The case $p=2$ is a famous theorem of Saloff-Coste, of which we give a shorter proof, without parabolic Moser iteration. We also give a more direct proof of the main result in \cite{HS}. This relies in particular on a new notion of $L^p$ Hölder regularity for a semigroup and on a characterization of $(P_2)$ in terms of harmonic functions.
Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in $L^2$ spaces, and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of $L^p$ spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator, to prove that they have a bounded holomorphic functional calculus in those $L^p$ spaces. We also obtain functional calculi results for restrictions to certain subspaces, for a larger range of $p$. This provides a framework for obtaining $L^p$ results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator $L$ with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and $L^p$ bounds on the square-root of $L$ by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in $L^2$ extends to $L^p$ for all $p \in (1,\infty)$, while the restrictions in $p$ come from the operator-theoretic part of the $L^2$ proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces, and about the relationship between conical and vertical square functions.
We give a new proof of a well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in $\R^n$ with small initial data in $BMO^{-1}(\R^n)$. The proof is formulated operator theoretically and does not make use of self-adjointness of the Laplacian.
Related to a semigroup of operators on a metric measure space, we define and study pseudodifferential operators (including the setting of Riemannian manifold, fractals, graphs ...). Boundedness on $L^p$ for pseudodifferential operators of order 0 are proved. Mainly, we focus on symbols belonging to the class $S^0_{1,δ}$ for $δ\in[0,1)$. For the limit class $S^0_{1,1}$, we describe some results by restricting our attention to the case of a sub-Laplacian operator on a Riemannian manifold.
Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. Associated to $L$ are certain approximations of the identity. We call an operator $T$ a non-integral operator if compositions involving $T$ and these approximations satisfy certain weighted norm estimates. The Davies-Gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on $T$ in Calderón-Zygmund theory. In this paper, we show, under the additional assumption that a vertical Littlewood-Paley-Stein square function associated to $L$ is bounded on $L^2(X)$, that a non-integral operator $T$ is bounded on $L^2(X)$ if and only if $T(1) \in BMO_L(X)$ and $T^{\ast}(1) \in BMO_{L^{\ast}}(X)$. Here, $BMO_L(X)$ and $BMO_{L^{\ast}}(X)$ denote the recently defined $BMO(X)$ spaces associated to $L$ that generalize the space $BMO(X)$ of John and Nirenberg. Generalizing a recent result due to F. Bernicot, we show a second version of a T(1)-Theorem under weaker off-diagonal estimates, which gives a positive answer to a question raised by him. As an application, we prove $L^2(X)$-boundedness of a paraproduct operator associated to $L$. We moreover study criterions for a $T(b)$-Theorem to be valid.
Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated to $L$. We show various boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO spaces $H^p_L(X)$ and $BMO_L(X)$. In generalization of standard paraproducts constructed via convolution operators, we show $L^2(X)$ off-diagonal estimates as a substitute for Calderón-Zygmund kernel estimates. As an application, we study differentiability properties of paraproducts in terms of fractional powers of the operator $L$. The results of this paper are fundamental for the proof of a T(1)-Theorem for operators beyond Calderón-Zygmund theory, which will be the subject of a forthcoming paper.