Strichartz Estimates for Non-selfadjoint Operators and Applications

Abstract

We prove Strichartz type estimates for non-selfadjoint pseudo-differential operators of order m with a curvature condition on the characteristic set. Local solvability results for the corresponding equations with L ^n/m potential (where n > m stands for the number of variables) and for the corresponding semi-linear equations with a non-linearity of order m/n ≤ α ≤ 2m/(n ? m) follow. This is achieved using a parametrix construction and a Brenner inequality for Fourier integral operators with complex phase.     

Decay of global solutions of two nonlinear evolution equations

1.IntroductionTherehavebeenconsiderableliteratuxeonthedecayofsolutionstothebestialvalueproblemsforsomenonlinearevolutionequations[3,4,6,7,161.Undercertainassumptions,LZdecayandLoodecayofsolutionstotheseproblemswereestablished.Thereadersinterestedcanfindsuchworksinourreferences.OurillterestisfocusedonthedecayofsolutionsoftheinitialvalueproblemsfornonlinearBenjamin--OnthBurgers(BOB)l"'19--21]andSchlodinger-Burgers(SB)equationwhereHisHilberttransform,definedbyWewallttoshowthattheLZandLoon…     

Uniform and optimal estimates for solutions to singularly perturbed parabolic equations

The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L²-estimates as well as certain decay properties of bounded classical solutions to it are established. These do not depend on the value of the small perturbation parameter, and allow to prove global in time existence of strong solutions to certain boundary-value problems for ultraparabolic equations with unbounded coefficients. Optimal decay estimates are proved for such solutions. All results concerning ultraparabolic equations apply, in particular, to the Kolmogorov equation for diffusion with inertia, to the (linear) Fokker-Planck equation, to the linearized Boltzmann equation, and to some nonlinear integro-differential ultraparabolic equations of the Fokker-Planck type, arising from biophysics. Optimal decay estimates are derived for global in time strong solutions to such equations.     

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

On quadratic hypersurfaces in $\mathbb {H}^2$, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □_b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in $\mathbb {H}^2$ cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.     

Global existence and asymptotic behavior for a coupled hyperbolic system with localized damping

In this paper we investigate the global existence and energy decay rate for the solution of a coupled hyperbolic system. The semi-explicit energy decay rate is established by using piecewise multiplier techniques and weighted integral inequality. We extend the energy decay result in Alabau-Boussouira [F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim. 51 (2005) 61-105] for a single equation to the coupled hyperbolic system.     

Oscillatory integral operators with homogeneous polynomial phases in several variables

We obtain L² decay estimates in λ for oscillatory integral operators Tλ whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decay rates obtained are optimal in the case of (2+2)-dimensions for any m, while in higher dimensions the result is sharp for m sufficiently large. The proof for large m follows from essentially algebraic considerations. For cubics in (2+2)-dimensions, the proof involves decomposing the operator near the conic zero variety of the determinant of the Hessian of the phase function, using an elaboration of the general approach of Phong and Stein [D.H. Phong, E.M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. (2) 140 (1994) 703-722].     

Estimates for oscillatory strongly singular integral operators

In this paper we consider oscillatory integral operators with strong singularities on . We obtain sharp decay estimates for L² operator norm. We also obtain H^p estimates for difference of oscillatory strongly singular integral operators and strongly singular integral operators, which gives L^p mapping properties of oscillatory strongly singular integral operators.     

On the global boundedness of Fourier integral operators

We consider a class of Fourier integral operators, globally defined on \mathbb R^d{{\mathbb R}^{d}} , with symbols and phases satisfying product type estimates (the so-called SG or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces M ^p . The minimal loss of derivatives is shown to be d|1/2 − 1/p|. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on L ^p spaces are presented.     

Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

For second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L²-based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue.     

OPTIMAL DECAY RATE OF THE COMPRESSIBLE QUANTUM NAVIER-STOKES EQUATIONS

For quantum fluids governed by the compressible quantum Navier-Stokes equations in $\Bbb R^3$ with viscosity and heat conduction, we prove the optimal $L^p-L^q$ decay rates for the classical solutions near constant states. The proof is based on the detailed linearized decay estimates by Fourier analysis of the operators, which is drastically different from the case when quantum effects are absent.     

Changes of variables in modulation and Wiener amalgam spaces

In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of such spaces and, as a consequence, we obtain several versions of local and global Beurling–Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on ${\mathscr F}L^q$. Finally, counterparts of these results are discussed for spaces on the torus.     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in where the linear term is given by Schr?dinger operators H = − Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|². Dedicated to Professor Jun Uchiyama on the occasion of his sixtieth birthday Received: May 4, 2004     

Quantization of Pseudo-differential Operators on the Torus

Pseudo-differential and Fourier series operators on the torus \mathbbTⁿ=(\BbbR/2p\BbbZ)ⁿ{{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n} are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on L ² under certain conditions on their phases and amplitudes.     

A weighted L q -approach to Stokes flow around a rotating body

Considering time-periodic Stokes flow around a rotating body in or we prove weighted a priori estimates in L ^q -spaces for the whole space problem. After a time-dependent change of coordinates the problem is reduced to a stationary Stokes equation with the additional term in the equation of momentum, where ω denotes the angular velocity. In cylindrical coordinates attached to the rotating body we allow for Muckenhoupt weights which may be anisotropic or even depend on the angular variable and prove weighted L ^q -estimates using the weighted theory of Littlewood-Paley decomposition and of maximal operators. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503, by the Grant Agency of the Academy of Sciences No. IAA100190505, and by the joint research project of DAAD (D/04/25763) and the Academy of Sciences of the Czech Republic (D-CZ 3/05-06).     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L² of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [10], [5], [4], and [14].     

SUPERCRITICAL SEMILINEAR WAVE EQUATION WITH NON-NEGATIVE POTENTIAL

We prove a weighted L ^∞ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.     

Parabolic boundary integral operators symbolic representation and basic properties

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ⁿ⁺¹× ⁿ⁺¹, and then develop a periodisation procedure for the calculus of symbols on the cylinder ×. We show Gårding's inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.     

Hardy Spaces and bmo on Manifolds with Bounded Geometry

We develop the theory of the “local” Hardy space and John-Nirenberg space when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include – duality, L ^p estimates on an appropriate variant of the sharp maximal function, and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on L ^p -Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.      

Weighted norm inequalities for convolution operators and links with the Weiss Conjecture

We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section 3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section 4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture (for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness results for certain convolution operators.