Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.

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Nonlinear dispersive equations : local and global analysis

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

The essential norm of a composition operator

The X s,b spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behavior of nonlinear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel's theorem and duality, these estimates reduce to estimating a weighted convolution integral in terms of the L 2 norms of the component functions. In this paper we systematically study weighted convolution estimates on L 2 . As a consequence we obtain sharp bilinear estimates for the KdV, wave, and Schrodinger X s,b spaces.

Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations

Suppose that A is a smooth measure on a curve in R3. It is shown that A * LP(R3) C Lq (R3) under certain conditions on A,p, and q. For 1 1, Young's inequality shows that for 1 p. One might say that such a measure is "LPimproving." If A is singular, it is still possible for A to be LP-improving when p > 1. The Cantor-Lebesgue measures on R are examples [1]. The purpose of this paper is to investigate similar behavior for a different class of singular measures, smooth measures supported on curves in R3. Our motivation comes from the following two results. THEOREM 1. Suppose 1 < k < n and K is a smooth k-dimensional surface in Rn. Suppose that A is a smooth measure on K. If A*LP C Lq, then (1/p, 1/q) lies in the closed triangle in R2 with vertices (0, 0), (1, 1), and (n/(2n-k), (n-k)/(2n-k)). PROOF. A theorem of H6rmander [3] implies that p < q. Estimating the norms of f and A * f when f is the characteristic function of a small ball shows that (l/p, 1/q) lies on or above the line L joining (1,1) and (n/(2n-k), (n-k)/(2n-k)). Since A * LP c Lq implies A * Lq c LP', where p' and q' are the conjugate indices, (1/q', l/p') also lies on or above L. This completes the proof. THEOREM 2. Suppose K is a smooth (n 1)-dimensional surface in Rn on which all n 1 principal curvatures are bounded away from zero. Suppose that A is a smooth finite measure compactly supported away from the boundary. Then A * LP c Lq if and only if (l/p, 1/q) lies in the closed triangle in R2 with vertices (0, 0), (1, 1), and (n/(n + 1), 1/(n + 1)). PROOF. A theorem of Littman [4] implies that A * L(n+l)/n C Ln+l. Interpolation with trivial cases completes half the proof, and Theorem 1 gives the rest. Theorems 1 and 2 lead naturally to the conjecture that if A is a nice measure supported on a nice k-dimensional surface in Rn, then A * LP c Lq precisely when (l/p, 1/q) is in the closed triangle of Theorem 1. This conjecture seems difficult even when k = 1, n = 3. Our purpose here is to prove a partial result for this case. We consider certain measures A on some curves in R3 and prove that A * LP c Lq whenever (1/p, 1/q) lies in the triangle of Theorem 1 and on or above the line Received by the editors October 23, 1985. 1980 Mathematics Subject Classfication (1985 Renision). Primary 43A22. Partially supported by the National Science Foundation. (?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

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Convolution estimates for some measures on curves

Lp-Lq mapping properties of the radon transform

We prove sharp endpoint results for the Fourier restriction
operator associated to nondegenerate curves in ${\Bbb R}^d$, $d\ge
3$, and related estimates for oscillatory integral operators.
Moreover, for some larger classes of curves in ${\Bbb R}^d$ we
obtain sharp uniform $L^p\to L^q$ bounds with respect to affine
arclength measure, thereby resolving a problem of Drury and
Marshall.

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Restriction of Fourier transforms to curves and related oscillatory integrals

and establish certain ( , ) estimates for them. The approach in [SW] is to obtain these estimates by studying the multipliers associated with the operators. This is done by employing the ”circle method” from number theory. The results obtained there continue a line of investigation dating back to Hardy and Littlewood and are extremely interesting in their own right. The purpose of this paper is to investigate the applicability to operators like Iλ and Jλ of two methods which have recently been used to study averaging operators associated with curves in R. These methods make no use of the Fourier transform and so the “circle method” will not figure in our proofs. Nevertheless, and unsurprisingly, there are still connections with number theory. One could abstract the ideas involved in the following proofs in order to fashion general theorems from which the results below would follow as special cases. Hoping to emphasize the simplicity of the current methods in this context, we have chosen not to do so. We begin with the operator Iλ. Here the results of [SW] are the next theorem and corollary.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2001/0008/0001/MRL-2001-0008-0001-a001.pdf

Two discrete fractional integrals

We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in $\Bbb R^d$, $d\ge 3$, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in $\Bbb R^d$ we obtain sharp uniform $L^p\to L^q$ bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall.

Daniel M. Oberlin

Papers

Convolution estimates for some measures on curves

Lp-Lq mapping properties of the radon transform

Restriction of Fourier transforms to curves and related oscillatory integrals

Two discrete fractional integrals

Restriction of Fourier transforms to curves and related oscillatory integrals