zbMATH — the first resource for mathematics

A Hardy space for Fourier integral operators. (English) Zbl 1031.42020
The author introduces a new subspace of $$L^1(\mathbb{R}^n)$$, denoted by $${\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)$$, on which the algebra of Fourier integral operators of order $$0$$, associated to local canonical transformations, acts continuously. This space is in many ways analogous to the local version of the real Hardy space, which can be characterized as the largest subspace of $$L^1(\mathbb{R}^n)$$ that is preserved by order $$0$$ pseudodifferential operators. A key role is played by the atomic and molecular decomposition theorems for $${\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)$$. The boundedness of Fourier integral operators on $${\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)$$ is established by showing that such an operator maps a molecule centered at one point in $$S^*(\mathbb{R}^n)= \mathbb{R}^n\times S^{n-1}$$ to a molecule centered at the image of that point under the associated canonical transformation. The author also establishes the embedding theorem ${\mathcal H}^1(\mathbb{R}^n) @>\langle D\rangle^{-{n-1\over 2}}>>{\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)@> I>>{\mathcal H}^1(\mathbb{R}^n),$ where $\|f\|_{{\mathcal H}^1(\mathbb{R}^n)}= \|(1- r(D))f\|_{H^1(\mathbb{R}^n)}+ \|r(D) f\|_{L^1(\mathbb{R}^n)}$ and $$r$$ is a function in $$C^\infty_c(\mathbb{R}^n)$$ such that $$r(\xi)= 1$$ if $$|\xi|\leq 1$$.

MSC:
 42B30 $$H^p$$-spaces 35S30 Fourier integral operators applied to PDEs 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47G30 Pseudodifferential operators 42B25 Maximal functions, Littlewood-Paley theory
Full Text:
References:
 [1] Beals, M.L p boundedness of fourier integral operators,Memoirs Amer. Math. Soc.,264, (1982). · Zbl 0508.42020 [2] Coifman, R.R. and Weiss, G. Analyse harmonique non-commutative sur certains espaces homogenes,Lecture Notes in Mathematics,242, Springer-Verlag, New York, (1971). · Zbl 0224.43006 [3] Cordoba, A. and Fefferman, C. Wave packets and Fourier integral operators,Comm. Partial Differential Equations,3(11), 979–1005, (1978). · Zbl 0389.35046 · doi:10.1080/03605307808820083 [4] Fefferman, C. A note on spherical summation multipliers,Israel J. Math.,15, 44–52, (1973). · Zbl 0262.42007 · doi:10.1007/BF02771772 [5] Fefferman, C. and Stein, E.M.H p spaces of several variables,Acta. Math.,129, 137–193, (1972). · Zbl 0257.46078 · doi:10.1007/BF02392215 [6] Hörmander, L.The Analysis of Linear Partial Differential Operators, Vols. I–IV, Springer-Verlag, New York, 1983. [7] Peral, J.L p estimates for the wave equation,J. Funct. Anal.,36, 114–145, (1980). · Zbl 0442.35017 · doi:10.1016/0022-1236(80)90110-X [8] Seeger, A., Sogge, C.D., and Stein, E.M. Regularity properties of Fourier integral operators,Annals Math.,133, 231–251, (1991). · Zbl 0754.58037 · doi:10.2307/2944346 [9] Stein, E.M.Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. · Zbl 0207.13501 [10] Stein, E.M.Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001 [11] Torchinsky, A.Real Variable Methods in Harmonic Analysis, Academic Press, New York, 1986. · Zbl 0621.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.