Second order elliptic operators with complex bounded measurable\n coefficients in L^p, Sobolev and Hardy spaces

2010/02/03 by Steve Hofmann, Svitlana Mayboroda, Hofmann, Steve +3 · 3 citations
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1002.0792

openalex publication_date 2010/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L be a second order divergence form elliptic operator with complex\nbounded measurable coefficients. The operators arising in connection with L,\nsuch as the heat semigroup and Riesz transform, are not, in general, of\nCalder 'on-Zygmund type and exhibit behavior different from their counterparts\nbuilt upon the Laplacian. The current paper aims at a thorough description of\nthe properties of such operators in L^p, Sobolev, and some new Hardy spaces\nnaturally associated to L.\n First, we show that the known ranges of boundedness in L^p for the heat\nsemigroup and Riesz transform of L, are sharp. In particular, the heat\nsemigroup e^-tL need not be bounded in L^p if p not\∈\n[2n/(n+2),2n/(n-2)]. Then we provide a complete description of it all\nSobolev spaces in which L admits a bounded functional calculus, in\nparticular, where e^-tL is bounded.\n Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces\nassociated to L, that serves the range of p beyond [2n/(n+2),2n/(n-2)].\nIt includes, in particular, characterizations by the sharp maximal function and\nthe Riesz transform (for certain ranges of p), as well as the molecular\ndecomposition and duality and interpolation theorems.\n

Citations

• Some new function spaces and their applications to Harmonic analysis

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