{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:5PZXBWEWTMSMD4BOABWNJN6OIO","short_pith_number":"pith:5PZXBWEW","schema_version":"1.0","canonical_sha256":"ebf370d8969b24c1f02e006cd4b7ce4389bba35643bacaffbd0b7494a43cdfa5","source":{"kind":"arxiv","id":"0804.0405","version":2},"attestation_state":"computed","paper":{"title":"Boundedness and convergence for singular integrals of measures separated by Lipschitz graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Pertti Mattila, Vasilis Chousionis","submitted_at":"2008-04-02T17:46:59Z","abstract_excerpt":"We shall consider the truncated singular integral operators T_{\\mu, K}^{\\epsilon}f(x)=\\int_{\\mathbb{R}^{n}\\setminus B(x,\\epsilon)}K(x-y)f(y)d\\mu y and related maximal operators $T_{\\mu,K}^{\\ast}f(x)=\\underset{\\epsilon >0}{\\sup}| T_{\\mu,K}^{\\epsilon}f(x)|$. We shall prove for a large class of kernels $K$ and measures $\\mu$ and $\\nu$ that if $\\mu$ and $\\nu$ are separated by a Lipschitz graph, then $T_{\\nu,K}^{\\ast}:L^p(\\nu)\\to L^p(\\mu)$ is bounded for $10}{\\sup}| T_{\\mu,K}^{\\epsilon}f(x)|$. We shall prove for a large class of kernels $K$ and measures $\\mu$ and $\\nu$ that if $\\mu$ and $\\nu$ are separated by a Lipschitz graph, then $T_{\\nu,K}^{\\ast}:L^p(\\nu)\\to L^p(\\mu)$ is bounded for $1