Xavier Tolsa
Smoothness of the Beurling transform in Lipschitz domains.
Пт, 01/27/2012 - 15:01 — SomNambulAuthors: Xavier Tolsa, Victor Cruz
Subjects: Classical Analysis and ODEs
AbstractLet D be a planar Lipschitz domain and consider the Beurling transform of the
characteristic function of D, B(1_D). Let 1<p<\infty and 0<a<1 with ap>1. In
this paper we show that if the outward unit normal N on bD, the boundary of D,
belongs to the Besov space B_{p,p}^{a-1/p}(bD), then the Beurling transform of
1_D is in the Sobolev space W^{a,p}(D). This result is sharp. Further, together
with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling
transform is bounded in W^{a,p}(D) if N belongs to B_{p,p}^{a-1/p}(bD),
assuming that ap>2.- 12 комментариев
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Calderon-Zygmund capacities and Wolff potentials on Cantor sets.
Втр, 01/19/2010 - 16:01 — SomNambulAuthors: Xavier Tolsa
Subjects: Classical Analysis and ODEs
AbstractWe show that, for some Cantor sets in R^d, the capacity g_s associated to the
s-dimensional Riesz kernel x/|x|^{s+1} is comparable to the capacity
C_{2(d-s)/3,3/2} from non linear potential theory. It is an open problem to
show that, when s is positive and non integer, they are comparable for all
compact sets in R^d. We also discuss other open questions in the area.Hausdorff measure of quasicircles.
Пт, 12/18/2009 - 16:01 — SomNambulAuthors: István Prause, Ignacio Uriarte-Tuero, Xavier Tolsa
Subjects: Complex Variables
AbstractS. Smirnov proved recently that the Hausdorff dimension of any K-quasicircle
is at most 1+k^2, where k=(K-1)/(K+1). In this paper we show that if $\Gamma$
is such a quasicircle, then $H^{1+k^2}(B(x,r)\cap \Gamma)\leq C(k) r^{1+k^2}$
for all x in \C and r>0, where H^s stands for the s-Haudorff measure. On a
related note we derive a sharp weak-integrability of the derivative of the
Riemann map of a quasidisk.
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