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Painlevé's problem and analytic capacity.

Xavier Tolsa (2006)

Collectanea Mathematica

In this paper we survey some recent results in connection with the so called Painlevé's problem and the semiadditivity of analytic capacity γ. In particular, we give the detailed proof of the semiadditivity of the capacity γ+, and we show almost completely all the arguments for the proof of the comparability between γ and γ+.

Para-accretive functions, the weak boundedness property and the Tb theorem.

Yongsheng Han, Eric T. Sawyer (1990)

Revista Matemática Iberoamericana

G. David, J.-L. Journé and S. Semmes have shown that if b1 and b2 are para-accretive functions on Rn, then the Tb theorem holds: A linear operator T with Calderón-Zygmund kernel is bounded on L2 if and only if Tb1 ∈ BMO, T*b2 ∈ BMO and Mb2TMb1 has the weak boundedness property. Conversely they showed that when b1 = b2 = b, para-accretivity of b is necessary for Tb Theorem to hold. In this paper we show that para-accretivity of both b1 and b2 is necessary for the Tb Theorem to hold in general. In...

Parabolic Marcinkiewicz integrals on product spaces and extrapolation

Mohammed Ali, Mohammed Al-Dolat (2016)

Open Mathematics

In this paper, we study the the parabolic Marcinkiewicz integral [...] MΩ,hρ1,ρ2 Ω , h ρ 1 , ρ 2 on product domains Rn × Rm (n, m ≥ 2). Lp estimates of such operators are obtained under weak conditions on the kernels. These estimates allow us to use an extrapolation argument to obtain some new and improved results on parabolic Marcinkiewicz integral operators.

Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces

Ferit Gurbuz (2016)

Open Mathematics

In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator which...

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