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We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the case 2 < p < 3 : ∂ t u + (− Δ) 2 u = − ∇ ⋅ (| ∇ u | p − 2 ∇ u). The study of this form of equations arises from mathematical models that simulate the epitaxial growth of the thin film. We prove the local existence of mild solutions for any initial data lies in L 2 in both cases. Our main result is: in the advective case, if the imposed advection is sufficiently mixing, then the global existence of solution can be proved, and the solution will converge exponentially to a homogeneous mixed state. While in the absence of advection, there exist initial data in H 2 ∩ W 1 , ∞ such that the solution will blow up in finite time. [ABSTRACT FROM AUTHOR]

Consider a bounded, strongly pseudoconvex domain D ⊂ C n with minimal smoothness (namely, the class C 2) and let b be a locally integrable function on D. We characterize boundedness (resp., compactness) in L p (D) , p > 1 , of the commutator [ b , P ] of the Bergman projection P in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on b. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to introduce Hilbert spaces of entire Dirichlet series with real frequencies and consider composition operators on these spaces. We establish necessary and sufficient conditions for such series to have Ritt order zero, as well as to have finite logarithmic orders. This allows us to apply the Polya theorem on composition of entire functions to consideration of composition operators on the Hilbert spaces of entire Dirichlet series. In particular, criteria for action, boundedness, compactness and compact difference of such operators are obtained. [Copyright &y& Elsevier]

The purpose of this paper is to study sparse domination estimates of composition operators in the setting of complex function theory. The method originates from proofs of the A 2 theorem for Calderón-Zygmund operators in harmonic analysis. Using this tool from harmonic analysis, some new characterizations are given for the boundedness and compactness of weighted composition operators acting between weighted Bergman spaces in the upper half plane. Moreover, we establish a new weighted type estimate for the holomorphic Bergman-class functions, for a new class of weights, which is adapted to Sawyer–testing conditions. We also extend our results to the unit ball B in C n. [ABSTRACT FROM AUTHOR]