Abstract. We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on L p , 1 < p < ∞. and of a perfect dyadic singular integral operator. This allows to deduce a local T (b) theorem for singular integral operators from the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Thiele, Tao and the first author.
In this article, with the help of the concept of the harmonic sequence of polynomials, the well known Hermite-Hadamard's inequality for convex functions is generalized to cases with bounded derivatives of nth order, including the so-called n-convex functions, from which Hermite-Hadamard's inequality is extended and refined.
Abstract. In this paper, the logarithmically complete monotonicity results of the functions [Γ(1 + x)] y /Γ(1 + xy) and Γ(1 + y)[Γ(1 + x)] y /Γ(1 + xy) are established.
In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results: Besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form a two-, three-or four-fold translative tiling in the Euclidean plane. However, there are convex octagons and decagons which can form five-fold translative tilings.
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