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In this article we develop the theory of one-sided versions of the g function of Littlewood and Paley, the area function S of Lusin and the that admit weighted norm estimates with weights belonging to the classes Ap+ of Sawyer.
      
Section 3 is devoted to the study of the one-sided version of the functions g and S.
      
In Section 4 we obtain a good λ estimate for the one-sided function, and in Sections 5 and 6 we apply the results already obtained to fractional integrals and multiplier operators.
      
Recent study on the subject is an indirect approach: in order to compute the Gabor coefficients, one needs to find an auxiliary bi-orthogonal window function γ.
      
With this construction, one is able to construct local sinusoidal bases and lapped orthogonal transforms (LOT) on arbitrarily shaped regions.
      
Moreover, we demonstrate that these wavelets do not behave like their one-dimensional couterparts.
      
In a much cited article, Yau [5] proved that when the Ricci curvature is bounded uniformly below, then the only bounded solution to the heat equation ?tμ=Δμ on [0, ∞) × M which vanishes at t=0 is the one which vanishes evarywhere.
      
We further show that every frame can be written as a (multiple of a) sum of two tight frames with frame bounds one or a sum of an orthonormal basis and a Riesz basis for H.
      
We also prove that among all matrix functions with these properties, there exists a distinguished one that maximizes the entropy.
      
For 0 ≤α >amp;lt; ∞ let Tαf denote one of the operators We characterize the pairs of weights (u, v) for which Tα is a bounded operator from Lp(v) to Lq(u), 0 >amp;lt;p ≤q >amp;lt; ∞.
      
Spherical harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform (FFT).
      
Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures.
      
We extend to general finite groups a well-known relation used for checking the orthogonality of a system of vectors as well as for orthogonalizing a nonorthogonal one.
      
This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series.
      
We show that several such actions may be considered, and investigate those which may be written as deformations of the canonical one.
      
The first one is based on the use of the generalized Calderón reproducing formula and multidimensional fractional integrals with a Bessel function in the kernel.
      
We extend to general finite groups a well-known relation used for checking the orthogonality of a system of vectors as well as for orthogonalizing a nonorthogonal one.
      
We study generalizations of this operator in one and several variables.
      
In this article we give sufficient conditions on a pair of weight (w, v) for some one-sided operators to be bounded from Lp (vp) to Lp (wp).
      
The operators we deal with include the one-sided fractional maximal operator and the one-sided singular integrals.
      
 

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