A Generalized Marcinkiewicz Interpolation Theorem and Its Applications

Using a simple method,we generalize Marcinkiewicz interpolation theorem to operators.on Orlicz space and apply it to several important theorems in harmonic analysis.

AN INTERPOLATION FORMULA OF THE DERIVATIVES OF HIGHER ORDER

In this article we shall obtain an interpolation formula passing given a serial points and satisfying initial values of the derivatives of higher order in preceding points Finally we shall give the erroneous estimate of the preceding interpolation formula.

An Off-Diagonal Marcinkiewicz Interpolation Theorem on Lorentz Spaces

Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,q（X） with p, q ￡ （0, ∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.： Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ￡ [1, ∞） and q ￡ [l,∞）, was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].

L^p MIXED-NORM ESTIMATES FOR CONVOLUTION OPERATOR DEFINED BY SINGULAR MEASURES

Let F be a C^∞ curve in IR^n and μ be the measure induced by Lebesgue measure on F, multiplied by a smooth cut-off function. In this paper, we will prove some mixednorm estimates based on the average decay estimates of the Fourier transform of μ.

A Growth Behavior of SzegöType Operators

We define new integral operators on the Haydy space similar to Szegö projection. We show that these operators map from Hp to H2 for some 1 ≤ p < 2, where the range of p is depending on a growth condition. To prove that, we generalize the Hausdorff-Young Theorem to multi-dimensional case.

Inequalities for Higher Order Riesz-Laguerre Transforms

The weak-type (1, 1) boundedness of the higher order Riesz-Laguerre transforms associated with the Laguerre polynomials and the boundedness for the Riesz-Laguerre transforms of order 2 are considered. We discuss a polynomial weight w that makes the Riesz-Laguerre transforms of order greater than or equal to 2 continuous from L1 (wdμα) into L1,∞ (dμα), under specific value α, where μα is the Laguerre measure.

A Note on Rough Parametric Marcinkiewicz Functions

In this note, we obtain sharp Lp estimates of parametric Marcinkiewicz integral operators. Our result resolves a long standing open problem. Also, we present a class of parametric Marcinkiewicz integral operators that are bounded provided that their kernels belong to the sole space L^1(S^n-1).