Generalized Littlewood-Paley functions on product spaces

On Bilinear Littlewood-paley Square Functions

On the real line, let the Fourier transform of kn be k̂n(ξ) = k̂(ξ−n) where k̂(ξ) is a smooth compactly supported function. Consider the bilinear operators Sn(f, g)(x) = ∫ f(x + y)g(x − y)kn(y) dy. If 2 ≤ p, q ≤ ∞, with 1/p + 1/q = 1/2, I prove that ∞ ∑ n=−∞ ‖Sn(f, g)‖2 ≤ C‖f‖p‖g‖q . The constant C depends only upon k.

Littlewood-Paley characterization for Qα(R) spaces

In Baraka’s paper [2], he obtained the Littlewood-Paley characterization of Campanato spaces L and introduced Lp,λ,s spaces. He showed that L2,λ,s = (−△)− s 2L for 0 ≤ λ < n+ 2. In [7], by using the properties of fractional Carleson measures, J Xiao proved that for n ≥ 2, 0 < α < 1. (−△)−α2 L is essential the Qα(R) spaces which were introduced in [4]. Then we could conclude that Qα(R) = L2,n−2α...

Littlewood-Paley Operators on Morrey Spaces with Variable Exponent

By applying the vector-valued inequalities for the Littlewood-Paley operators and their commutators on Lebesgue spaces with variable exponent, the boundedness of the Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g μ *-functions, and their commutators generated by BMO functions, is obtained on the Morrey spaces with variable exponent.

Remarks on Square Functions in the Littlewood-paley Theory

We prove that certain square function operators in the Littlewood-Paley theory defined by the kernels without any regularity are bounded on L p w , 1 < p < ∞, w ∈ Ap (the weights of Muckenhoupt). Then, we give some applications to the Carleson measures on the upper half space.

Commutators of Littlewood-Paley Operators on the Generalized Morrey Space