We introduce a space DHH =D（R2） H2H1D（R2）, where D（R2） is the testing function space whose functions are infinitely differentiable and have bounded support, and H2H1D（R2） is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from DHH onto itself.
Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.
In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.
It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,｜∫Rn ∫Rn f（x）g（y）/｜x｜^α｜x-y｜^λ｜y｜^β dxdy｜≤ B（p,q,α,λ,β ,n）｜｜f｜｜Lp（Rn）｜｜g｜｜Lq（Rn）.The main purpose of this paper is to give the sharp constants B（p,q,,,,n）for the above inequality for three cases：（i）p=1 and q=1;（ii）p=1 and 1q 6∞,or 1p 6∞and q=1;（iii）1p,q∞and 1p＋1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1/p＋1/q〉1.
We characterize a class of piecewise linear spectral sequences. Associated with the spectral sequence, we construct an orthonormal exponential bases for L2（[0,1）d）, which are called generalized Fourier bases. Moreover, we investigate the convergence of Bochner-Riesz means of the generalized Fourier series.
