We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of in...We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.展开更多
For double Walsh–Fourier series and with f∈L2([0,1)×[0,1))we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f(x,y).Assumptions are N(x,y)non-decrea...For double Walsh–Fourier series and with f∈L2([0,1)×[0,1))we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f(x,y).Assumptions are N(x,y)non-decreasing as a function of x and of y and,roughly speaking,partial derivatives with approximately constant ratio■≌2n0 for all x and y,where n0 is any fixed non-negative integer.Estimates,independent of N(x,y)and n0,are then extended to Lr,1<r<2.We give an application to the family N(x,y)=λxy on[0,1)×[0,1),anyλ>10.展开更多
文摘We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.
基金Supported by MIUR Excellence Department Project awarded to the Department of Mathematics,University of Rome Tor Vergata(Grant No.CUP E83C18000100006)。
文摘For double Walsh–Fourier series and with f∈L2([0,1)×[0,1))we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f(x,y).Assumptions are N(x,y)non-decreasing as a function of x and of y and,roughly speaking,partial derivatives with approximately constant ratio■≌2n0 for all x and y,where n0 is any fixed non-negative integer.Estimates,independent of N(x,y)and n0,are then extended to Lr,1<r<2.We give an application to the family N(x,y)=λxy on[0,1)×[0,1),anyλ>10.
