Let q greater than or equal to 2,f is measurable function on R-n such that f(x)\x\(n(1-2/q)) is an element of L-q(R-n), then its Fourier transform (f) over cap can be defined and there exists a constant A(q) such that...Let q greater than or equal to 2,f is measurable function on R-n such that f(x)\x\(n(1-2/q)) is an element of L-q(R-n), then its Fourier transform (f) over cap can be defined and there exists a constant A(q) such that the inequality parallel to (f) over cap parallel to(q) less than or equal to A(q) parallel to f\.\(n(1-2/q))parallel to(q) holds. This is the Hardy-Littlewood Theorem. This paper considers the corresponding result for the Fourier-Bessel transform and Fourier-Jacobi transform. It is interesting that we can deal with these two cases in the same way, and the function corresponding to \x\(n) is tw(t), where w(t) is the weight, w(t) = t(2 alpha+1) for Fourier-Bessel transform, and w(t) = (2 sinh t)(2 alpha+1)(2 cosh t)(2 beta+1) for Fourier-Jacobi transform.展开更多
文摘Let q greater than or equal to 2,f is measurable function on R-n such that f(x)\x\(n(1-2/q)) is an element of L-q(R-n), then its Fourier transform (f) over cap can be defined and there exists a constant A(q) such that the inequality parallel to (f) over cap parallel to(q) less than or equal to A(q) parallel to f\.\(n(1-2/q))parallel to(q) holds. This is the Hardy-Littlewood Theorem. This paper considers the corresponding result for the Fourier-Bessel transform and Fourier-Jacobi transform. It is interesting that we can deal with these two cases in the same way, and the function corresponding to \x\(n) is tw(t), where w(t) is the weight, w(t) = t(2 alpha+1) for Fourier-Bessel transform, and w(t) = (2 sinh t)(2 alpha+1)(2 cosh t)(2 beta+1) for Fourier-Jacobi transform.
