Weighted norm inequalities for the conjugate function and Hilbert transform

Abstract

The principal problem considered is the determination of all nonnegative functions W(x) with period 277 such that i" \Jie)\pwie)de<c C \fie)\pwie)de J-n J-7r where 1 < p < °°, f has period 277, C is a constant independent of /, and / is the conjugate function defined by fj8-<p)d(f> The main result is that W(x) is such a function if and only if 1/7-1 7(0)= iim-f fie-4>)d4> * Am.+ 77 Jes|</>|<rr 2 tan 0/2

Weighted Weak Behaviour of Fourier‐Jacobi Series

“…The boundedness of the Hilbert transform can be stated in terms of Muckenhoupt's classes of weights A p (see [3] and [8]; throughout this paper, these will be A p classes on the interval [−1, 1]): if u, v are two weights, 1 < p < ∞ and…”

Weighted $L^p$-boundedness of Fourier series with respect to generalized Jacobi weights

“…b) Now, assume that the operators Sn are uniformly bounded . Let 'us prove that (2), (3) and (4) v-1 E L9 (dp) ;…”

“…A + (a+ 1)(P -1 2 4 ) < min{1,~} (2) B + (,d + 1) (P -2) < min{4,~} Gi+(yi+1)(P-12 2 ) < min{1,~} (i=1, . .…”

“…The objective of this paper is to show that the result remains true without the restriction yi >_ 0 and that conditions (2), (3) and (4) For the sake of completeness, we give a brief sketch of the proof of theorem 1 (see also [10]) . By using Pollard's decomposition of the kernels K,, (x, y) (see [8], [5]), the uniform boundedness of S , can be reduced to that of the Hilbert transform with pairs of weights and…”

Pointwise Multipliers of Weighted BMO Spaces

“…That's the approach we take to the weighted problem. The Hubert transform is bounded on Lp(w) if and only if w G A [6]. In p the unweighted case, 77 can be characterized by all functions for which / and 77/ G L (R).…”