It is shown that the operator below maps ${L}^{p}$ into itself for 1 < p < ∞.
$Cf\left(x\right):=su{p}_{a,b}|p.v.\int f(x-y){e}^{i(ay\xb2+by)}dy/y|$.
The supremum over b alone gives the famous theorem of L. Carleson [2] on the pointwise convergence of Fourier series. The supremum over a alone is an observation of E. M. Stein [12]. The method of proof builds upon Stein’s observation and an approach to Carleson’s theorem jointly developed by the author and C. M. Thiele [7].

Let f ∈ L^{∞} and g ∈ L^{2} be supported on [0,1]. Then the principal value integral below exists in L^{1}.
p.v. ∫ f(x + y) g(x - y) dy / y.

Carleson's Theorem from 1965 states that the partial Fourier sums of a square integrable function converge pointwise. We prove an equivalent statement on the real line, following the method developed by the author and C. Thiele. This theorem, and the proof presented, is at the center of an emerging theory which complements the statement and proof of Carleson's theorem. An outline of these variations is also given.

On the real line, let the Fourier transform of k_{n} be k'_{n}(ξ) = k'(ξ-n) where k'(ξ) is a smooth compactly supported function. Consider the bilinear operators S_{n}(f, g)(x) = ∫ f(x+y)g(x-y)k_{n}(y) dy. If 2 ≤ p, q ≤ ∞, with 1/p + 1/q = 1/2, I prove that
Σ^{∞}
_{n=-∞} ||S_{n}(f,g)||_{2}
^{2}...

We study twoweight inequalities in the recent innovative language of ‘entropy’ due to Treil-Volberg. The inequalities are extended to Lp, for 1 < p ≠ 2 < ∞, with new short proofs. A result proved is as follows. Let ℇ be a monotonic increasing function on (1,∞) which satisfy [...] Let σ and w be two weights on Rd. If this supremum is finite, for a choice of 1 < p < ∞, [...] then any Calderón-Zygmund operator T satisfies the bound [...]

We study twoweight inequalities in the recent innovative language of ‘entropy’ due to Treil-Volberg. The inequalities are extended to Lp, for 1 < p ≠ 2 < ∞, with new short proofs. A result proved is as follows. Let ɛ be a monotonic increasing function on (1,∞) which satisfy [...] Let σ and w be two weights on ℝd. If this supremum is finite, for a choice of 1 < p < ∞, [...] then any Calderón-Zygmund operator T satisfies the bound ||Tof||Lp(w) ≲ ||f|| Lp(o).

Download Results (CSV)