We prove that the L2 regularity problem is solvable for the elliptic equation ∑n j,k=1 ∂j(ajk∂ku) = 0 when ∑ i,j,k |∂iajk(x)|xndx is a Carleson measure with a sufficiently small constant, ∑ i,j,k |∂iajk(x)| ≤ C/xn and the bottom row of the coefficient matrix has the particular form (0, 0, . . . , 0, 1). This is done in any dimension n. This was proved in the case n = 2 earlier in [9] without th...

We prove the following extension of the Wiener–Wintner Theorem and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows (X,μ, Tt) and f ∈ L(X,μ), there is a set Xf ⊂ X of probability one, so that for all x ∈ Xf we have lim s↓0 ∫ s<|t|<1/s e f(Tt x) dt t exists for all θ. The proof is by way of establishing an appropriate oscillation inequality which ...

In this work, we give new sufficient conditions for Littlewood-Paley-Stein square function and necessary and sufficient conditions for a Calderón-Zygmund operator to be bounded on Hardy spaces H p with indices smaller than 1. New Carleson measure type conditions are defined for Littlewood-Paley-Stein operators, and the authors show that they are sufficient for the associated square function to ...

The study of exceptional sets for nontangential limits of harmonic functions has led to the problem of the equivalence of harmonic measure and surface measure on the boundary of Lipschitz domains. In this note we will review the development of these results. We will see how estimates of Green's function can be used to relate the measures and look at methods of obtaining the desired estimates. I...

We provide some new sharp embeddings for p-Carleson measures and some related measures in the unit disk of the complex plane.

We study the weighted boundedness of the Cauchy singular integral operator SΓ in Morrey spaces L(Γ) on curves satisfying the arc-chord condition, for a class of ”radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L(0, ...

For/e ¿"(R2), let (7£/)"(£) = (1 |£|2«2)î/(£). It is a well-known theorem of Carleson and Sjölin that T" defines a bounded operator on Z,4 if a > 0. In this paper we obtain an explicit weighted inequality of the form / sup \T%f(x)\2w(x)dxti( \f\2Paw(x)dx, 0 0. This strengthens the above theorem of Carleson and Sjölin. The method gives information on the maximal...

I spent most of my research time this summer studying the basic methods of harmonic analysis. My goal was to write an explanation and elaboration of Fefferman’s 1973 paper, Pointwise Convergence of Fourier Series. I was attempting to fill in the details and clarify things to the degree that the paper would be accessible at the level of a beginning graduate student, and to add enough background ...