Lp estimates for strongly singular convolution operators in Rn

Inequalities for Strongly Singular Convolution Operators

Strongly Singular Convolution Operators on the Heisenberg Group

We consider the L mapping properties of a model class of strongly singular integral operators on the Heisenberg group H; these are convolution operators on H whose kernels are too singular at the origin to be of Calderón-Zygmund type. This strong singularity is compensated for by introducing a suitably large oscillation. Our results are obtained by utilizing the group Fourier transform and unif...

Estimates in Lp for Magnetic Schrödinger Operators

We study the magnetic Schrödinger operator H(a,V ) in R, n ≥ 3. The L (1 < p < ∞) and weak-type (1,1) estimates are obtained under certain conditions, given in terms of the reverse Hölder inequality, on the magnetic field B = curl a and the electrical potential V . In particular, we show that the L and weak-type (1,1) estimates hold if the components of a are polynomials, and V is a nonnegative...

SHARP Lp ESTIMATES FOR SINGULAR TRANSPORT EQUATIONS

We prove that L estimates for a singular transport equation are sharp by building what we call a cascading solution; the equation we consider studies the combined effect of multiplying by a bounded function and application of the Hilbert transform. Along the way we prove an invariance result for the Hilbert transform which could be of independent interest. Finally, we give an example of a bound...

TWO-SIDED Lp ESTIMATES OF CONVOLUTION TRANSFORMS

where ψ{v) = u, u. R. O'Neil obtained sharp upper and lower estimates when ψ(u) — u and n — 2, [3, Lemma 1.5]. Our results coincide with his for this case. We were able to apply our estimates for the case ψ(u) = u (^-arbitrary) to classical Fourier transform inequalities of Hardy and Little wood. The main problem of this paper is to determine whether or not one can obtain the same types of uppe...