L^p-L^qestimates for some convolution operators with singular measures on the Heisenberg group

Abstract

Abstract. We consider the Heisenberg group H n = C n ×R. Let ν be the Borel measure on H n defined by ν(E) = C n χ E (w, ϕ(w)) η(w)dw, where ϕ(w) = n j=1 a j |w j | 2 , w = (w 1 , ..., wn) ∈ C n , a j ∈ R, and η(w) = η 0 |w| 2 with η 0 ∈ C ∞ c (R). In this paper we characterize the set of pairs (p, q) such that the convolution operator with ν isWe also obtain L p -improving properties of measures supported on the graph of the function ϕ(w) = |w| 2m .

Convolution operators with singular measures of fractional type on the Heisenberg group

“…where T µ0 is the operator defined by (2), taking there γ = 0 and ϕ(w) = n j=1 a j |w j | 2 . Then Theorem 1 will follow from Theorem 1 in [2], the Riesz-Thorin convexity Theorem and Lemma 4. Thus T µ k p,q ≤ c2 kγ T µ0 p,q , from Theorem 1 in [2] it follows that…”

“…where D is a closed disk in R 2n contained in the unit disk centered in the origin such that the origin not belongs to D, we observe that the argument utilized in the proof of Lemma 4 in [2] works in this setting so we get the others two inequalities.…”

“…Let T µγ be the right convolution operator by µ γ , defined by (2) We are interested in studying the type set…”

“…We say that that the measure µ γ defined in (1) is L p -improving if E µγ does not reduce to the diagonal 1/p = 1/q. This problem is well known if in (2) we consider γ = 0 and replace the Heisenberg group convolution with the ordinary convolution in R 2n+1 . If the graph of ϕ has non-zero Gaussian curvature at each point, a theorem of Littman (see [4]) implies that E ν is the closed triangle with vertices (0, 0), (1,1), and 2n+1 2n+2 , 1 2n+2 (see [5]).…”

“…Indeed, the first inequality follows from Lemma 3 in [2], replacing the sets A δ and F δ,x in the proof of Lemma 4 in [2] by the sets…”