$L_p-L_q$ Estimates for the Bochner-Riesz Operator of Complex Order

Abstract

We describe convex sets on the (1 p , 1 q)-plane for which the well-known Bochner-Riesz operator with the symbol (1 − |ξ| 2) −α + (0 < Re α < n+1 2) is bounded from L p into L q .

On the boundedness of some potential‐type operators with oscillating kernels

“…As was proved in [8] (x) / ∈ L q for the mentioned q. In the case a) of item 1), the mentioned statement follows from Theorems 3.7 and 4.7 due to convexity and duality and the fact that the L-characteristic L K α,β,γ does not contain the points of the set L 2 (β +1, n)\L 2 (β, n) (as was proved above).…”

“…The following theorem was proved in [8] (see also Figures 1 and 2 for the cases (n − 1)/2 < Re α < n and 0 < Re α ≤ (n − 1)/2, respectively). …”

“…Here we deal with the case γ = n and β < 1, which includes Strichartztype potentials over R n with oscillating characteristics (see [12], Remark 5.3); the reader is also refered to the papers [1,8,6,13] for the cases of the Bochner-Riesz operators and acoustic potentials.…”

“…Nevertheless, the investigation of potentials with oscillating kernels is at the very beginning (see the survey paper [14] and the aforementioned papers [1,8,6,13] for some special-type potentials). Paper [5] is devoted to L p → L q -estimates for the potential (1.1) in the case γ = α ∈ (0, n) and 0 < β < 1.…”

Bochner–Riesz operators in grand lebesgue spaces

“…There exists a huge numbers of works devoted to the p, q estimates for Bochner-Riesz operators B α R , as a rule for the case q = p, see e.g. [23, chapter5], [9,10,13,29,35,42,43], etc. The boundedness of these operators in Morrey-Lorentz spaces and in L p spaces with variable exponent has been investigated in [25] and [8], respectively.…”